text:
Control Systems/Routh-Hurwitz Criterion

Stability Criteria
The Routh-Hurwitz stability criterion provides a simple algorithm to decide whether or not the zeros of a polynomial are all in the left half of the complex plane (such a polynomial is called at times "Hurwitz"). A Hurwitz polynomial is a key requirement for a linear continuous-time invariant to be stable (all bounded inputs produce bounded outputs).

If any of them fails - the polynomial is not stable. However, they may all hold without implying stability.
 * Necessary stability conditions: Conditions that must hold for a polynomial to be Hurwitz.
 * Sufficient stability conditions: Conditions that if met imply that the polynomial is stable. However, a polynomial may be stable without implying some or any of them.

The Routh criteria provides condition that are both necessary and sufficient for a polynomial to be Hurwitz.

Routh-Hurwitz Criteria
The Routh-Hurwitz criteria is comprised of three separate tests that must be satisfied. If any single test fails, the system is not stable and further tests need not be performed. For this reason, the tests are arranged in order from the easiest to determine to the hardest.

The Routh Hurwitz test is performed on the denominator of the transfer function, the characteristic equation. For instance, in a closed-loop transfer function with G(s) in the forward path, and H(s) in the feedback loop, we have:

 * $$T(s) = \frac{G(s)}{1 + G(s)H(s)}$$

If we simplify this equation, we will have an equation with a numerator N(s), and a denominator D(s):

 * $$T(s) = \frac{N(s)}{D(s)}$$

The Routh-Hurwitz criteria will focus on the denominator polynomial D(s).

Routh-Hurwitz Tests
Here are the three tests of the Routh-Hurwitz Criteria. For convenience, we will use N as the order of the polynomial (the value of the highest exponent of s in D(s)). The equation D(s) can be represented generally as follows:

 * $$D(s) = a_0 + a_1s + a_2s^2 + \cdots + a_Ns^N$$

We will explain the Routh array below.

The Routh's Array
The Routh array is formed by taking all the coefficients ai of D(s), and staggering them in array form. The final columns for each row should contain zeros:

 * $$\begin{matrix}s^N \\ s^{N-1} \end{matrix}

\begin{vmatrix}a_N & a_{N - 2} & \cdots & 0 \\ a_{N-1} & a_{N-3} & \cdots & 0 \end{vmatrix}$$

Therefore, if N is odd, the top row will be all the odd coefficients. If N is even, the top row will be all the even coefficients. We can fill in the remainder of the Routh Array as follows:

 * $$\begin{matrix}s^N \\ s^{N-1} \\ \\  \\ s^0 \end{matrix}

\begin{vmatrix}a_N & a_{N - 2} & \cdots & 0 \\ a_{N-1} & a_{N-3} & \cdots & 0 \\ b_{N-1} & b_{N-3} & \cdots \\ c_{N-1} & c_{N-3} & \cdots \\ \cdots \end{vmatrix}$$

Now, we can define all our b, c, and other coefficients, until we reach row s0. To fill them in, we use the following formulae:

 * $$b_{N-1} = \frac{-1}{a_{N-1}}\begin{vmatrix}a_N & a_{N-2} \\ a_{N-1} & a_{N-3}\end{vmatrix}$$

And

 * $$b_{N-3} = \frac{-1}{a_{N-1}}\begin{vmatrix}a_N & a_{N-4} \\ a_{N-1} & a_{N-5}\end{vmatrix}$$

For each row that we are computing, we call the left-most element in the row directly above it the pivot element. For instance, in row b, the pivot element is aN-1, and in row c, the pivot element is bN-1 and so on and so forth until we reach the bottom of the array.

To obtain any element, we negate the determinant of the following matrix, and divide by the pivot element:

 * $$\begin{vmatrix}k & m \\ l & n \end{vmatrix}$$

Where:
 * k is the left-most element two rows above the current row.
 * l is the pivot element.
 * m is the element two rows up, and one column to the right of the current element.
 * n is the element one row up, and one column to the right of the current element.

In terms of k l m n, our equation is:

 * $$v = \frac{(lm) - (kn)}{l}$$

Special Case: Row of All Zeros
If, while calculating our Routh-Hurwitz, we obtain a row of all zeros, we do not stop, but can actually learn more information about our system.

If we have a row of all zeros, the row directly above it is known as the Auxiliary Polynomial, and can be very helpful. The roots of the auxiliary polynomial give us the precise locations of complex conjugate roots that lie on the jω axis. However, one important point to notice is that if there are repeated roots on the jω axis, the system is actually unstable. Therefore, we must use the auxiliary polynomial to determine whether the roots are repeated or not.

The auxiliary equation is to be differentiated with respect to s and the coefficients of this equation replaces the all zero row. Routh array can be further calculated using these new values.

Special Case: Zero in the First Column
In this special case, there is a zero in the first column of the Routh Array, but the other elements of that row are non-zero. Like the above case, we can replace the zero with a small variable epsilon (ε) and use that variable to continue our calculations. After we have constructed the entire array, we can take the limit as epsilon approaches zero to get our final values. If the sign coefficient above the (ε) is the same as below it, this indicates a pure imaginary root.

title: Control Systems/Routh-Hurwitz Criterion


text:
Umbraco/GetMedia

Umbraco.library.GetMedia
The method is used to get information of files in the media library - such as filename, size, height, width.

Parameters

GetMedia(int MediaId, bool Deep);

Integer  MediaId Boolean  Deep

Return value

The method returns a XPathNodeIterator with the information of selected file.

Usage

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Umbraco/GetMedia


text:
Control Systems/Nyquist Stability Criteria

Nyquist Stability Criteria
The Nyquist Stability Criteria is a test for system stability, just like the Routh-Hurwitz test, or the Root-Locus Methodology. However, the Nyquist Criteria can also give us additional information about a system. Routh-Hurwitz and Root-Locus can tell us where the poles of the system are for particular values of gain. By altering the gain of the system, we can determine if any of the poles move into the RHP, and therefore become unstable. The Nyquist Criteria, however, can tell us things about the frequency characteristics of the system. For instance, some systems with constant gain might be stable for low-frequency inputs, but become unstable for high-frequency inputs.

Also, the Nyquist Criteria can tell us things about the phase of the input signals, the time-shift of the system, and other important information.

Contours
A contour is a complicated mathematical construct, but luckily we only need to worry ourselves with a few points about them. We will denote contours with the Greek letter Γ (gamma). Contours are lines, drawn on a graph, that follow certain rules:

 * 1) The contour must close (it must form a complete loop)
 * 2) The contour may not cross directly through a pole of the system.
 * 3) Contours must have a direction (clockwise or counterclockwise, generally).
 * 4) A contour is called "simple" if it has no self-intersections. We only consider simple contours here.

Once we have such a contour, we can develop some important theorems about them, and finally use these theorems to derive the Nyquist stability criterion.

Argument Principle
Here is the argument principle, which we will use to derive the stability criterion. Do not worry if you do not understand all the terminology, we will walk through it:

When we have our contour, Γ, we transform it into $$\Gamma_{F(s)}$$ by plugging every point of the contour into the function F(s), and taking the resultant value to be a point on the transformed contour.

The Nyquist Contour
The Nyquist contour, the contour that makes the entire nyquist criterion work, must encircle the entire unstable region of the complex plane. For analog systems, this is the right half of the complex s plane. For digital systems, this is the entire plane outside the unit circle. Remember that if a pole to the closed-loop transfer function (or equivalently a zero of the characteristic equation) lies in the unstable region of the complex plane, the system is an unstable system.

 * Analog Systems:The Nyquist contour for analog systems is an infinite semi-circle that encircles the entire right-half of the s plane. The semicircle travels up the imaginary axis from negative infinity to positive infinity. From positive infinity, the contour breaks away from the imaginary axis, in the clock-wise direction, and forms a giant semicircle.
 * Digital Systems:The Nyquist contour in digital systems is a counter-clockwise encirclement of the unit circle.

Nyquist Criteria
Let us first introduce the most important equation when dealing with the Nyquist criterion:

 * $$N = Z - P$$

Where:

 * N is the number of encirclements of the (-1, 0) point.
 * Z is the number of zeros of the characteristic equation.
 * P is the number of poles in the of the open-loop characteristic equation.

With this equation stated, we can now state the Nyquist Stability Criterion:

In other words, if P is zero then N must equal zero. Otherwise, N must equal P. Essentially, we are saying that Z must always equal zero, because Z is the number of zeros of the characteristic equation (and therefore the number of poles of the closed-loop transfer function) that are in the right-half of the s plane.

Keep in mind that we don't necessarily know the locations of all the zeros of the characteristic equation. So if we find, using the nyquist criterion, that the number of poles is not equal to N, then we know that there must be a zero in the right-half plane, and that therefore the system is unstable.

Nyquist ↔ Bode
A careful inspection of the Nyquist plot will reveal a surprising relationship to the Bode plots of the system. If we use the Bode phase plot as the angle θ, and the Bode magnitude plot as the distance r, then it becomes apparent that the Nyquist plot of a system is simply the polar representation of the Bode plots.

To obtain the Nyquist plot from the Bode plots, we take the phase angle and the magnitude value at each frequency ω. We convert the magnitude value from decibels back into gain ratios. Then, we plot the ordered pairs (r, θ) on a polar graph.

Nyquist in the Z Domain
The Nyquist Criteria can be utilized in the digital domain in a similar manner as it is used with analog systems. The primary difference in using the criteria is that the shape of the Nyquist contour must change to encompass the unstable region of the Z plane. Therefore, instead of an infinitesimal semi-circle, the Nyquist contour for digital systems is a counter-clockwise unit circle. By changing the shape of the contour, the same N = Z - P equation holds true, and the resulting Nyquist graph will typically look identical to one from an analog system, and can be interpreted in the same way.


text:
Umbraco/Reference/umbraco.library

Umbraco.library v4.0.1

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text:
Umbraco/Reference/umbraco.library/GetMedia

Umbraco.library.GetMedia
The method is used to get information of files in the media library - such as filename, size, height, width.

Parameters

GetMedia(int MediaId, bool Deep);

Integer  MediaId Boolean  Deep

Return value

The method returns a XPathNodeIterator with the information of selected file.

Usage

Getting the url of a media file

Getting the url of a media file with the new xml structure


text:
Control Systems/Transfer Functions

Transfer Functions
A Transfer Function is the ratio of the output of a system to the input of a system, in the Laplace domain considering its initial conditions and equilibrium point to be zero. This assumption is relaxed for systems observing transience. If we have an input function of X(s), and an output function Y(s), we define the transfer function H(s) to be:

 * $$H(s) = {Y(s) \over X(s)}$$

Readers who have read the Circuit Theory book will recognize the transfer function as being the impedance, admittance, impedance ratio of a voltage divider or the admittance ratio of a current divider.

Impulse Response
For comparison, we will consider the time-domain equivalent to the above input/output relationship. In the time domain, we generally denote the input to a system as x(t), and the output of the system as y(t). The relationship between the input and the output is denoted as the impulse response, h(t).

We define the impulse response as being the relationship between the system output to its input. We can use the following equation to define the impulse response:

 * $$h(t) = \frac{y(t)}{x(t)}$$

Impulse Function
It would be handy at this point to define precisely what an "impulse" is. The Impulse Function, denoted with δ(t) is a special function defined piece-wise as follows:

 * $$\delta(t) = \left\{

\begin{matrix} 0, & t < 0 \\ \mbox{undefined}, & t = 0 \\ 0, & t > 0 \end{matrix}\right. $$

The impulse function is also known as the delta function because it's denoted with the Greek lower-case letter δ. The delta function is typically graphed as an arrow towards infinity, as shown below:

It is drawn as an arrow because it is difficult to show a single point at infinity in any other graphing method. Notice how the arrow only exists at location 0, and does not exist for any other time t. The delta function works with regular time shifts just like any other function. For instance, we can graph the function δ(t - N) by shifting the function δ(t) to the right, as such:

An examination of the impulse function will show that it is related to the unit-step function as follows:

 * $$\delta(t) = \frac{du(t)}{dt}$$

and

 * $$u(t) = \int \delta(t) dt$$

The impulse function is not defined at point t = 0, but the impulse must always satisfy the following condition, or else it is not a true impulse function:

 * $$\int_{-\infty}^\infty \delta(t)dt = 1$$

The response of a system to an impulse input is called the impulse response. Now, to get the Laplace Transform of the impulse function, we take the derivative of the unit step function, which means we multiply the transform of the unit step function by s:

 * $$\mathcal{L}[u(t)] = U(s) = \frac{1}{s}$$

 * $$\mathcal{L}[\delta(t)] = sU(s) = \frac{s}{s} = 1$$

This result can be verified in the transform tables in The Appendix.

Step Response
Similar to the impulse response, the step response of a system is the output of the system when a unit step function is used as the input. The step response is a common analysis tool used to determine certain metrics about a system. Typically, when a new system is designed, the step response of the system is the first characteristic of the system to be analyzed.

Convolution
However, the impulse response cannot be used to find the system output from the system input in the same manner as the transfer function. If we have the system input and the impulse response of the system, we can calculate the system output using the convolution operation as such:

 * $$y(t) = h(t) * x(t)$$

Where " * " (asterisk) denotes the convolution operation. Convolution is a complicated combination of multiplication, integration and time-shifting. We can define the convolution between two functions, a(t) and b(t) as the following:

 * $$(a*b)(t) = (b*a)(t) = \int_{-\infty}^\infty a(\tau)b(t - \tau)d\tau$$

(The variable τ (Greek tau) is a dummy variable for integration). This operation can be difficult to perform. Therefore, many people prefer to use the Laplace Transform (or another transform) to convert the convolution operation into a multiplication operation, through the Convolution Theorem.

Time-Invariant System Response
If the system in question is time-invariant, then the general description of the system can be replaced by a convolution integral of the system's impulse response and the system input. We can call this the convolution description of a system, and define it below:

 * $$y(t) = x(t) * h(t) = \int_{-\infty}^\infty x(\tau)h(t - \tau)d\tau$$

Convolution Theorem
This method of solving for the output of a system is quite tedious, and in fact it can waste a large amount of time if you want to solve a system for a variety of input signals. Luckily, the Laplace transform has a special property, called the Convolution Theorem, that makes the operation of convolution easier:

The Convolution Theorem can be expressed using the following equations:

 * $$\mathcal{L}[f(t) * g(t)] = F(s)G(s)$$
 * $$\mathcal{L}[f(t)g(t)] = F(s) * G(s)$$

This also serves as a good example of the property of Duality.

Using the Transfer Function
The Transfer Function fully describes a control system. The Order, Type and Frequency response can all be taken from this specific function. Nyquist and Bode plots can be drawn from the open loop Transfer Function. These plots show the stability of the system when the loop is closed. Using the denominator of the transfer function, called the characteristic equation, roots of the system can be derived.

For all these reasons and more, the Transfer function is an important aspect of classical control systems. Let's start out with the definition:

If the complex Laplace variable is s, then we generally denote the transfer function of a system as either G(s) or H(s). If the system input is X(s), and the system output is Y(s), then the transfer function can be defined as such:

 * $$H(s) = \frac{Y(s)}{X(s)}$$

If we know the input to a given system, and we have the transfer function of the system, we can solve for the system output by multiplying:

 * $$Y(s) = H(s)X(s)$$

Frequency Response
The Frequency Response is similar to the Transfer function, except that it is the relationship between the system output and input in the complex Fourier Domain, not the Laplace domain. We can obtain the frequency response from the transfer function, by using the following change of variables:

 * $$s = j\omega$$

Because the frequency response and the transfer function are so closely related, typically only one is ever calculated, and the other is gained by simple variable substitution. However, despite the close relationship between the two representations, they are both useful individually, and are each used for different purposes.

title: Control Systems/Transfer Functions


text:
Control Systems/Block Diagrams

When designing or analyzing a system, often it is useful to model the system graphically. Block Diagrams are a useful and simple method for analyzing a system graphically. A "block" looks on paper exactly what it means: <!--another block of unformatted text- If system H has a time-domain impulse response h(t), we can express y(t) as: y(t) = x(t) * h(t) Where the asterisk ( * ) denotes convolution. If system H has a Laplace-domain transfer function H(s), we show the relationship between the input and the output as:

Y(s) = X(s)H(s) [edit]Systems in Series

When two or more systems are in series, they can be combined into a single representative system, with a transfer function that is the sum of the individual systems.

If we have two systems, f(t) and g(t), we can put them in series with one another so that the output of system f(t) is the input to system g(t). Now, we can analyze them depending on whether we are using our classical or modern methods. If we define the output of the first system as h(t), we can define h(t) as: h(t) = x(t) * f(t) Now, we can define the system output y(t) in terms of h(t) as: y(t) = h(t) * g(t) We can expand h(t): y(t) = [x(t) * f(t)] * g(t) But, since convolution is associative, we can re-write this as: y(t) = x(t) * [f(t) * g(t)] Our system can be simplified therefore as such:

[edit]Series Transfer Functions If two or more systems are in series with one another, the total transfer function of the series is the product of all the individual system transfer functions.

In the time domain we know that: y(t) = x(t) * [f(t) * g(t)] But, in the frequency domain we know that convolution becomes multiplication, so we can re-write this as: Y(s) = X(s)[F(s)G(s)] We can represent our system in the frequency domain as:

[edit]Series State Space If we have two systems in series (say system F and system G), where the output of F is the input to system G, we can write out the state-space equations for each individual system. System 1: xF' = AFxF + BFu yF = CFxF + DFu System 2: xG' = AGxG + BGyF yG = CGxG + DGyF And we can write substitute these equations together form the complete response of system H, that has input u, and output yG:

[Series state equation]

[Series output equation]

[edit]Systems in Parallel

Blocks may not be placed in parallel without the use of an adder. Blocks connected by an adder as shown above have a total transfer function of: Y(s) = X(s)[F(s) + G(s)] Since the Laplace transform is linear, we can easily transfer this to the time domain by converting the multiplication to convolution: y(t) = x(t) * [f(t) + g(t)]

[edit]State Space Model

The state-space equations, with non-zero A, B, C, and D matrices conceptually model the following system:

In this image, the strange-looking block in the center is either an integrator or an ideal delay, and can be represented in the transfer domain as: or Depending on the time characteristics of the system. If we only consider continuous-time systems, we can replace the funny block in the center with an integrator:

[edit]In the Laplace Domain The state space model of the above system, if A, B, C, and D are transfer functions A(s), B(s), C(s) and D(s) of the individual subsystems, and if U(s) and Y(s) represent a single input and output, can be written as follows:

We will explain how we got this result, and how we deal with feedforward and feedback loop structures in the next chapter. [edit]Adders and Multipliers

Some systems may have dedicated summation or multiplication devices, that automatically add or multiply the transfer functions of multiple systems together [edit]Simplifying Block Diagrams

Block diagrams can be systematically simplified. Transformation	Equation	Block Diagram	Equivalent Block Diagram 1	Cascaded Blocks 2	Combining Blocks in Parallel 3	Removing a Block from a Forward Loop 4	Eliminating a Feedback Loop 5	Removing a Block from a Feedback Loop 6	Rearranging Summing Junctions

7	Moving a Summing Juction in front of a Block 8	Moving a Summing Juction beyond a Block 9	Moving a Takeoff Point in front of a Block 10	Moving a Takeoff Point beyond a Block 11	Moving a Takeoff Point in front of a Summing Junction 12	Moving a Takeoff Point beyond a Summing Junction -This book needs fixing per 24 March 2011>

Systems in Series
When two or more systems are in series, they can be combined into a single representative system, with a transfer function that is the product of the individual systems.

If we have two systems, f(t) and g(t), we can put them in series with one another so that the output of system f(t) is the input to system g(t). Now, we can analyze them depending on whether we are using our classical or modern methods.

If we define the output of the first system as h(t), we can define h(t) as:

 * $$h(t) = x(t) * f(t)$$

Now, we can define the system output y(t) in terms of h(t) as:

 * $$y(t) = h(t) * g(t)$$

We can expand h(t):

 * $$y(t) = [x(t) * f(t)] * g(t)$$

But, since convolution is associative, we can re-write this as:

 * $$y(t) = x(t) * [f(t) * g(t)]$$

Our system can be simplified therefore as such:

Series Transfer Functions
If two or more systems are in series with one another, the total transfer function of the series is the product of all the individual system transfer functions.

In the time-domain we know that:

 * $$y(t) = x(t) * [f(t) * g(t)]$$

But, in the frequency domain we know that convolution becomes multiplication, so we can re-write this as:

 * $$Y(s) = X(s)[F(s)G(s)]$$

We can represent our system in the frequency domain as:

Series State Space
If we have two systems in series (say system F and system G), where the output of F is the input to system G, we can write out the state-space equations for each individual system.

System 1:
 * $$x_F' = A_Fx_F + B_Fu$$
 * $$y_F = C_Fx_F + D_Fu$$

System 2:
 * $$x_G' = A_Gx_G + B_Gy_F$$
 * $$y_G = C_Gx_G + D_Gy_F$$

And we can write substitute these equations together form the complete response of system H, that has input u, and output yG:

 * $$\begin{bmatrix}x_G' \\ x_F'\end{bmatrix}

= \begin{bmatrix}A_G & B_GC_F \\ 0 & A_F\end{bmatrix} \begin{bmatrix}x_G \\ x_F\end{bmatrix} + \begin{bmatrix}B_GD_F \\ B_F\end{bmatrix}u$$

 * $$\begin{bmatrix}y_G \\ y_F\end{bmatrix}

= \begin{bmatrix}C_G & D_GC_F \\ 0 & C_F\end{bmatrix} \begin{bmatrix}x_G \\ x_F\end{bmatrix} + \begin{bmatrix}D_GD_F \\ D_F\end{bmatrix}u$$

Systems in Parallel

Blocks may not be placed in parallel without the use of an adder. Blocks connected by an adder as shown above have a total transfer function of:

 * $$Y(s) = X(s) [F(s) + G(s)]$$

Since the Laplace transform is linear, we can easily transfer this to the time domain by converting the multiplication to convolution:

 * $$y(t) = x(t) * [f(t) + g(t)]$$

State Space Model
The state-space equations, with non-zero A, B, C, and D matrices conceptually model the following system:

In this image, the strange-looking block in the center is either an integrator or an ideal delay, and can be represented in the transfer domain as:

 * $$\frac{1}{s}$$ or $$\frac{1}{z}$$

Depending on the time characteristics of the system. If we only consider continuous-time systems, we can replace the funny block in the center with an integrator:

In the Laplace Domain
The state space model of the above system, if A, B, C, and D are transfer functions A(s), B(s), C(s) and D(s) of the individual subsystems, and if U(s) and Y(s) represent a single input and output, can be written as follows:

 * $$\frac{Y(s)}{U(s)} = B(s)\left(\frac{1}{s - A(s)}\right)C(s) + D(s)$$

We will explain how we got this result, and how we deal with feedforward and feedback loop structures in the next chapter.

Adders and Multipliers
Some systems may have dedicated summation or multiplication devices, that automatically add or multiply the transfer functions of multiple systems together

Simplifying Block Diagrams
Block diagrams can be systematically simplified. Note that this table is from Schaum's Outline: Feedback and Controls Systems by DiStefano et al

title: Control Systems/Block Diagrams


text:
Control Systems/Feedback Loops

Feedback
A feedback loop is a common and powerful tool when designing a control system. Feedback loops take the system output into consideration, which enables the system to adjust its performance to meet a desired output response.

When talking about control systems it is important to keep in mind that engineers typically are given existing systems such as actuators, sensors, motors, and other devices with set parameters, and are asked to adjust the performance of those systems. In many cases, it may not be possible to open the system (the "plant") and adjust it from the inside: modifications need to be made external to the system to force the system response to act as desired. This is performed by adding controllers, compensators, and feedback structures to the system.

Basic Feedback Structure

This is a basic feedback structure. Here, we are using the output value of the system to help us prepare the next output value. In this way, we can create systems that correct errors. Here we see a feedback loop with a value of one. We call this a unity feedback.

Here is a list of some relevant vocabulary, that will be used in the following sections:

 * Plant:The term "Plant" is a carry-over term from chemical engineering to refer to the main system process. The plant is the preexisting system that does not (without the aid of a controller or a compensator) meet the given specifications. Plants are usually given "as is", and are not changeable. In the picture above, the plant is denoted with a P.
 * Controller:A controller, or a "compensator" is an additional system that is added to the plant to control the operation of the plant. The system can have multiple compensators, and they can appear anywhere in the system: Before the pick-off node, after the summer, before or after the plant, and in the feedback loop. In the picture above, our compensator is denoted with a C.

 * Summer:A summer is a symbol on a system diagram, (denoted above with parenthesis) that conceptually adds two or more input signals, and produces a single sum output signal.
 * Pick-off node:A pickoff node is simply a fancy term for a split in a wire.
 * Forward Path:The forward path in the feedback loop is the path after the summer, that travels through the plant and towards the system output.
 * Reverse Path:The reverse path is the path after the pick-off node, that loops back to the beginning of the system. This is also known as the "feedback path".
 * Unity feedback:When the multiplicative value of the feedback path is 1.

Negative vs Positive Feedback
It turns out that negative feedback is almost always the most useful type of feedback. When we subtract the value of the output from the value of the input (our desired value), we get a value called the error signal. The error signal shows us how far off our output is from our desired input.

Positive feedback has the property that signals tend to reinforce themselves, and grow larger. In a positive feedback system, noise from the system is added back to the input, and that in turn produces more noise. As an example of a positive feedback system, consider an audio amplification system with a speaker and a microphone. Placing the microphone near the speaker creates a positive feedback loop, and the result is a sound that grows louder and louder. Because the majority of noise in an electrical system is high-frequency, the sound output of the system becomes high-pitched.

Feedback Loop Transfer Function
We can solve for the output of the system by using a series of equations:

 * $$E(s) = X(s) - Y(s)$$

 * $$ Y(s) = G(s)E(s)$$

and when we solve for Y(s) we get:

 * $$ Y(s) = X(s) \frac{Gp(s)}{1 + Gp(s)}$$

The reader is encouraged to use the above equations to derive the result by themselves.

The function E(s) is known as the error signal. The error signal is the difference between the system output (Y(s)), and the system input (X(s)). Notice that the error signal is now the direct input to the system G(s). X(s) is now called the reference input. The purpose of the negative feedback loop is to make the system output equal to the system input, by identifying large differences between X(s) and Y(s) and correcting for them.

Example: Elevator
Here is a simple example of reference inputs and feedback systems:

State-Space Feedback Loops
In the state-space representation, the plant is typically defined by the state-space equations:

 * $$x'(t) = Ax(t) + Bu(t)$$
 * $$y(t) = Cx(t) + Du(t)$$

The plant is considered to be pre-existing, and the matrices A, B, C, and D are considered to be internal to the plant (and therefore unchangeable). Also, in a typical system, the state variables are either fictional (in the sense of dummy-variables), or are not measurable. For these reasons, we need to add external components, such as a gain element, or a feedback element to the plant to enhance performance.

Consider the addition of a gain matrix K installed at the input of the plant, and a negative feedback element F that is multiplied by the system output y, and is added to the input signal of the plant. There are two cases:
 * 1) The feedback element F is subtracted from the input before multiplication of the K gain matrix.
 * 2) The feedback element F is subtracted from the input after multiplication of the K gain matrix.

In case 1, the feedback element F is added to the input before the multiplicative gain is applied to the input. If v is the input to the entire system, then we can define u as:

 * $$u(t) = Fv(t) - FKy(t)$$

In case 2, the feeback element F is subtracted from the input after the multiplicative gain is applied to the input. If v is the input to the entire system, then we can define u as:

 * $$u(t) = Kv(t) - Fy(t)$$

Open Loop vs Closed Loop

Let's say that we have the generalized system shown above. The top part, Gp(s) represents all the systems and all the controllers on the forward path. The bottom part, Gb(s) represents all the feedback processing elements of the system. The letter "K" in the beginning of the system is called the Gain. We will talk about the gain more in later chapters. We can define the Closed-Loop Transfer Function as follows:

 * $$ H_{cl}(s) = \frac{KGp(s)}{1 + Gp(s)Gb(s)}$$

If we "open" the loop, and break the feedback node, we can define the Open-Loop Transfer Function, as:

 * $$H_{ol}(s) = KGp(s)$$

We can redefine the closed-loop transfer function in terms of this open-loop transfer function:

 * $$ H_{cl}(s) = \frac{H_{ol}(s)}{1 +Gp(s)Gb(s)}$$

These results are important, and they will be used without further explanation or derivation throughout the rest of the book.

Placement of a Controller
There are a number of different places where we could place an additional controller.

 * {| class="wikitable"

 * [[Image:System 5 Positions.png|400px|center]]
 * 1) In front of the system, before the feedback loop.
 * 2) Inside the feedback loop, in the forward path, before the plant.
 * 3) In the forward path, after the plant.
 * 4) In the feedback loop, in the reverse path.
 * 5) After the feedback loop.
 * }
 * 1) In the feedback loop, in the reverse path.
 * 2) After the feedback loop.
 * }

Each location has certain benefits and problems, and hopefully we will get a chance to talk about all of them.

Second-Order Systems
The general expression of the transfer function of a second order system is given as:

$$

\frac{\omega_n^2}{s^2 + 2\zeta\omega_ns + \omega_n^2}

$$

where $$\zeta$$ and $$\omega_n$$ are damping ratio and natural frequency of the system respectively.

Damping Ratio
The damping ratio is defined by way of the sign $$\zeta$$. The damping ratio gives us an idea about the nature of the transient response detailing the amount of overshoot & oscillation that the system will undergo. This is completely regardless of time scaling.

If :
 * $$\zeta$$ = zero, the system is undamped;
 * $$\zeta$$ < 1, the system is underdamped;
 * $$\zeta$$ = 1, the system is critically damped;
 * $$\zeta$$ > 1, the system is overdamped.

$$\zeta$$ is used in conjunction with the natural frequency to determine system properties. To find the zeta value you must first find the natural response!

Natural Frequency
Natural Frequency, denoted by $$\omega_n$$ is defined as the frequency with which the system would oscillate if it were not damped and we define the damping ratio as $$ \zeta = \frac{\sigma}{\omega_n} $$.

title: Control Systems/Feedback Loops


text:
Control Systems/Poles and Zeros

Poles and Zeros
Poles and Zeros of a transfer function are the frequencies for which the value of the denominator and numerator of transfer function becomes infinite and zero respectively. The values of the poles and the zeros of a system determine whether the system is stable, and how well the system performs. Control systems, in the most simple sense, can be designed simply by assigning specific values to the poles and zeros of the system.

Physically realizable control systems must have a number of poles greater than the number of zeros. Systems that satisfy this relationship are called Proper. We will elaborate on this below.

Time-Domain Relationships
Let's say that we have a transfer function with 3 poles:

 * $$H(s) = \frac{a}{(s - l)(s - m)(s - n)}$$

The poles are located at s = l, m, n. Now, we can use partial fraction expansion to separate out the transfer function:

 * $$H(s) = \frac{a}{(s - l)(s - m)(s - n)} = \frac{A}{s-l} + \frac{B}{s-m} + \frac{C}{s-n}$$

Using the inverse transform on each of these component fractions (looking up the transforms in our table), we get the following:

 * $$h(t) = Ae^{lt}u(t) + Be^{mt}u(t) + Ce^{nt}u(t)$$

But, since s is a complex variable, l m and n can all potentially be complex numbers, with a real part (σ) and an imaginary part (jω). If we just look at the first term:

 * $$Ae^{lt}u(t) = Ae^{(\sigma_l + j\omega_l)t}u(t) = Ae^{\sigma_l t}e^{j\omega_l t}u(t)$$

Using Euler's Equation on the imaginary exponent, we get:

 * $$Ae^{\sigma_l t}[\cos(\omega_l t) + j\sin(\omega_l t)]u(t)$$

If a complex pole is present it is always accompanied by another pole that is its complex conjugate. The imaginary parts of their time domain representations thus cancel and we are left with 2 of the same real parts. Assuming that the complex conjugate pole of the first term is present, we can take 2 times the real part of this equation and we are left with our final result:

 * $$2Ae^{\sigma_l t}\cos(\omega_l t)u(t)$$

We can see from this equation that every pole will have an exponential part, and a sinusoidal part to its response. We can also go about constructing some rules:

 * 1) if σl = 0, the response of the pole is a perfect sinusoidal (an oscillator)
 * 2) if ωl = 0, the response of the pole is a perfect exponential.
 * 3) if σl < 0, the exponential part of the response will decay towards zero.
 * 4) if σl > 0, the exponential part of the response will rise towards infinity.

From the last two rules, we can see that all poles of the system must have negative real parts, and therefore they must all have the form (s + l) for the system to be stable. We will discuss stability in later chapters.

What are Poles and Zeros
Let's say we have a transfer function defined as a ratio of two polynomials:

 * $$H(s) = {N(s) \over D(s)}$$

Where N(s) and D(s) are simple polynomials. Zeros are the roots of N(s) (the numerator of the transfer function) obtained by setting N(s) = 0 and solving for s.

Poles are the roots of D(s) (the denominator of the transfer function), obtained by setting D(s) = 0 and solving for s. Because of our restriction above, that a transfer function must not have more zeros than poles, we can state that the polynomial order of D(s) must be greater than or equal to the polynomial order of N(s).

Effects of Poles and Zeros
As s approaches a zero, the numerator of the transfer function (and therefore the transfer function itself) approaches the value 0. When s approaches a pole, the denominator of the transfer function approaches zero, and the value of the transfer function approaches infinity. An output value of infinity should raise an alarm bell for people who are familiar with BIBO stability. We will discuss this later.

As we have seen above, the locations of the poles, and the values of the real and imaginary parts of the pole determine the response of the system. Real parts correspond to exponentials, and imaginary parts correspond to sinusoidal values. Addition of poles to the transfer function has the effect of pulling the root locus to the right, making the system less stable. Addition of zeros to the transfer function has the effect of pulling the root locus to the left, making the system more stable.

Second-Order Systems
The canonical form for a second order system is as follows:

 * $$H(s) = \frac{K\omega^2}{s^2 + 2\zeta\omega s + \omega^2}$$

Where K is the system gain, ζ is called the damping ratio of the function, and ω is called the natural frequency of the system. ζ and ω, if exactly known for a second order system, the time responses can be easily plotted and stability can easily be checked. More information on second order systems can be found here.

Damping Ratio
The damping ratio of a second-order system, denoted with the Greek letter zeta (ζ), is a real number that defines the damping properties of the system. More damping has the effect of less percent overshoot, and slower settling time. Damping is the inherent ability of the system to oppose the oscillatory nature of the system's transient response. Larger values of damping coefficient or damping factor produces transient responses with lesser oscillatory nature.

Natural Frequency
The natural frequency is occasionally written with a subscript:

 * $$\omega \to \omega_n$$

We will omit the subscript when it is clear that we are talking about the natural frequency, but we will include the subscript when we are using other values for the variable ω. Also, $$\omega~=~\omega_n$$ when $$\zeta~=0$$.

title: Control Systems/Poles and Zeros


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Control Systems/Stability

Stability
When a system is unstable, the output of the system may be infinite even though the input to the system was finite. This causes a number of practical problems. For instance, a robot arm controller that is unstable may cause the robot to move dangerously. Also, systems that are unstable often incur a certain amount of physical damage, which can become costly. Nonetheless, many systems are inherently unstable - a fighter jet, for instance, or a rocket at liftoff, are examples of naturally unstable systems. Although we can design controllers that stabilize the system, it is first important to understand what stability is, how it is determined, and why it matters.

The chapters in this section are heavily mathematical and many require a background in linear differential equations. Readers without a strong mathematical background might want to review the necessary chapters in the Calculus and Ordinary Differential Equations books (or equivalent) before reading this material.

For most of this chapter we will be assuming that the system is linear and can be represented either by a set of transfer functions or in state space. Linear systems have an associated characteristic polynomial which tells us a great deal about the stability of the system. If any coefficient of the characteristic polynomial is zero or negative then the system is either unstable or at most marginally stable. It is important to note that even if all of the coefficients of the characteristic polynomial are positive the system may still be unstable. We will look into this in more detail below.

BIBO Stability
A system is defined to be BIBO Stable if every bounded input to the system results in a bounded output over the time interval $$[t_0, \infty)$$. This must hold for all initial times to. So long as we don't input infinity to our system, we won't get infinity output.

A system is defined to be uniformly BIBO Stable if there exists a positive constant k that is independent of t0 such that for all t0 the following conditions:

 * $$\|u(t)\| \le 1$$

 * $$ t \ge t_0$$

implies that

 * $$\|y(t)\| \le k$$

There are a number of different types of stability, and keywords that are used with the topic of stability. Some of the important words that we are going to be discussing in this chapter, and the next few chapters are: BIBO Stable, Marginally Stable, Conditionally Stable, Uniformly Stable, Asymptotically Stable, and Unstable. All of these words mean slightly different things.

Determining BIBO Stability
We can prove mathematically that a system f is BIBO stable if an arbitrary input x is bounded by two finite but large arbitrary constants M and -M:
 * $$-M < x \le M$$

We apply the input x, and the arbitrary boundaries M and -M to the system to produce three outputs:

 * $$y_x = f(x)$$
 * $$y_M = f(M)$$
 * $$y_{-M} = f(-M)$$

Now, all three outputs should be finite for all possible values of M and x, and they should satisfy the following relationship:

 * $$y_{-M} \le y_x \le y_M$$

If this condition is satisfied, then the system is BIBO stable.

A SISO linear time-invariant (LTI) system is BIBO stable if and only if $$g(t)$$ is absolutely integrable from [0,∞] or from:

 * $$\int_{0}^{\infty} |g(t)| \,dt \leq M < {\infty}$$

Poles and Stability
When the poles of the closed-loop transfer function of a given system are located in the right-half of the S-plane (RHP), the system becomes unstable. When the poles of the system are located in the left-half plane (LHP) and the system is not improper, the system is shown to be stable. A number of tests deal with this particular facet of stability: The Routh-Hurwitz Criteria, the Root-Locus, and the Nyquist Stability Criteria all test whether there are poles of the transfer function in the RHP. We will learn about all these tests in the upcoming chapters.

If the system is a multivariable, or a MIMO system, then the system is stable if and only if every pole of every transfer function in the transfer function matrix has a negative real part and every transfer function in the transfer function matrix is not improper. For these systems, it is possible to use the Routh-Hurwitz, Root Locus, and Nyquist methods described later, but these methods must be performed once for each individual transfer function in the transfer function matrix.

Poles and Eigenvalues
The  poles of the transfer function, and the eigenvalues of the system matrix A are related. In fact, we can say that the eigenvalues of the system matrix A are the poles of the transfer function of the system. In this way, if we have the eigenvalues of a system in the state-space domain, we can use the Routh-Hurwitz, and Root Locus methods as if we had our system represented by a transfer function instead.

On a related note, eigenvalues and all methods and mathematical techniques that use eigenvalues to determine system stability only work with time-invariant systems. In systems which are time-variant, the methods using eigenvalues to determine system stability fail.

Transfer Functions Revisited
We are going to have a brief refesher here about transfer functions, because several of the later chapters will use transfer functions for analyzing system stability.

Let us remember our generalized feedback-loop transfer function, with a gain element of K, a forward path Gp(s), and a feedback of Gb(s). We write the transfer function for this system as:

 * $$ H_{cl}(s) = \frac{KGp(s)}{1 + H_{ol}(s)}$$

Where $$H_{cl}$$ is the closed-loop transfer function, and $$H_{ol}$$ is the open-loop transfer function. Again, we define the open-loop transfer function as the product of the forward path and the feedback elements, as such:

 * $$H_{ol}(s) = KGp(s)Gb(s)$$    <---Note this definition now contradicts the updated definition in the Feedback Loops section.

Now, we can define F(s) to be the characteristic equation. F(s) is simply the denominator of the closed-loop transfer function, and can be defined as such:

 * $$F(s) = 1 + H_{ol} = D(s)$$

We can say conclusively that the roots of the characteristic equation are the poles of the transfer function. Now, we know a few simple facts:

 * 1) The locations of the poles of the closed-loop transfer function determine if the system is stable or not
 * 2) The zeros of the characteristic equation are the poles of the closed-loop transfer function.
 * 3) The characteristic equation is always a simpler equation than the closed-loop transfer function.

These functions combined show us that we can focus our attention on the characteristic equation, and find the roots of that equation.

State-Space and Stability
As we have discussed earlier, the system is stable if the eigenvalues of the system matrix A have negative real parts. However, there are other stability issues that we can analyze, such as whether a system is uniformly stable, asymptotically stable, or otherwise. We will discuss all these topics in a later chapter.

Marginal Stability
When the poles of the system in the complex s-domain exist on the imaginary axis (the vertical axis), or when the eigenvalues of the system matrix are imaginary (no real part), the system exhibits oscillatory characteristics, and is said to be marginally stable. A marginally stable system may become unstable under certain circumstances, and may be perfectly stable under other circumstances. It is impossible to tell by inspection whether a marginally stable system will become unstable or not.

We will discuss marginal stability more in the following chapters.

title: Control Systems/Stability


text:
User:Pengrate~enwikibooks

Eh, I'll maybe fill this in later.

title: User:Pengrate~enwikibooks


text:
Organic Chemistry/Alkenes/Cycloalkenes

Cycloolefin
A cycloalkene (also sometimes called a cycloolefin) is a type of alkene hydrocarbon which contains a closed ring of carbon atoms. The prefix cyclo- comes from ancient Greek and in this case means round. Whenever the ends of a carbon chain are joined together, that molecule is said to be cyclic, and alkenes are no different than other carbon chains in that respect.

Cycloalkenes are perhaps among the most important organic substances for biological and industrial purposes because they are used in the production of molecules essential to a broad spectrum of applications. Understanding cycloalkenes is a critical part of understanding organic chemistry and its applications to biology, medicine, industry and every other relevant field.

What They Are
As stated above, a cycloalkene is merely a chain of carbon molecules containing at least one double bond between two carbon atoms that has joined two ends to form some type of ring structure. The number of carbons involved and the number of double bonds shared affect molecule stability and other properties, but those factors do not affect the definition of what a cycloalkene is.

If an organic chemist knows the empirical formula of a molecule and also knows that it is a cyclic molecule, she can derive the shape of the molecule from this information. She does so using the concept of sites of unsaturation. As you know from the chapter on alkanes, a saturated hydrocarbon chain is one that has only single bonds between carbon atoms and enough hydrogens attached to give every single carbon a total of four molecular bonds. If there are four carbon atoms, then using the saturation formula below we can calculate that there will be ten hydrogen atoms attached.

 * nC * 2 + 2 = nH

Where nC equals the number of carbon atoms and nH equals the number of hydrogen atoms.

But what if the empirical formula we are given does not show ten atoms of hydrogen for every four atoms of carbon? What if it only shows eight? You might guess that there was an error, but it is more likely that the molecule features one site of unsaturation where either a double bond has formed (as in an alkene) or the molecule is cyclic (where the two ends have joined together to form a ring). The formula for calculating the number of double bonds in an alkene, and for accounting for cyclic structures, is below:

 * ALKENES: (nC * 2 + 2) - (2 * nDBL) = nH
 * CYCLOALKENES: (nC * 2 + 2) - (2 * nDBL) - 2 = nH

Where nC equals the number of carbon atoms, nH equals the number of hydrogen atoms, and nDBL equals the number of double bonds. For cycloalkenes, the - 2 just before the equal sign signifies the fact that two hydrogen atoms are displaced when a carbon chain joins from head to tail in order to form a ring.

Physical Properties
Cycloalkenes have slightly fewer bonds than equivalent straight-chain alkenes, which means that they have less internal energy. The additional site of unsaturation due to the ring means that there are two fewer hydrogen atoms on the molecule, which is why there are fewer bonds in cycloalkenes than in straight chains.

On the macroscopic scale, cycloalkenes are most often found as liquids at standard temperature and pressure. This is because a certain number of carbons are needed to make ring structures, which means the molecules are generally too heavy to be gaseous at STP. Likewise, too many carbon atoms makes for too large a ring, so cycloalkenes are very rarely solid at STP as well.

Depending on which functional groups and side-chains a molecule may possess, some cycloalkenes can polymerize at STP thus forming a solid substance, but this should be considered a reasonable exception to the generalities listed above.

Chemical Properties
Alkenes contain double bonds by definition, and cycloalkenes benefit from enhanced stability if the double bonds are conjugated. Cyclohexene is one such case, where if it contains a total of three double bonds it is no longer called cyclohexene. A six-carbon ring with three alternating (and therefore conjugated) double bonds is called benzene, and benzene is such a stable molecule that it displays completely different chemistry than other non-aromatic cycloalkenes.

Star of the Show: Benzene
Benzene is critical molecule to the study of organic chemistry. It is a colorless and flammable carcinogenic liquid at STP, with a pleasant, sweet smell.

Benzene presents a special problem in that, to account for all the bonds, there must be alternating double carbon bonds. Using X-ray diffraction, researchers in the early 20th century discovered that all of the carbon-carbon bonds in benzene are of the same length, but it was well known at the time that a single bond should be longer than a double bond. In benzene, the distance between the two bonded atoms was greater than the length of a double bond, but shorter than the length of a single bond. There seemed to be, in effect, a bond and a half between each carbon!

Today this fact can be explained by the concept of electron delocalization. Essentially, the electrons are not held tightly by one carbon atom or another, but instead they are somewhat "smeared" out, like cream cheese around the edges of a bagel. The actual physics of the situation are far more complex than breakfast-food analogies could do justice, but the essential concept is the same. The electrons - and thus, the electronegativity - of a benzene ring are spread evenly around the molecule.

This electron delocalization gives benzene many interesting and useful properties, as you will see in later chapters. Of course benzene isn't the only molecule in organic chemistry with "smeared" electrons; the delocalization of electron density is also useful in discussing other molecules, which are called aromatics.

Aromaticity
Aromaticity is a chemical property in which a conjugated ring of unsaturated bonds, lone electron pairs, or empty electron orbitals exhibit a stabilization stronger than would be expected by other factors alone. Aromaticity can also be considered a manifestation of resonance, in that multiple possible conformations of the aromatic molecule contribute to its stability.

An aromatic compound contains a set of covalently-bound atoms with very specific characteristics, namely: a delocalized conjugated pi system, a coplanar structure, atoms arranged in one or more rings, and a number of pi delocalized electrons that is even but not a multiple of four.

The permissible numbers of pi electrons in an aromatic structure are generally the even numbers not divisible by four, which would include 6, 10, 14, and so on. Aromaticity is most often estimated using Hückel's rule:

 * A cyclic ring molecule should be aromatic when the number of π electrons equals 4n + 2, where n is any positive integer or zero.

But there have been exceptions found to this rule in more recent decades.

Aromatic molecules display special reactivity in organic reactions -- for example electrophilic aromatic substitution and nucleophilic aromatic substitution -- which can be used distinguish them from other cycloalkenes.

Uses of Benzene
Benzene is an important industrial solvent and a precursor in the industrial production of many types of drugs, plastics, synthetic rubbers, and dyes. It is used in most organic compounds of commercial value because it is so very versatile and has many desirable chemical properties. It is also used as a solvent in certain organic chemistry laboratory experiments, but great care should be taken when using it due to its carcinogenic properties.

Sources of Benzene
Benzene can be found naturally occurring in crude oil, but today it is also often synthesized from other compounds which can be found in unrefined petroleum. Another source of benzene is benzoic acid and certain other biological compounds, but the first industrial source of benzene was distillation from coal tar.

title: Organic Chemistry/Alkenes/Cycloalkenes


text:
Blender 3D: Noob to Pro/Every Material Known to Man/Human Skin

Method 1
Material Col: R=0.94 G=0.82 B=0.73 Spe: R=0.94 G=0.82 B=0.73 Mir: R=0.94 G=0.82 B=0.73 Shaders [Oren-Nayar] Reflection: 0.95 Roughness:0.3 [CookTorr] Specularity: 0.05 Hardness: 10 Translucency 0.00 Ambient: 0.500 Emit: 0.000

SSS-based method with Nodes
This method will use material nodes and SSS (SubSurfaceScattering) to create a more realistic slightly tanned caucasian skin.

Nodes
Just enable it as a Nodes material and create two material nodes, referring the first as Node1 and the second as Node2. Skip this section if you are not a "noob". First off, under the "Materials" section select {Add new} and under "Links and Pipeline" tab select "Nodes".Now go to the "Window Type", click, it and from the drop-down menu select "Node Editor". You should have a Node marked as Out and another as Material.Right-click on an empty space and select Add>Input>Material. Link the top-right {Color} (yellow circle), from the node you've created and link it to the bottom-left {Color} of the other Material node. Click on {Add new} in each Material node.

Node1
Select Node1 On "Materials Button" Shaders tab Enable {Cubic} Oren-Nayar: Ref-0.8; Rough:0.3 CookTorr: Spec-0.15; Hard:50 SSS tab Enable {Subsurface Scattering} Scale: 0.13 Radius R:4.821 Radius G:1.694 Radius B:1.090 IOR (Index of Refraction):1.3 (Note: you can change the IOR to give a lighter,+, or darker,-, skin) Front:1.15 Back:1 Go to the color picker (not Col slider), Hex:E9B08F

Node2
Select Node2 On the "Materials button" Shaders tab Enable {cubic} Minnaert: Ref-0.71; Dark-0.77 Cooktorrent: Spec-0.19; Hard-8 On the "Textures Button" Add new>Stucci Enable {Plastic}, and {Soft noise} NoiseSize: 0.001 Turbulence: 3.4 Noise Basis>Original Perlin Again on the "Materials Button" Map Input tab Enable {Orco} and {Sphere} Map to tab Only enable Nor, and leave the rest unselected Nor slider: 0.4 (Author Note: Change everything as needed if you feel it gives a better look or a more realistic one, also, this article may require some reformatting, since i'm a newbie in wikibooks.)

Method: General Caucasian

Color Shaders
 * Col
 * R- 0.902
 * G- 0.773
 * B- 0.694
 * Spec
 * R- 0.834
 * G- 0.760
 * B- 0.679
 * Diffuse Type- Oren-Nayar
 * Ref- 0.85
 * Rough- 0.3
 * Specular Type- CookTorr
 * Spec- 0.17
 * Hard- 30

Method: Pale-Style
Color Shaders
 * Col
 * R- 0.889
 * G- 0.788
 * B- 0.788
 * Spec
 * R- 0.723
 * G- 0.692
 * B- 0.657
 * Diffuse Type- Oren-Nayar
 * Ref- 0.85
 * Rough- 0.3
 * Specular Type- CookTorr
 * Spec- 0.17
 * Hard- 30

Method: Black-Style
Col: R:350 G:244 B:194

Spe: R:834 G:760 B:679

Shaders

Oren-Nayer Ref:0.85 Rough:0.3

Cooktorr Spec:0.17 Hard:30

Method: Brown-Style
Col: R:0.584 G:0.494 B:0.246

Spe: R:0.834 G:0.76 B:0.679

Shaders

Oren-Nayer: Ref:0.85 Rough:0.3

Spec:0.18 Hard:30

Method: South-Asian
Col: R=0.75 G=0.52 B=0.4 Spe: R=0.75 G=0.52 B=0.4 Mir: R=0.75 G=0.52 B=0.4 (note: the intensity of light and the type of lamp{and other light settings} will affect the material preview ,press f5 ,under the lamp buttons make sure that:-  energy="1" r,g,b all are set to "1" under preview pane lamp is selected )

Shaders

[Oren-Nayar] Reflection: 0.95 Roughness:0.3 [CookTorr] Specularity: 0.05 Hardness: 10 Translucency 0.00 Ambient: 0.500 Emit: 0.000

Preview

(Note:in this image the actual color of the material is not so clearly visible as it is highly compressed,click on the image to see the image in original size and actual shades{color})

Human Skin Black Skin White Skin Brown Skin South Asian Skin


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A-level Physics/Forces, Fields and Energy/Capacitors

Introduction When two conductive materials are separated by an insulating material, then it will behave as a Capacitor with associated Capacitance in the units of Farads (Coulombs/Volt).

Intuitively, Capacitance can be interpreted as "How much charge can I shove into a material if I apply a certain voltage?"Capacitors are useful because it can store energy momentarily and dissipate the energy later, and with combination of a resistor, it is capable of "delaying" a signal.

Definition of Capacitance/Capacitor
A capacitor is usually made from two sheets of metal separated by an insulating material (such as air or ceramics). If we apply a voltage between the two sheets, there will be an associated electric field generated, and charges will accumulate on each side of the plates. We define Capacitance ($$C$$) to be $$C = \frac{Q}{V}$$ where Q is the charge that accumulates on the plate when voltage V is applied. This can be concluded intuitively. A capacitor of impossibly high quality will be able to accumulate the maximum amount of charge with the least amount of voltage applied. The unit of capacitance is in Farad, or F for short.

Another definition for capacitance is area of the plates times the permittivity, divided by the distance of the two plates from the insulating material. In other words, $$C =\frac{\epsilon A}{D}$$. This makes intuitive sense - if we make the plates bigger, we can store more charge, and if we bring the plates closer, the tendency for the charges to attract increase, thereby increasing the electric field generated. Further, because of the nature of a capacitor, as the ability for the dielectric (in between the two plates) to restrict electrical fields (permittivity) goes up, so does the capacitance,

Now, it does not mean that capacitance is a property that appears only on two sheets of metallic sheet. In fact, any piece of wire or metal would have small but non-zero associated capacitance with it. Calculating such capacitances and either exploiting them or taking necessary measures to counteract it is a big deal in engineering electric circuits.

Capacitors connected in Parallel and Series
Let's find the equivalent capacitance of capacitors in series and in parallel.

Capacitors in parallel

Capacitance in two capacitors connected in parallel adds up, i.e.

$$C_{equivalent}=C_{1}+C_{2}$$

Not-so-rigorous proof
When two capacitors are connected in parallel, then the terminals of capacitor will have the same voltage. So, if we swap the capacitors in parallel with some equivalent capacitor, it should have the same voltage drop as the either one of the parallel capacitors had. If we count the charges accumulated on the capacitors in parallel, they add up (If one capacitor had Q1 charges accumulated and the other Q2 then the equivalent charges accumulated is Q1+Q2). That means that the charges in the equivalent capacitor is the sum of charges accumulated... which means:

 * $$Q_{equivalent}=Q_{1}+Q_{2}...$$

 * $$V_{equivalent} = V_{1} = V_{2} $$ They are all equal, so let's call it "V".

 * So,

 * $$\frac{Q_{equivalent}}{V_{equivalent}}=\frac{Q_{1}+Q_{2}}{V}$$

 * $$\frac{Q_{equivalent}}{V_{equivalent}}=\frac{Q_{1}}{V_{1}}+\frac{Q_{2}}{V_{2}}$$

 * Therefore,

$$C_{equivalent}=C_{1}+C_{2}$$

...We can generalize this for more than 2 capacitors - just keep adding the effective capacitance of each capacitor. So for five capacitors, the total capacitance would be
 * $$C_{equivalent}=C_{1}+C_{2}+C_{3}+C_{4}+C_{5}$$

Capacitors in series

Reciprocal of Capacitance adds up for capacitor connected in series., i.e.

$$\frac{1}{C_{equivalent}}=\frac{1}{C_{1}}+\frac{1}{C_{2}}...$$

(Once again) Not-So-Rigorous Proof
It's the exact opposite of parallel circuit.

First, the voltage drop must add up (for example, if two series capacitors C1 and C2 had voltage drop of 3V and 1V, then the equivalent capacitor had better have voltage drop of 4V).

What about the charge, however? The charges must remain the same in the equivalent capacitor. To illustrate, suppose two capacitors C1 and C2 are connected in series. Then if charge Q accumulates on one plate of C1, then charge of -Q would accumulate on the other plate. Conservation of charge dictates that the '-Q' must come from somewhere. That 'somewhere' is the top plate of C2.

So, the top plate of C2 loses '-Q' charge, which is essentially saying that C2 accumulates charge of 'Q'. Then, the other side of C2 will have a charge '-Q'. So, if we view the system holistically, the magnitude of charge accumulated on top of C1 is the magnitude of charge on bottom of C2)...

In short,

 * $$V_{equivalent}=V_{1}+V_{2}$$

 * $$Q_{equivalent} = Q_{1} = Q_{2} $$ They are all equal, so let's call it "Q".

 * So,

 * $$\frac{V_{equivalent}}{Q_{equivalent}}=\frac{V_{1}+V_{2}}{Q}$$

 * $$\frac{V_{equivalent}}{Q_{equivalent}}=\frac{V_{1}}{Q_{1}}+\frac{V_{2}}{Q_{2}}$$

 * Therefore,

 * $$\frac{1}{C_{equivalent}}=\frac{1}{C_{1}}+\frac{1}{C_{2}}$$

Once again, we can generalize this rule for more than 2 capacitors - just add the reciprocals!

Capacitor as an energy storage element
A capacitor can be considered as a device that stores energy in the electric field by applying voltage across it. If we calculate the energy stored a capacitor (E) of capacitance C when voltage V is applied, we find that $$E=\frac{C V^2}{2} (J)$$ (By the way, the magnetic analogue of this is called the inductor, and it possesses surprisingly similar characteristic with surprisingly similar equations.)

Not-so-rigorous Proof
The power dissipated for an electric component was defined to be P=v i where v=voltage and i=current. Current is change of charge over time, or dQ/dt. We have defined C=Q/V, so Q=CV. Since C is constant, i = dQ/dt = C dV/dt. Plug this into the equation for power, and we get:
 * $$P=C V \frac{dV}{dt}$$

Because power is rate at which energy is changing, (P=dW/dt), to find work W, we have to integrate with respect to time. This gives us:
 * $$W=\int P dt=\int C V \frac{dV}{dt} dt$$

Though mathematicians will be infuriated by what I'm about to say now, it usually works for most cases. If we consider derivatives like a fraction, then we note that the 'dt's will cancel out, giving us:
 * $$W=\int P dt=\int C V dV$$
 * which gives us:
 * $$W=\frac{C V^2}{2}$$

...which is the work required to store charges in a capacitor with voltage V applied, which is the energy stored in the capacitor when we apply a voltage V.

Capacitor with a Resistor (RC Circuits)
When we have a circuit with resistor and a capacitor, we have what is known as a RC circuit, which appears all the time in any electric system. It can be used to delay a signal or filter unwanted signals.

Derivation
Let's consider a case where a resistor with resistance R is connected in series with a capacitor with capacitance C and a voltage source with voltage $$-V_{source}$$. Assume that the Capacitor at time=0 has potential difference $$V_{0}$$

If we take Kirchoff's Voltage Law for this circuit, what we will get is the following:

$$ -V_{source} + i R + V_{capacitor} = 0 $$

We know that the current flowing through the resistor is same as the current flowing through the capacitor. Because Q=CV for capacitor, the current i is $$\frac{dQ}{dt}=C V_{capacitor}'$$. Replacing i, we get:

$$ -V_{source} + C V_{capacitor}' R + V_{capacitor} = 0 $$

Add $$V_{source}$$ and divide by RC to get:

$$ V_{capacitor}' + \frac{V_{capacitor}}{RC} = \frac{V_{source}}{RC} $$

This is a first order differential equation. Solving this, we get:

$$ V_{capacitor} = A e^{-\frac{t}{RC}} + B $$

If we then differentiate this we get:

$$V'_{capacitor} = \frac{-A}{RC}e^{-\frac{t}{RC}}$$

And limiting t towards infinity gives:

$$V'_{capacitor} = 0$$

$$V_{capacitor} = 0 + B$$

If we plug this into the differential equation mentioned above, we will get:

$$ 0 + \frac{B}{RC} = \frac{V_{source}}{RC} $$

Thus, $$B=V_{source}$$

Now, plug in the equation that we've found for V(capacitor) for time=0 to find:

$$ V_{0} = A + V_{source} $$

Which gives us $$ V_{0} - V_{source} = A $$. Combined, we get:

$$ V_{capacitor} = (V_{0}-V_{source}) e^{-\frac{t}{RC}} + V_{source} $$

We will call $$\tau = RC$$ which gives us:

$$ V_{capacitor} = (V_{0}-V_{source}) e^{-\frac{t}{\tau}} + V_{source}; \tau=RC $$

Interpretation of the Equation
At t=0, we can see that voltage of the capacitor is equal to its initial condition. We can also notice that as time approaches infinity, the exponential term gets smaller and smaller, which gives us voltage of the source. The nature of the function does not allow discontinuity, so that means that the function is slowly making a transition from V(0) to V(source). How fast? Just take the derivative.

With this interpretation, RC circuit is a 'circuit that makes a smooth transition from one voltage level to another in an exponential fashion.'

Time Constant
$$\tau$$ is what is known as the "Time constant" of the RC circuit. It is a magnitude that indicates how slowly the circuit voltage is decreasing or increasing. Larger T implies longer transition between the two states.

Practical uses for Time Constant
RC circuits are mainly used to create delays and filters.

Delay
Let's say you were making a switch where the user had to press a button for more than three seconds. Say this device was connected to some other machinery that considered anything higher than 4.5V as "ON." Also suppose you had a 5V voltage source. With just these information, you will be able to construct a RC circuit with appropriate time constant to achieve this effect. If we assume the capacitor is initially discharged (Vc(0)=0V), then it becomes a problem of mere algebraic manipulation.

Filter
Remember that high RC meant smoother transition. If the voltage source was changing (as in signals that comes in from a microphone), then what would happen?

Well, from waves we know that low sounds have low frequency. Low frequency means that it takes more time to change from one value to another. The opposite of that is high frequency, which changes its values rapidly. If our RC term is very high, then the RC circuit won't be able to "catch up" with the rapid transition of the high frequency. This means that the circuit will pass the low frequency signals better than the higher frequency ones.

Such use of RC circuit is called a Low-pass Filter and it has important applications in signal processing.


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Accountancy/Books of Prime Entry

An alternative introduction is under the journal entry. A journal is known as primary book. Books of Prime Entry are a more efficient variation on double-entry accounting. In basic double entry, a double entry is made in the general journal, which is posted in the general ledger accounts. Originally, the Venetian method also suggested a preceding diary step, which makes sense as no thinking is required in double entry, so it may have been faster. In a manual system, books of prime entry act as the speed entry step: instead of trying to remember which accounts to debit and which to credit, and writing the names down for each entry for each transaction in the general journal, the general journal is reserved for infrequent accrual entries; the more frequent cash entries, and the most frequent accrual entries are divided into specialized journals of cash receipts and cash payments; credit sales journal and credit purchases journal (credit means 'on credit' here); and for medium frequency accrual entries, sales returns and purchase returns journal. Apart from not having to write account names each time, the column layout in these specialized journal help systemize the double entry rules; most of them can be totaled at the end of each month to provide monthly entries into control account ledgers, as well as reconciliation with summary monthly totals when a schedule of subsidiary ledger accounts is created. Cash reconciliation also is a monthly task, which is made easier by tracking with numbered transactions such as numbered cheque books where cheque numbers can be entered in the cash payments journal; for tracking cash receipts, it is recommended banking occurs daily so that end of month bank reconciliation is easier. Bank reconciliation involves looking at outstanding items from the last reconciliation then seeing which of these occurred in this period's bank statement; then a search is made for unpresented cheques, and unrecorded receipts, and then payments and receipts that occurred through the bank and not through the business. This makes the cash receipts and cash payments journal essential for reconciliation.

Books of prime entry OR books of original entry are books where transactions are first recorded. These may or may not be part of the double entry system.

The main books of prime entry are:

 * Sales day book
 * Purchase day book
 * Sales returns day book
 * Purchases returns day book
 * Bank Book
 * Cash Receipts Book
 * Cash Payments Book
 * Petty Cash Receipts Book
 * Petty Cash Payments Book
 * Journal

Sales Day Book
This is the book of prime entry for credit sales, where all credit sales of the day are listed and totaled. The total is then used as a single posting entry to the sales ledger and also posted to a sales control account in a single total to tally with the underlying sales ledger. However, individual debits are posted separately in the respective sales ledger (or debtors or receivables ledger accounts). An excerpt might look like this:

SALES LEDGER         TOTAL AMOUNT       (ADDITIONAL COLUMNS OF ANALYSIS DATE     INVOICE    CUSTOMER                        FOLIO                 INVOICED           INTO DEPARTMENTS ETC) 2006 April 2 002345     Alexander & Co                  0016                  $  4,257.50 002346    Benjamin Consulting             0168                  $  5,200.00 002347    ABS Company                     0027                  $  6,800.00 002348    Butler W Company                0278                  $  1,680.40 August 4 002349     Dubula v company                0486                  $  80,000.00

Books for prime entry are synonymous with manual accounting system of special journals and subsidiary ledgers
Another description for books of prime entry, are a manual accounting system of special journals and subsidiary ledgers.

In a basic accounting system, or one that follows the original Venetian method, is to write an English statement of a transaction at time of occurrence in a diary. Then, a this entry , which has at least a detail of a date, an amount, a sentence describing what happened , is then transferred to the general journal as an entry as soon as possible, the entry will be a first step in accounting classification, naming accounts to debit and amounts to debit, and accounts to credit and amounts to credit, where sum of debits equals sum of credits, and the entry is dated. There may be a further description written in brackets under the debits and credits, which may describe more information, such as quantities sold, and to whom, or quantities bought, and from whom, and receipt numbers ).

The general journal entry is then latter entered in the appropriate general ledger accounts, with a date and possibly a back reference to the journal page, and forward references in the journal entry to the ledger account ID numbers to indicate posting from the general journal to the general ledger of accounts.

In a special journal / subsidiary ledger system, or books of prime entry system, several constraints are added:

Instead of making an entry for every transaction in the general journal, there are several journals as named above.

The posting rules are:

 * credit sales are entered in the day sales journals. Likewise, sales returns are entered in the day sales returns journals.
 * purchases on credit are entered in the day purchases journal. Likewise, purchase returns are entered in the purchase returns journal.
 * cash payments i.e. payments by cheque, or by eftpos, or via direct bank transactions such as bank fees and interest paid, are made to the cash payments journal.
 * cash receipts i.e. payments by cheque, direct deposits, visa receipts with reference numbers ,are made in the cash receipts journal.
 * At the end of each day, any entries into credit sales, sales returns, credit purchases, purchase returns, cash receipts, and cash payments, are posted to the relevant subsidiary ledger accounts. There will be subsidiary accounts receivables for each regular customer, and subsidiary accounts payable for each regular supplier.
 * In general, credit sales are posted as debits to the relevant subsidiary account receivables, and cash receipts from that customer are posted as credits that account. Credit purchases are credits in the subsidiary accounts payables, and cash payments to the same supplier are debits to that same subsidiary accounts payable.
 * At the end of each month, each column of the credit sales journal, credit purchases journal, cash receipts journal, and cash payments journal, as well as the sales returns and purchase returns journal, is totaled at the bottom of each column, and then posted to the relevant control ledger account. For instance, a more elaborate credit sales journal for GST collections ,  will have a debit column for accounts receivables, whose end of month total is posted to accounts receivables control, as well as a credit column for sales ledger account, which  is not posted daily but totaled for monthly posting, and a credit column for GST collections, which is also posted as a monthly credit to the GST collections account (liability).  Likewise,credit purchase journal will have a debit column for purchases , a debit column for GST paid, and a credit column for accounts payable. The credits for accounts payable are posted daily to subsidiary accounts payable(creditors) , and the monthly total of accounts payable as a credit to accounts payable control. The GST paid debit column is posted as a monthly total to GST paid (a contra-liability account). The purchases are totaled monthly , and posted to the purchases account , which in the periodic inventory system , is , at period's end, added to the beginning inventory in the income statement to give cost of goods available for sale, and the stocktake determined ending inventory subtracted to find the cost of goods sold, which can be subtracted from gross revenue, to get gross profit.
 * the reason for the daily posting to subsidiary ledger accounts, and the monthly posting to the control ledger accounts, is that a monthly cross check can be made by doing a monthly  schedule of accounts receivables, and a schedule of accounts payable : the sum of the ending balances of one type of subsidiary ledger accounts should equal the balance of the control account after the monthly control posting e.g. subsidiary accounts receivable balance sum equals accounts receivable control balance.
 * the cash receipts journal and the cash payments journal have the expected main case of debit and credit columns, but have additional columns for special cases:
 * cash receipts journal has a debit column for cash in bank, and a credit column for accounts receivable ( because of cash payments from customers honoring credit terms ) . However, there will be other debit and credit columns for other uses of cash receipts :-
 * cash sales will need a credit column for sales, and a credit column for GST collections.
 * Discounts allowed for credit sales early repayment will need a debit column for discount allowed (financial) expense account, and a debit column for gst collections, in order to reverse previously accrued GST collection equal to 10% of the discount allowed.
 * A sundry credit column may be needed for cash receipts such as loans (a credit to a liability account ), or sale of non-current/ non-inventory assets.
 * cash payment journal, for the main function of paying suppliers/creditors, has a debit column for accounts payable, a debit column for GST outlays (refundable asset),  and a credit column for cash in bank. Cash in bank is the source account for cash, and each credit to this asset account reduces its balance.
 * For the extended supplier case of early payment within discounted credit terms, there will be a discount received credit column (other income credit), and a credit for GST outlays ( a reduction in refund  for 10% of the discount allowed).
 * For the secondary cases of payment of bills, insurances, rents, rates and for cash payments for supplies and equipment, a debit column for other or sundry cash payments ( debits for assets purchased, or debits for expense accounts).
 * The general journal is for other entries that are not related to cash or credit, such as period-opening, reversing entries; period-ending, adjusting entries, ; and closing entries transferring income and expense account balances to the summary profit and loss temporary account.
 * Closing of temporary income and expense accounts can be to a temporary profit and loss summary account, made initially in the general journal, and then posted to the income accounts, expense accounts, and profit and loss summary account.
 * petty cash payments may operate with the interest system, where there is a fixed interest amount for which a petty cash fund is reimbursed to, and on reimbursement, an entry is made in the cash payments journal, with debits for the previously petty cash expenses recorded in the petty cash voucher book, and a credit to cash in bank for the cheque that is used to reimburse the petty cash. (If there are also receipts of petty cash, then reimbursement might consist of the sum of unaccounted petty cash payments vouchers, less the sum of unaccounted petty cash receipt vouchers, with possibly separate voucher books for payments and receipts).

Example daily posting of example Sales entries, to subsidiary accounts receivable accounts
referring to the example above of credit sales journal entry, at the end of the day, the journal entries are posted to the subsidiary receivable account ledgers.

SALES LEDGER         TOTAL AMOUNT DATE    INVOICE    CUSTOMER                        FOLIO                 INVOICED 2006 April 2 002345     Alexander & Co                  0016                  $  4,257.50

Post as

Accounts Receivable Ledger AR-0016 CUSTOMER - Alexander & Co DATE    INVOICE    POST-REF DEBIT       CREDIT        BALANCE ..                                                    4,520.00 DR 2 Apr 06  002345    SJ       4,257.50                  8,777.50 DR  --- 002346    Benjamin Consulting             0168                            $  5,200.00

Post as

Accounts Receivable Ledger AR-0168 CUSTOMER - Benjamin Consulting DATE    INVOICE    POST-REF DEBIT       CREDIT        BALANCE ..                                                    2,345.00 DR 2 Apr 06  002346    SJ       5,200.00                  7,545.50 DR  --- 002347    ABS Company                     0027                  $  6,800.00 Post as

Accounts Receivable Ledger AR-0027 CUSTOMER - ABS Company DATE    INVOICE    POST-REF DEBIT       CREDIT        BALANCE ..                                                        0.00 DR 2 Apr 06  002347    SJ       6,800.00                  6,800.00 DR -

002348    Butler W Company                0278                  $  1,680.40 Post as

Accounts Receivable Ledger AR-0028 CUSTOMER - Butler W Company DATE    INVOICE    POST-REF DEBIT       CREDIT        BALANCE, ..                                                      500.00 DR 2 Apr 06  002348    SJ       1,680.40                  2,180.40 DR

 * Any sales returns journal entries, are also recorded as credits daily in the relevant subsidiary account receivables ledgers.
 * Any cash receipts journal entries related to trade debtor settlements are recorded as credits daily in the relevant subsidiary accounts receivables ledgers.
 * At the end of the month, a schedule of subsidiary accounts receivable balances is made, and the sum of the balances noted.
 * At the end of the month, the total of entries in the sales journal, less the total in the sales returns journal, less the total in the accounts receivable credit column of the cash receipts journal, should equal the total of the schedule of subsidiary accounts receivable.
 * credit purchase journal entries are similarly posted in the subsidiary accounts payable daily, as well as cash payment journal entries related to settlement of trade creditor accounts ; similarly monthly total of trade creditor account ending balances should equal totals of credit purchase journal, less credit purchase return journal, less cash payment journal purchase column total.
 * any credit purchase or sale of non-current assets such as office furniture and equipment should not be entered in the credit purchases or credit sales journal, which are reserved for recording sale of trade inventory. Since they are non-frequent, they are entered in the general journal instead, and when paid for, are entered in the sundry debit column of the cash payments journal, or the sundry credit column of the cash receipts journal.

title: Accountancy/Books of Prime Entry


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Harmonica/Types of harmonica

There are three basic types of harmonica; chromatic, diatonic, and the Asian tremolo.

Chromatic harmonica
This type of harmonica can play semi-tones easily. In this book's context, it refers to either the traditional version (using a slide), or the Tombo's slideless versions. Horn harmonica is a different category altogether and, as of now, it will not be covered (since it's only popular in East Asia, and for Orchestra purposes only). In general, materials for chromatic harmonicas is lacking, regardless of location, and as such, it is usually recommended to be played after one has learned the basics of the 10-hole diatonic or Tremolo harmonica. However, if one manages to find good materials, chromatic harmonicas are just as easy to learn.

Diatonic
It has ten holes which offer the player 19 notes (10 holes times a draw and a blow for each hole minus one repeated note) in a three octave range. The standard diatonic harmonica is designed to allow a player to play chords and melody in a single key. Because they are only designed to be played in a single key at a time, diatonic harmonicas are available in all keys.

It is most commonly used in North America and Europe and is a main feature of Blues music.

Variants

 * Valved Diatonic - one of the most common ways to let a diatonic play chromatic notes is to place windvalves over hole 1-6 draw reeds and hole 7-10 blow reeds. Therefore, one can bend on all blows and on draws to get the chromatic notes.
 * Suzuki Overdrive - the Overdrive can be seen as an overblow trainer, in which precisely placed holes on both covers allow the player to stop appropriate reeds, allowing the harmonica player to overblow easily.
 * Hohner XB-40 - the XB-40 is basically a combination between a chromatic harmonica and a valved diatonic. It has 40 reeds; 20 of them activate when playing normally, and 20 others operate when bending occurs. The chromatic scale can be achieved by bending.

East Asian Tremolo
This version of a tremolo harmonica is the standard variant of diatonic harmonicas in East Asia and have more materials. As such, for beginners in East Asia, this type of harmonica is the best.

Other types
Other types of harmonica are the European Tremolo, the Octave harmonica, the very rare Chord harmonica and the Bass harmonica. These are designed for ensemble playing. A beginner would not need to worry about learning these types of harmonicas just yet.

Playing chromatic note on a harmonica — which is better?
In general, there are five main ways to get chromatic notes: use a chromatic harmonica, overbend on a diatonic, valve diatonics, use the Hohner XB-40, and using two 24-hole tremolo stacked together. As to which is better, here are the following benefits:

As a beginner, one does not need to worry about playing chromatic notes just yet. However, if you're seeking more in-depth information on which type of harmonica is the best for performing them, see below.

Due to the table, if one wants an instrument that is capable of thickening the sound and playing many tricks and basically "one-size-fits-all", but incapable of playing chords and performing a "bluesy" wail (a bane of classical music), then a chromatic harmonica, especially a four-octaves version, is the only way to go. This is because the lowest octave actually allows tranposing to lower octaves; additionally it will make the music sound closer to saxophone in lower registers.

However, if one wants to gain that certain feeling in blues and jazz (growling and wailing, for example), it is a good idea to either get the valved harmonica or learn overbends; however, much learning is needed since those methods depend on proper embouchure.

Alternatively, the XB-40 is quite easy to bend, but perhaps too easy to bend; the player needs precise control of the bending in order to only bend a semi-tone down instead of a whole tone down. However, it has the tonal quality and precision of both diatonics and chromatics if the player is trained properly; adding to the fact is that it is capable to do some true tones and double reed bends because of the additional reeds. Thus, it can be very suited for hands-free playing at the very least, and is a definite must for people interested in playing music that have a bluesy or jazzy feeling; lastly, since there are many ways to play the same note, it is easier to play legato on XB-40.

Still, it is not without problems; the C-key model, for example, often warrants complaints about its third octave, in which bending (blow and draw) is difficult to perform. Also, it requires precise tonal control (not unlike playing trumpet), as it is easy to miss the required note during bending and accidentally bend down a whole tone. Lastly, it is limited to three octaves, making it restricting if players truly want to play in numerous keys. (Of course, Toots use a standard 3 octaves harmonica to play jazz, so maybe it's not much of a problem.)

Stacked tremolos, as common as it is in East Asia, just do not cut it, since switching the notes in a key is very slow. If one must have the tremolo sound, get Suzuki's chromatic tremolo, which plays just like a conventional 16-hole chromatic.

Conclusion
All in all, it depends on how a player approaches playing music. Keep in mind that just because a piano is designed to play classical music doesn't mean a jazz player cannot play piano jazz with the proper feeling. Likewise, a chromatic harp can also play jazz harp, as long as the musician is skilled enough. On the other hand, classical music has such a rigid requirement that some view XB-40 as inappropriate; to those classical musicians, they will simply call it "classical music play in jazz style", as they will deem the wailing effect not proper. Others, however, believe that it's fine, so long as the player can hit the notes precisely.

There is a strong incentive among North American players to play chromatic-capable harmonica in positions only, and thus treat it as diatonic. This is inappropriate, as it defeats the purpose of a chromatic-capable harmonica. They should instead train themselves so that they can play with only one or two harmonicas (e.g.: chromatic and XB-40)

title: Harmonica/Types of harmonica


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User:Everlong/Sandbox 1

Interactive Self-test Template
I made a template called "Peekaboo" to provide a degree of interactive, self-test abilility to Wikibooks. A question is placed in the first box, then by clicking on the [ Show ] button to the right, users can reveal the answers in the second box without leaving the page. Everything prints out as currently displayed on-screen. Hopefully this will be useful to a few of you and help some Wikibooks. -- Everlong 02:17, 12 April 2006 (UTC)

Example:

Example code:

New dictionary entry format
bad adjective   = huài 坏 (壞)

Endnotes
Using code:  text to only appear on pages titled "Language/Dialogues" and "Language/Lesson 1," but not on any other page

title: User:Everlong/Sandbox 1


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Blender 3D: Noob to Pro/Every Material Known to Man/Marble

Marble 1
There are many different marbles out there, here is an OK one found by me. Color: Red=0 Green=0 Blue=0 Under the "Map To" tab change the color to white(Red=1 Green=1 Blue=1) Texture Type: Marble Marble Settings: Sharper Harder Sin

Marble / Alabaster
Here is a marble I made to use with a project. This one uses a perlin marble for COL and for NOR which gives both color and bump. Crucial to realistic marble of course is the subsurface scattering (SSS). I also added a bit of reflection which you can find on some marble, but usually not alabaster unless it has been very well polished.

The top row are the texture buttons, but bottom two rows are tabs from the material buttons, pulled apart so you can see all of the values.

Adding more layers
One way to make the marble look more realistic is to add more layers of the marble texture with slightly different characteristics. For instance, create another texture: Marble Settings - Sharper - Soft Noise - Tri (Same settings otherwise, or add your own slight variation) You can try using a different Noise Basis, such as Original Perlin or Voronoi F3.

Marble


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Optics/Huygens’ Principle

Huygens' Principle recognizes that each point of an advancing wave front is in fact the center of a fresh disturbance and the source of a new train of waves; additionally, the advancing wave as a whole may be regarded as the sum of all the secondary waves arising from points in the medium already traversed. This view of wave propagation helps better understand a variety of wave phenomena, such as diffraction.

Fermat's Principle, the principle of least time, can be derived from it.

title: Optics/Huygens’ Principle


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User:Too Old~enwikibooks/recipes

Potato Latkes
As for most traditional foods, there are many recipes for Potato Latkes, which can be eaten at any time, but are especially associated with Chanuka. I give the way I make them, which is probably the simplest.

Ingedients

 * Potatoes, preferably thin skinned.
 * Onions, about 1/3 as much, by bulk, as potatoes.
 * Salt to taste
 * Flour for thickening
 * Oil for frying

Instructions

 * 1) Finely grate the potatoes and onions, you will get a very watery batter. You can use a food processor.
 * 2) Add salt to taste.
 * 3) Thicken with flour, so that the batter holds together when frying.
 * 4) Heat oil in frying pan.
 * 5) Using a large cooking spoon, drop spoonfulls of batter into hot oil, adding more flour to the batter if necessary. The latkes should be about 3 or 4 inches long by 1-1/2 to 2 inches wide.
 * 6) Fry on each side until latke is thoroughly brown.

Serving
Serve hot, with applesauce or sour cream, to taste.

title: User:Too Old~enwikibooks/recipes


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User:Lzj~enwikibooks

fr:Utilisateur:Zj

I speaks Chinese, English, French fluently. Chinese is my mother tongue, and French is the language I used in University.

I live in France, in the city of Paris.

I am here to study Arabic :

Here is my program :

Vocabulary Exercise : There are many people who speak Arabic in Paris. I have the chance to have a few colleagues from Arabic countries. Imouna is now exchanging with me : i teach her Chinese and She teaches me Arabic. I am leaning every day a few Arabic words, i jut not them here and try to master them.

Before meeting Imouna, I studied with Leïla and Noura. They are also very kind. That's one of the reasons why I love so much Arabic language. But most importantly, I love calligraphy and Arabic culture in attracting me, I want to read it in it's own language.

I am aware that all this will take a lot of time.

and I am contributing to : |Chinese Methode in French version.

title: User:Lzj~enwikibooks