domain: phys spi

text:
L Guidoni
email: luca.guidoni@univ-paris-diderot.fr

C S P Tan

Zhao

H Y Li

W W Seit
email: seitww@ime.a-star.edu.sg

Y D Lim

T Henner
email: theo.henner@univ-paris-diderot.fr

J P Likforman

Student Member, IEEE Peng Zhao

Yu Hong

Theo Li

Jean-Pierre Henner
email: jean-pierre.likforman@univ-paris-diderot.fr

Yu Dian Likforman

Lim

Xing Liang

Wen Wei Hu

Guidoni Seit

Chuan Seng Luca

Fellow, IEEE Q Tan

Simplified Assembly of through silicon via Integrated Ion Traps

Keywords: Ceramic Pin Grid Array, Redistribution Layer, Surface Electrode Ion Trap, TSV I. INTRODUCTION

The scalability of surface electrode ion traps has been progressively improved with the on-chip integration of conventional bulk components. Based on the development of through silicon via (TSV) integrated ion trap, in this work, we further simplify the back-end assembly process by patterning a redistribution layer (RDL) onto customized ceramic pin grid array (CPGA) package. This RDL has internal connection with CPGA pin on the back side, rerouting the electrical signal and facilitating the direct bonding of TSV trap and CPGA. The patterned neighboring RDL has an averaged resistance of 4.5 × 10 12 ohm, three orders of magnitude higher than that of TSV trap itself. As compared to the previous assembly (glass interposer wire bonded to a conventional CPGA), a small parasitic capacitance increase of 0.1 pF of the customized CPGA is observed. Radio frequency (RF) electrical tests indicate that the trap on customized CPGA will not induce additional power loss. Meanwhile, due to the high thermal conductivity of ceramic materials, the heat dissipation capability of the customized CPGA is boosted. The functionality of TSV trap on customized CPGA is also demonstrated by trapping and laser-cooling 88 Sr + ions. The measured heating rate (310±60 mK/s) and lifetime (30 minutes) are comparable with the trap assembled in previous approach. This indicates that the CPGA with built-in RDL is fully compatible with ion trapping applications, providing a new approach towards compact, flexible, and robust ion trap assembly.

intricate, in which the bonded trap (i.e., trap on interposer) was first required to be pasted onto a glass spacer (thickness ~750 μm). This was to raise the trap surface and make it higher than the CPGA, since the CPGA has a sunken cavity with a depth of ~1400 μm. The paste used should also be UHV compatible. Subsequently, to connect the RDL on glass interposer and lead frame on CPGA, extra wire bonding was still essential, though it was not directly on the surface electrode. A long electrical signal transmission path from CPGA pin to surface electrode was formed, via CPGA lead frame, wire bonding, RDL on glass interposer, microbump and TSV.

To simplify the assembly process between trap and socket, we explore the elimination of the glass interposer and wire bondings. By patterning an RDL layer onto the CPGA and building internal connections from this RDL to the CPGA pin, the direct bonding of TSV trap and CPGA can be achieved (Fig. 1(b)). The abovementioned electrical signal transmission path is consequently shortened. In this simplified design, the CPGA is not only a packaging apparatus, but also an essential part of the ion trap, determining the delicate rerouting of the electric signal. Indeed, pin grid arrays (PGA) have been widely used for decades to bridge the silicon die and organic motherboard, due to its high I/O density, fan-out circuitry and demountable capability [START_REF] Puttlitz | Area array interconnection handbook[END_REF]. CPGA is one of the most popular candidates amid various PGA products, in view of the low dielectric constant, high thermal conductivity and cost-effective manufacturing of ceramic materials (e.g., alumina, aluminum nitride, etc.). It is also expected that the direct bonding onto CPGA will promote the heat dissipation from RF traps. In brief, locating a TSV integrated ion trap on a CPGA with built-in RDL facilitates a flexible, robust, and ultra-compact trap-package architecture. On top of that, this architecture provides a possible route towards the goal to build a compact UHV ion trapping system to simultaneously maximize the system flexibility and the mechanical stability [START_REF] Wilpers | A compact UHV package for microfabricated ion-trap arrays with direct electronic air-side access[END_REF][START_REF] Aikyo | Vacuum characterization of a compact roomtemperature trapped ion system[END_REF].

The pioneering customization of array packages was reported in 2001, where a pin grid array package and a ball grid array package were designed for the application of microwave and millimeter waves [START_REF] Liang | Development of microwave and millimeter-wave pin grid array and ball grid array packages[END_REF]. Similarly, in year 2004, a dielectric chip antenna was seamlessly integrated in a ceramic ball grid array package [START_REF] Zhang | Integrated circuit ceramic ball grid array package antenna[END_REF]. The obtained bandwidth, gain and radiation efficiency of the integrated antenna suggested that the antenna-package codesign introduced negligible effects to the antenna performance. In the context of ion trap applications, in year 2013, a conventional ceramic leadless chip carrier (LCC) was cut using water-jet-laser to create an orifice for vertically optical access and vacuum conductivity [START_REF] Wilpers | A compact UHV package for microfabricated ion-trap arrays with direct electronic air-side access[END_REF]. Besides, in the work reported by Sandia National Laboratories in 2020, a bowtie shape CPGA was customized to preserve the optical access to the trap isthmus region. However, no direct metal layer patterning was made onto the package and wire bonding was still heavily used [START_REF] Revelle | Phoenix and Peregrine ion traps[END_REF].

In Section I, we first introduce the fundamental principles of surface ion traps. We also highlight the motivation and necessity of the trap assembly process simplification. The customized CPGA design is given in Section II. The design rules and considerations are discussed. Section III provides the electrical characterization results and analysis of the customized CPGA before and after it is bonded to the TSV integrated trap. In Section IV the ion trapping related setup and the performance of TSV trap on this customized CPGA is given in more details. The key findings and conclusions are highlighted in Section V.

II. CUSTOMIZED CPGA DESIGN

The first concern about the customized CPGA carrier is the form factor limitation. Since the trap will be directly located on a CPGA (i.e., in the absence of a smaller glass interposer), the focused Gaussian beams are more likely to collide with the CPGA surface along the propagation path (especially for those of small beam waist), when the carrier dimension is enlarged. Therefore, in the new design, the sunken cavity in conventional CPGA is removed, and the RDL is at the same height as CPGA surface (Fig. 2 (a) and (b)). In this case, the laser beam properties (i.e., wavelength 𝜆 and beam waist 𝑤 ! ) together with ion height (𝑤) will determine the maximum allowable carrier dimension (𝑧):

𝑤 " (𝑧) = 𝑤 ! " '1 + * # # ! +,, where 𝑧 ! (𝑧 ! = $% ! " & )
is the Raleigh length. Note 𝑤 is the sum of height from trapped ion to surface electrode (~76 μm) and height from surface electrode to CPGA surface (~125 μm).

In our ion trapping experiment, five lasers with different wavelengths and beam waists are used, as shown in Table I. Their specific functions are also listed accordingly. The incident directions of laser beams are marked in Fig. 2 (b). Table I shows that the maximum substrate dimension is 65.8 × 65.8 mm (assuming a square substrate shape), which is determined by the laser beam of 1092 nm wavelength. Based on that, the dimension of customized CPGA is set as 14 × 14 mm, leaving a sufficiently large threshold to prevent laser collision issues. Meanwhile, the minimum allowable beam waist with this substrate dimension (14 × 14 mm) is also calculated. The obtained 12 μm beam waist is compatible with general inter-ion distance when individual addressing is needed in a long ion string. Since the inter-pin distance of CPGA should be kept at 0.1 inch, the further CPGA size reduction can only be achieved by sacrificing the pin number of CPGA backside.

The RDL on CPGA has a similar geometry design as the ion trap electrodes (Fig. 2 (c)). 12 metal pads are specifically defined to locate microbumps, building connections with corresponding surface electrodes on the top. Each pad has a small tail, for the internal connection to the routing circuit inside the CPGA, and finally to the pin at CPGA backside. This tail can also be used for wire bonding connections if needed. In total, 25 pins (5 × 5 pin array) are built, enabling a fan-out circuitry for up to 25 RDL pads (ignoring the possible signal multiplex for DC electrodes). In the application of current TSV trap, 9 of 25 CPGA pins are respectively connected to 9 of 12 RDL pads. It should be noted that the remaining 3 RDL pads are kept floating (i.e., not connected with pin, C-DC-1, RF1-2 and RF2-2 in Fig. 2 (c)). This is because the signal feedthrough for the central three electrodes from one end is sufficient. In addition, two CPGA pins are reserved for the surface grounding plane connections. The unoccupied 14 pins are floating.

To locate the microbumps with a diameter of 30 μm, the width of the RDL pads for the central three electrodes is set as 80/50/80 μm. The gap width in between is 50 μm. These dimensions are close to the fabrication limits offered by Kyocera. The alignment markers on customized CPGA (for subsequent trap-to-CPGA bonding process) are also simplified to hollow squares with a diameter of 150 μm (insight of Fig. 2 (b)). These markers can be matched with those on the backside of the TSV trap.

We illustrate the fundamental fabrication steps of a customized CPGA, and more details can be found in [START_REF] Puttlitz | Area array interconnection handbook[END_REF]. First, the ceramic green sheet is formed by casting the ceramic slurry (A440, Kyocera). After cut into square shape, it is mechanically punched to create vias. The vias are subsequently filled by tungsten paste. Next, the routing circuit is patterned using screen printing. Following that, multiple green sheets are stacked, laminated, and shaped. The chip carriers are then co-fired to form the electrical and mechanical integrity. After inserting CPGA pins (Fe-Ni-Co alloy), the last step is the electroplating process of ~2 μm Ni and ~2 μm Au, in which the patterned RDL pads and the grounding plane are defined. The measured root-mean-square roughness (𝑅 ' ) of RDL is ~80 nm. We note that the CPGA fabrication is provided by Kyocera.

III. ELECTRICAL CHARACTERIZATION AND DISCUSSION

A. Electrical Tests before Trap Bonding

The following tests are carried out on a bare customized CPGA, before its bonding to the TSV trap. First, the internal connection between each RDL pad and its respective CPGA pin is examined and no open circuit is found. Subsequently, to evaluate the insulation resistance between two arbitrary RDL pads, an I-V test is performed using a probe station. Two probes (one signal, one ground) are used to connect with the RDL pads. A voltage sweep from 0 to 10 V with a step of 0.2 V is applied and the leakage current in between is measured. As shown in Fig. 3, the resistance varies from 2 to 7 × 10 12 Ω, due to the size and distance difference of both RDL pad and internal routing circuit. We note that the resistances of the floating pads (C-DC-1, RF1-2 and RF2-2) are the highest. The averaged resistance value of RDL pad (4.5 × 10 12 ohm) is three orders of magnitude higher than that of TSV trap (in the range of 10 9 ohm). This indicates that the final resistance between electrodes after bonding will be determined by the TSV trap, and that the customized CPGA itself has excellent insulation property.

The capacitance between RF and central DC RDL pads of the customized CPGA is also measured. One probe is contacted with a RF pad (RF1 or RF2) and signal is applied, while the other probe is contacted with central DC pad (C-DC) and set as ground (see Fig. 2 (c)). The applied DC voltage dual sweep is from -5 to 5 V with a step of 0.1 V, while the AC signal has an amplitude of 30 mV and frequency of 100 KHz. The obtained capacitance of customized CPGA is benchmarked to that of previous glass interposer, before and after its wire bonding on conventional CPGA. It should be highlighted that the capacitance is from the single end of central three RDL pads. As shown in Fig. 4 (a), the customized CPGA has a capacitance of ~1 pF, which is higher than that of the glass interposer itself (~0.4 pF). However, after the interposer is connected to the conventional CPGA by wire bonding, the total capacitance will rise to ~0.9 pF. On the other hand, the obtained high capacitance of the customized CPGA is largely due to the relatively high dielectric constant of A440 alumina (9.8), though the overall signal routing path is shortened. In comparison, the dielectric constant of glass is 3.9. Besides, due to the carrier form factor constraint, the closer routing circuits inside CPGA may also increase the mutual capacitance.

B. Test after Trap Bonding

TSV trap is independently fabricated using a via-first approach with standard CMOS back end of line process. The process details are illustrated in [START_REF] Zhao | TSV-integrated surface electrode ion trap for scalable quantum information processing[END_REF][START_REF] Zhao | RF Performance Benchmarking of TSV Integrated Surface Electrode Ion Trap for Quantum Computing[END_REF]. The cross-sectional SEM image Fig. 5 (a) shows the vertical connection from surface electrode to TSV and finally to backside metal layer and microbump of fabricated TSV trap. The TSV trap is then bonded to the customized CPGA using a flip-chip process. Flux is first dispensed onto the RDL pad surface, which is followed by TSV die placement. The alignment is facilitated by locating the cross markers on the TSV die backside into the hollow squares on RDL pads of CPGA (Fig. 2 (b) and Fig. 5 (c)). Next, the assembled chip is put into a reflow oven (~4 mins, with reflow temperature of up to ~240 ℃). Xray is used to morphologically check the alignment and bonding quality, as shown in Fig. 5 (c). An overall wire bonding-free assembly is achieved for DC and RF signal feedthrough. However, for DC electrodes, bonding wires are still used to connect with thin-film capacitors for RF pick-up filtering. Again, wire bonding end close to the trap is located on the RDL pad tail (Fig. 5 (b)). To avoid degassing issue in the UHV chamber during ion trapping operations, no underfill material is introduced between TSV die and CPGA. As the TSV die is bonded to the RDL pad in the absence of specifically defined under bump metallization (UBM) (see Fig. 5 (c)), it is necessary to evaluate the resultant bonding strength. A sheer test is performed in accordance with MIL-STD-883E method 2019.5 [START_REF]Test Method Standard for Microcircuits[END_REF]. It is found that the fracture is at the interface of backside metal layer and microbump of TSV die, instead of the interface of microbump and CPGA RDL. The averaged maximum shear force for a single bump is ~0.07 N, corresponding to a shear stress of ~100 MPa (given the diameter of microbump is 30 μm). This is sufficiently high and indicates that a robust bonding is achieved [START_REF] Bertheau | Effect of intermetallic compound thickness on shear strength of 25 μm diameter Cu-pillars[END_REF].

During ion trapping tests, the trap is connected to an external resonant transformer to step up the RF voltage. A resistorinductor-capacitor (RLC) circuit is formed: the resistor and inductor come from the transformer, whereas the capacitor (between RF electrodes and ground) is largely determined by the packaged trap itself. Therefore, we first evaluate the capacitance of packaged trap after the bonding process, followed by a resonator test emulating the actual setup during ion trap operation.

The capacitance is between one single RF electrode and the ground of CPGA is first measured. The measurement setup is similar to that described in Section III. As shown in Fig. 4 (b), the obtained a trap capacitance on the customized CPGA of ~10.8 pF, which is ~0.3 pF higher than that of the trap in a conventional CPGA with glass interposer. Next, the two RF electrodes are shortened by a wire bonding. The total RF-GND capacitance is measured again, and a difference of ~1 pF is observed.

Next, the packaged traps are respectively connected to a homemade resonator box composed of inductors and resistors. A -10 dBm RF signal sweeping from 10 to 100 MHz from a RF generator is applied to the resonator. A spectrum analyzer is used to monitor the output signal from the ion trap. The overlapped resonance curves of different traps are shown in Fig. 6, indicating that TSV trap on customized CPGA and conventional CPGA with glass interposer have similar RF performance. Both have a Q factor of ~30. The small deviation of resonance frequency (51.9 and 52.3 MHz) is also consistent with the RF-GND capacitance difference mentioned above (𝑓 = ( "$√*+

).

A separate S-parameter test is also performed. Two 3-pin probes are placed at the two ends of the central RF-DC-RF electrodes, establishing a two-port circuit network. Note this test is also performed on traps that are packaged into different CPGA (i.e., ready for ion trapping test). The RF signal sweep is from 10 to 200 MHz. As shown in Fig. 7, at a lower frequency range (<50 MHz), the traps on different CPGA have a similar insertion loss (S21) and reflection loss (S11). However, when the frequency is higher than 100 MHz, the trap performance on the customized CPGA becomes more favorable. Its smooth curves also suggest that a good stability of the components is achieved across the entire frequency range.

Although both traps have similar RF losses, the high thermal conductivity (14 W/(m•K)) of ceramic material (alumina A440) is able to facilitate high efficiency heat dissipation, maintaining the trap temperature in an acceptable range. Based on finite element modelling results, it is found that at a given power loss of 0.1 W, the trap on customized CPGA has a temperature increase of ~3 K, significantly lower than that of a trap on glass interposer (> 30 K).

IV. ION TRAPPING PERFORMANCE

The trap is tested in a stainless-steel ultra-high vacuum (UHV) chamber with a set of viewports and flanges for photon detection, laser input/output, lateral imaging, pumping and electrical feedthrough. During trap operation the chamber is pumped by an ion pump (IP) and a Titanium sublimator pump (TSP) with a base pressure of 3×10 -10 mbar. The customized-CPGA holder includes a home-made socket, machined in a Peek plate. Adjustable DC voltages for the segmented electrodes are generated by a DAC card (Measurement Computing PCI-DAC6703), filtered by an out-of-the-chamber low-pass passive inductor-capacitor-resistor filter and then steered to CPGA socket using screened cables. RF voltage (frequency of 31.1 MHz) is supplied by a digital RF generator, amplified in a 10 W, 50 Ohms amplifier, adapted to the high impedance of the trap by a toroidal resonant transformer (step-up of 9) and steered to the socket with a coaxial cable. The maximum RF voltage amplitude at the trap is around 100 V. An atomic beam of neutral Sr is generated by sublimation of a Sr dendrite inserted in a tungsten filament. Two-color CW photoionisation [18] allows us to create Sr + ions in the center of the trap: the loading-time for an ion is typically around 5 seconds. The 88 Sr + ions are Doppler-cooled addressing the 422 nm 5 2 S1/2 → 5 2 P1/2 transition with a free-space beam from a GaN extended-cavity laser diode. Two additional free-space lasers beams ("repumpers") address the 1003 nm 4 2 D3/2 → 5 2 P3/2 and the 1033 nm 4 2 D5/2 → 5 2 P3/2 transition to avoid optical pumping. All the beams are coupled in single mode optical fibers and then focused at the ion position with typical spot sizes reported in table 1. The photons scattered by the ion during the cooling process are collected by an objective (numerical aperture of 0.4), spatially filtered in a 150 µm diameter pinhole, spectrally filtered by an interference filter and detected by a photomultiplier in photoncounting mode. The overall photon collection efficiency [19] is 2.20±0.05x10 -3 . We observe stable trapping of ions (lifetime of the order of 30 minutes) in a range of RF voltage amplitudes between 35 and 65 V. Amplitudes exceeding 60 V result in a base pressure increase in the vacuum cell (pressure increase of 2.5x10-11 mbar for a RF amplitude of 75 V) probably induced by residual device heating induced by RF dissipation.

We measured the motional frequencies corresponding to trapping voltage amplitudes using a "tickle" technique coupled to a sequential fluorescence acquisition. The agreement between the measured and the calculated frequencies (according to an analytical gap-less model [20]) is within 12%, as usual in surface traps of this size. The heating rate of the trap has been characterised using the Doppler re-cooling method [21]. For this purpose we operate the trap with an axial frequency of 300 kHz and radial frequencies around 2.3 MHz. We expect the re-cooling dynamics to be dominated by the temperature of the lower frequency axial mode. Incoherent repumping allows us to operate in the two-levelatom approximation needed for the analysis of the acquired data [21]. From the analysis of several experimental sets, the heating rate of the trap is evaluated at 310±60 mK/s that corresponds to 21 axial quanta per millisecond. This result corresponds to a normalized electric-field spectral noise density, wwz*SE(wwz), of 1.9 10 -4 (V/m) 2 , similar to what we obtained in a TSV trap with glass interposer [START_REF] Zhao | TSV-integrated surface electrode ion trap for scalable quantum information processing[END_REF].

V. DISCUSSION AND CONCLUSION

A. Further Improvement

First, to further boost the thermal dissipation capability of the package, a ceramic material with even higher thermal conductivity (e.g., aluminum nitride, 150 W/(m•K)) can be selected. Meanwhile, with current CPGA manufacturing technologies, the feature size and pitch of CPGA RDL are hard to be reduced to less than 50 μm. Mismatch between microbumps and CPGA RDL will be induced when TSVs/microbumps are of high density. This poses a challenge to the metal layer routing on the TSV trap backside, which bridges the via and microbump. Delicate routing shall be implemented to guarantee the precise connection between bump and RDL, without sacrificing the TSV density. In addition, the CPGA size/the backside pin number limitation considering the laser beam propagation can be fundamentally solved once silicon photonics devicesare integrated on-chip allowing laser light to be locally routed and emitted [START_REF] Mehta | Integrated optical addressing of an ion qubit[END_REF][START_REF] Niffenegger | Integrated multi-wavelength control of an ion qubit[END_REF]. Therefore, a complex ion trap design with a high number of electrodes can be achieved. Finally, a flexible CPGA socket is in need, to facilitate frequent CPGA size and CPGA pin arrangement adjustments.

B. Conclusion

Fig. 1 .

 1 Fig. 1. The development of TSV integrated ion trap assembly method. (a) Conventional method: TSV trap + glass interposer + glass spacer + CPGA. (b) New method: TSV trap + customized CPGA with patterned RDL. Capacitors for RF filtering are not included in the schematic.

Fig. 2 .

 2 Fig. 2. (a) Conventional CPGA that has lead frame and a sunken cavity. (b) Customized CPGA where patterned RDL pads are on the substrate surface. (c) Schematic of patterned RDL pads, where functions are labelled.

Fig. 3 .

 3 Fig. 3. Cumulative probability of measured resistance between arbitrary RDL pads of customized CPGA.

Fig. 4 .

 4 Fig. 4. (a) The RF-central DC capacitance comparison among customized CPGA, glass interposer assembled on conventional CPGA, and glass interposer only. The images of these three candidates are shown as insets. (b) RF-GND capacitance of trap on customized CPGA and trap on conventional CPGA with glass interposer.

Fig. 5 .

 5 Fig. 5. (a) Cross-sectional SEM of fabricated TSV trap, before its bonding onto CPGA. (b) TSV trap bonded onto customized CPGA. (c) X-ray image of TSV trap bonded on customized CPGA (view from backside). A zoom-in X-ray image showing the connections between TSVs and RDLs is on the right.

Fig. 6 .

 6 Fig. 6. The resonance curve of TSV trap on customized CPGA, with a reference curve of TSV trap with glass interposer wire bonded with conventional CPGA. The curve of TSV trap on customized CPGA has higher peak and smaller 3 dB bandwidth.

Fig. 7 .

 7 Fig. 7. The insertion loss (S21) and reflection loss (S11) comparison of TSV traps respectively packaged on customized CPGA and on conventional CPGA with glass interposer.

ACKNOWLEDGMENT

We thank Kyocera Japan for the support on the CPGA fabrication. We acknowledge the assistance from Mr. Ng Vynx and Ms. Wang Runa in Kyocera Singapore on CPGA design discussion.

This work is supported by National Research Foundation, Singapore, under its ANR-NRF Joint Grant Call (NRF2020-NRF-ANR073 HIT).


Shuang-Ren Zhao

Discussion on the Correction of Classical Electromagnetic Wave Theory through Transmission Lines

Keywords: Maxwell's equation, Poynting, retarded wave, advanced wave, retarded potential, advanced potential, antenna, transformer, electromagnetic wave, electromagnetic field, energy flow, quasi-static state, transmission line

In recent years, the author has made appropriate revisions to Maxwell's classical electromagnetic field theory. This correction includes considering that the receiving antenna radiates advanced waves; The law that radiation does not overflow the universe; The law of localized energy conservation and the theorem of mutual energy flow are valid. The Poynting vector of far field of a antenna transmits reactive power. The electric and magnetic fields that radiate the far field maintain a 90 degree phase difference instead of being in phase. In the author's revised electric field theory, the Maxwell equation still holds, but its solution has been reinterpreted. This article applies the author's theory to transmission line problems.

Transmission line problems are often solved using circuit methods. The voltage and current propagate along the transmission line in phase. Our textbooks often believe that there is only one type of electric field on a transmission line, which is in phase with the voltage and therefore with the current. Some scholars have also calculated the electric and magnetic fields based on Maxwell's electromagnetic theory, and obtained results consistent with the circuit method. There is only one type of electric field on the transmission line. According to the author's electromagnetic theory, there are two types of electric fields on transmission lines: electrostatic fields and induced electric fields. The electrostatic field is in phase with the voltage on the transmission line. The phase of the induced electric field lags 90 degrees behind the current. This is consistent with a simple circuit. Therefore, for transmission lines, the author's theory contradicts the conclusions of our existing textbooks. Therefore, transmission 1 1 INTRODUCTION 2 lines are a good example, which can be used to distinguish between two theories. The author also proposed a method for experimental verification.

Introduction 1.Maxwell's Electromagnetic Theory

The author found that Maxwell's electric field theory is a theory of retarded potential, and its key is that the vector potential in the frequency domain is,

A(x, ω) ≡ µ 0 4π ˆV J(x ) r dV (1)
A (r) (x, ω) ≡ µ 0 4π

ˆV J(x ) exp(-jkr)

r dV (2)
The superscript (r) indicates retardation. A (r) (x, ω) is a vector potential with retardation, while A(x, ω) is a vector potential without considering retardation. Electric and magnetic fields are defined as,

E = -∇φ (r) - ∂ ∂t A (r) (3)
B = ∇ × A (r) (4)
The above φ (r) is determined by the Lorenz gauge condition.

∇ • A (r) = -jωµ 0 0 φ (r) (5)
Here we assume that all quantities have a time factor exp(jωt). We omitted this factor. The key to this theory is to ensure that the following generalizations have been made when transitioning from quasi-static electromagnetic field theory or magnetic quasi-static electromagnetic field theory to radiation electromagnetic field theory considering retardation,

A(x, ω) → A (r) (x, ω) (6)
This promotion ensures that lim kr→0 A (r) (x, ω) = A(x, ω)

kr → 0 [START_REF] Liang | Fundamentals of microwave theory[END_REF] i.e., 2π λ r → 0 (9)

r λ (10)
r = ||x -x || (11)
Is the distance from the field point to the origin. If this distance is much smaller than the wavelength λ, the retarded potential A (r) degenerates into a non retarded vector potential A. The Maxwell equation is completely equivalent to the retarded potential theory above. The theory of retarded potential was first completed by Lorentz in 1867 [?], and the theory of Lorenz retarded potential was only 5 years later than the Maxwell equation (completed in 1861-2). Maxwell did not solve the Maxwell equation, but Lorenz's theory is not only equivalent to Maxwell's equation, but also provides a general form of solution.

The author found that this condition is too simplistic for electromagnetic theory. There are other conditions that should be considered when transitioning from retarded potential theory to non retarded electromagnetic theory.

The author's electromagnetic theory

The author's electromagnetic theory has made revisions to Maxwell's classical electromagnetic field theory. This correction is based on the following considerations:, 1.Not only the retarded potential (wave), but also the advanced potential (wave).

2. Radiation does not overflow the universe, where Γ is a sphere with an infinite radius, assuming that the source of the electromagnetic field is the current J near the center of the sphere. ˆ∞ t=-∞ dt

" Γ (E × H) • ndΓ = 0 ( 12
)
No matter should overflow the universe, including electromagnetic waves. It is wrong for classical electromagnetic theory to allow electromagnetic waves to overflow the universe.

3. Ensure that the law of conservation of energy holds

N i=1 N j=i,j =i ˆ∞ t=-∞ dt ˆV (E i • J j )dV = 0 (13)
This law tells us that if there are only two currents J 1 , J 2 . Current J 1 give a portion of the energy to the current J 2 . So current J 1 is a primary coil, current J 2 is a secondary coil. The energy loss of J 1 is exactly equal to the energy obtained by the current J 2 . The algebraic sum of the energy lost by the two coils together is 0. This indicates the conservation of energy. We cannot derive this law of energy conservation from our classical electromagnetic theory today. We can only take a step back and prove this self proclaimed energy law as the energy theorem or reciprocity theorem.

4.The current generates half retarded and half advanvced waves. Therefore, when the retarded wave and advanced wave degenerate into a non retarded and non advanced electromagnetic field, this electromagnetic field must be superimposed, that is, the superposition of the electric field and the magnetic field is also superimposed. No offsetting can occur. Our classical electromagnetic theory today does not meet this requirement 5.For discrete systems, such as systems composed of primary and secondary transformers, the energy of the system composed of transmitting and receiving antennas is transmitted by mutual energy flow

The first of the five above is the characteristic of the author's electromagnetic field theory. This theory is an electromagnetic field theory that includes advanced waves. The theory of advanced waves, also known as the absorber theory, was first proposed by Wheeler Feynman in 1945[?, ?].This Classical electromagnetic field theory cannot be directly applied to form a theory that includes advanced waves, which can be obtained by advanced potentials

A (a) (x, ω) ≡ µ 0 4π ˆV J(x ) exp(+jkr) r dV (14)
E = -∇φ - ∂ ∂t A (a) (15)
B = ∇ × A (a) (16)
The electromagnetic field obtained in this way cannot be directly used, as the original Maxwell electric field theory is difficult to reconcile with after the introduction of advanced waves. The conservation of energy theorem is the first challenge encountered. Because once there is an advanced wave, the retarded wave radiates energy, and the advanced wave also radiates energy (this energy is negative). However, the author found that there is also an interaction between the retarded wave and the advanced wave at this time. This interaction also transfers energy. The energy transferred by this interaction is exactly the energy from the source to the sink. Here, the source refers to the source of electromagnetic waves, including transmitting antennas, primary coils of transformers, light sources, etc. A sink is a device that receives electromagnetic waves, including a receiving antenna, a secondary coil of a transformer, a screen for receiving light, and so on.

Since the energy flow of interaction, also known as the mutual energy flow, can transfer energy from the source to the sink. So the energy transmitted by the advanced and retarded waves themselves is redundant. In these specific problems, the author found that if Maxwell's electromagnetic theory is used to calculate the electric and magnetic fields, and then the mutual energy flow is calculated, the mutual energy flow is often an imaginary number, so it transfers reactive power, that is, it does not transfer energy. Therefore, time reversal waves cannot truly solve the problem and complicate it.

The author proposes the concept that both retaded wave and advanced wave are reactive power waves, and reactive power waves are waves that maintain a 90 degree phase difference between the electric and magnetic fields. On average, the energy transmitted by this wave is 0. Therefore, it is no longer transmitting energy. If both the retarded wave and the advanced wave are reactive power waves, they themselves will not transmit energy.

Time reversal waves are no longer needed. The reactive power wave can completely replace the original retarded wave and advanced wave, and also includes the time reversal wave of retarded wave and advanced wave. The author found that if the retarded wave and advanced wave are reactive power, the mutual energy flow formed by them happens to be active power.

The 1-5 items mentioned earlier have been perfectly implemented.

According to Maxwell's electromagnetic field theory, the far-field electric and magnetic fields of the antenna remain in phase, therefore they are not reactive power waves. However, the author found that Maxwell's electromagnetic field theory is actually a retarded potential theory, which can ensure that the electric field is a retarded field, but cannot guarantee that the magnetic field is also a retarded field. Therefore, the author made corrections to the magnetic field obtained from Maxwell's classical electromagnetic theory [?, ?, ?][?, ?, ?, ?, ?, ?][?, ?, ?, ?, ?].

Here we summarize the conclusions from these literature as follows.

Assuming we calculate the magnetic field H M axwell according to retarded potential of the electromagnetic field theory, the magnetic field corrected by the author is

E (r) i = E (r) iM axwell (17)
H (r) = (-j)H (r) M axwell (18)
H is the true magnetic field of the system. The above correction is for the electromagnetic wave part, that is, for the far-field correction. No correction is needed for near field. The above correction factor (-j) holds for the retarded wave. If it is a advanced wave, the factor is (j).

E (a) i = E (a) iM axwell (19)
H (a) = (j)H (a) M axwell (20)
Above, we are correcting the magnetic field. Sometimes, the correct magnetic field is already known, and it is necessary to correct the induced electric field. At this time, we also add a phase factor to the above correction formula, such as adding (j) to the retarded wave

E (r) i = jE (r) iM axwell (21)
H (r) = j(-j)H (r) M axwell = H (r) M axwell (22)
For advanced waves, the correction factor is (-j),

E (a) i = -jE (a) iM axwell (23)
H (a) = -j(j)H (a)
M axwell = H (a) M axwell [START_REF] Shuang Ren Zhao | Experiment to prove the existence of the advanced wave and experiment to prove the wrong definition of magnetic field in maxwell's theory[END_REF] Note that the superscript (r) means retarded wave, the superscript (a) means advanced wave, and the subscript i means induced electric field. The subscript M axwell refers to the meaning calculated according to Maxwell's equation.

The author's theory has achieved the following results:

1. for the independent systems, such as simple circuit, transmit energy through the self energy flow corresponding to the Poynting vector. For the discrete systems, such as the primary and secondary coils of transformers, transmitting and receiving antennas, and the energy flow from the light source to the light sink are mutual energy flows.

2. The law of conservation of energy and the theorem of mutual energy flow hold:

-

ˆV1 E 1 • J * 2 dV = (ξ 1 , ξ 2 ) = ˆV1 E * 2 • J 1 dV (25)
(ξ 1 , ξ 2 ) = " Γ (E 1 × H * 2 + E * 2 × H 1 ) • ndΓ ( 26
)
The author has verified the above laws for systems with dual planar transformers, a system with planar transmitting antenna and a planar receiving antenna, and a system with a dipole transmitting antenna and a dipole receiving antenna. Note that the above theorem is actually the law of conservation of energy, and it is a localized law of conservation of energy. (Please note that the above law of conservation of energy is not valid in Maxwell's electromagnetic theory. As mentioned earlier, the mutual energy flow calculated based on Maxwell's equations for electric and magnetic fields is often a pure imaginary number and therefore does not transfer energy. The above mutual energy flow theorem is valid in the author's revised electromagnetic theory.)

3. The mutual energy flow is generated at the source and annihilated at the sink, thus possessing the properties of photons. Therefore, the author believes that photons are mutual energy flows. The mutual energy flow is composed of retarded waves and advanced waves, which have the properties of waves. The mutual energy flow has the properties of particles, especially at the light source and light sink, where it gathers into a point and carries energy and momentum. Therefore, like particles. The mutual energy flow itself is both a retarded wave and an advanced wave, therefore it is also a wave. This explains the problem of wave particle duality.

4. The author's electromagnetic field theory further supports Cramer's quantum mechanical trading interpretation [?, ?]. Therefore, the interpretation of quantum mechanics through transactional interpretation is also applied to the author's electromagnetic theory. 2 Plane waves 2.1 Solving plane waves according to Maxwell's equation

The focus of this article is on transmission lines

Solving plane waves according to Maxwell's equation

∇ • E = 0 ( 27
)
∇ • B = 0 (28) ∇ × E = -jωµ 0 H (29) ∇ × H = jω 0 E (30)
The above formula is the Maxwell' equations without sources.

Wave equation

∇ × ∇ × E = -jωµ 0 ∇ × H = -jωµ 0 (jω 0 E) (31)
∇ × ∇ × E = ω 2 µ 0 0 E (32) ∇(∇ • E) -∇ 2 E = ω 2 µ 0 0 E (33) Considering ∇ • E = 0, ∇ 2 E + ω 2 µ 0 0 E = 0 (34) ∇ 2 E + k 2 E = 0 (35) wherein k 2 = ω 2 µ 0 0 (36)

Wave equation solving

E = E 0 exp(-jk • x) (37)
Consider ∇ • E = 0, there is,

-jk • E = 0 (38) or k • E = 0 (39)
therefore,

E ⊥ k (40)
To simplify the problem, we can assume that,

k = kx (41)
This indicates that the wave propagates along the x-axis direction, so there is,

E = E 0 exp(-jkx) (42)
wherein

E 0 = E 0 (-ẑ) (43) E = E 0 exp(-jkx)(-ẑ) (44)
We calculate the magnetic field using Faraday's law (29)

H = 1 -jωµ 0 ∇ × E = 1 -jωµ 0 ∇ × (E 0 exp(-jkx)(-ẑ)) = 1 -jωµ 0 (-jkx) × (E 0 exp(-jkx)(-ẑ)) = k ωµ 0 x × (E 0 exp(-jkx)(-ẑ)) = k ωµ 0 E 0 exp(-jkx)ŷ = 1 η 0 E 0 exp(-jkx)ŷ ( 45
)
η 0 = ωµ 0 k = µ 0 ω √ µ 0 0 = µ 0 0 (46)
In this way, we obtain a plane invariant solution, which is the solution of the retarded wave. Therefore, we add a superscript (r) to them,

E (r) = E 0 exp(-jkx)(-ẑ) (47)
H (r) = 1 η 0 E 0 exp(-jkx)ŷ (48)
Based on Maxwell's theory, the electric and magnetic fields of retarded waves were obtained.

Advanced wave

The above is the retarded wave. Considering the electric field of the advanced wave,

E = E 0 exp(+jkx)(-ẑ) (49)
H = 1 -jωµ 0 ∇ × E = 1 -jωµ 0 ∇ × (E 0 exp(+jkx)(-ẑ)) = 1 -jωµ 0 jkx × (E 0 exp(+jkx)(-ẑ)) = - 1 ωµ 0 kx × (E 0 exp(+jkx)(-ẑ)) = - 1 η 0 E 0 exp(+jkx)ŷ (50)
Therefore, we obtain the advanced wave solution,

E (a) = E 0 exp(+jkx)(-ẑ) (51)
H (a) = - 1 η 0 E 0 exp(+jkx)ŷ (52)
We obtain the electric and magnetic fields of the advanced wave based on Maxwell's electromagnetic theory

Sources of plane waves

Assuming there is a constant AC current, this current is on an infinite flat plate. This current is Considering the retarded vector potential,

J (t) = J 0 exp(jωt)ẑ = J(t)ẑ (53)
A = µ 0 4π ˆσ J r exp(-jkr)dσẑ (54)
Consider that the flat plate current is a circle with a radius of R.

R = y 2 + z 2 (55)
Consider,

r = √ x 2 + R 2 (56) A(t, x) = µ 0 4π J ˆ2π o dθ ˆR o 1 √ x 2 + R 2 exp(-jk √ x 2 + R 2 )RdR (57) = µ 0 4π 2πJ ˆR o 1 √ x 2 + R 2 exp(-jk √ x 2 + R 2 ) 1 2 dR 2 (58) Consider, d √ x 2 + R 2 = (x 2 + R 2 ) -1 2 1 2 dR 2 (59) A(t, x) = µ 0 4π 2πJ ˆR o exp(-jk √ x 2 + R 2 )d √ x 2 + R 2 = µ 0 2 J -jk ˆR o d exp(-jk √ x 2 + R 2 ) = µ 0 2 J -jk exp(-jk √ x 2 + R 2 )| R 0 = µ 0 2 J -jk (exp(-jk √ x 2 + R 2 -exp(-jk √ x 2 )) = 1 jk µ 0 J 2 (exp(-jkx) -exp(-jk √ x 2 + R 2 )) (60)
From this, the electric field can be calculated

E = - ∂ ∂t A = -jωA = = -jω 1 jk µ 0 J 2 (exp(-jkx) -exp(-jk √ x 2 + R 2 ))ẑ = - ωµ 0 k J 2 (exp(-jkx) -exp(-jk √ x 2 + R 2 ))ẑ ( 61
)
ωµ 0 k = ωµ 0 ω √ µ 0 0 = µ 0 0 = η 0 (62) therefore E = -η 0 J 2 (exp(-jkx) -exp(-jk √ x 2 + R 2 ))ẑ (63)
Consider,

E ∞ = lim R→∞ E E ∞ = lim R→∞ (-η 0 J 2 (exp(-jkx) -exp(-jk √ x 2 + R 2 )))ẑ (64)
The latter term in the above equation diverges, but the average value of the latter term is 0. Therefore, it can also be considered as 0 this way

E ∞ = η 0 J 2 exp(-jkx)(-ẑ) (65)
From this, the magnetic field can be calculated,

H = 1 µ 0 ∇ × A(t, x) (66)
H = 1 µ 0    x ŷ ẑ ∂ ∂x ∂ ∂y ∂ ∂z A x A y A z    = 1 µ 0    x ŷ ẑ ∂ ∂x 0 0 0 0 A z    = - 1 µ 0 ∂ ∂x A z ŷ = - 1 µ 0 ∂ ∂x ( 1 jk µ 0 2 J(exp(-jkx) -exp(-jk √ x 2 + R 2 )))ŷ = - 1 µ 0 ( 1 jk µ 0 2 J(-jk exp(-jkx) -(-jk) R √ x 2 + R 2 exp(-jk √ x 2 + R 2 )))ŷ = ( J 2 (exp(-jkx) - R √ x 2 + R 2 exp(-jk √ x 2 + R 2 )))ŷ ( 67
)
H ∞ = lim R→∞ H = J 2 (exp(-jkx))ŷ (68)
We can also calculate the electric field from the magnetic field,

∂ ∂t 0 E ∞ = ∇ × H ∞ =    x ŷ ẑ ∂ ∂x ∂ ∂y ∂ ∂z H ∞ x H ∞ y H ∞ z    =    x ŷ ẑ ∂ ∂x 0 0 0 H ∞ y 0    = ẑ ∂ ∂x H ∞ y = ẑ ∂ ∂x J 2 exp(-jkx) = -jkẑ J 2 exp(-jkx) (69)
Or

jω 0 E ∞ = -jkẑ J 2 exp(-jkx) (70) E ∞ = - k ω 0 J 2 exp(-jkx)ẑ (71) k ω 0 = ω √ µ 0 0 ω 0 = µ 0 0 = η 0 (72) E ∞ = -η 0 J 2 exp(-jkx)ẑ (73)
Considering that we are calculating the retarded potential,

E r∞ = η 0 J 2 exp(-jkx)(-ẑ) (74)
H r∞ = lim R→∞ H = J 2 (exp(-jkx))ŷ (75)
The superscript r represents retardation, while the superscript ∞ represents the electromagnetic field of an infinite plate current. The above calculation results (74,75) are consistent with the previous formulas (47,48). Just consider,

E 0 = η 0 J 2 (76)
It is worth mentioning that in the above calculation, if the magnetic field is calculated first, and then the electric field is calculated by the magnetic field, the divergence problem encountered in calculating the electric field can be avoided. However, the results are consistent with our direct neglect of divergent items with an average of 0.

Advanced potential

Considering the advanced potential,

A a = µ 0 4π ˆσ J r exp(jkr)dσẑ (77)
Similar to the previous method, the advanced potential can be obtained,

A a (t, x) = - 1 jk µ 0 J 2 (exp(+jkx) -exp(+jk √ x 2 + R 2 )) (78)
E a = -jωA a (t, x) (79)
E a∞ = lim R→0 E a∞ (80)
The second term diverges, but it averages 0. Ignore the second term with an average of zero,

E a∞ = +η 0 J 2 exp(+jkx)ẑ (81)
(29)Strictly speaking, divergent terms cannot be ignored. But it is also possible to handle it this way in engineering. Consider Faraday's law (29)

H = 1 -jωµ 0 ∇ × E = 1 -jωµ 0    x ŷ ẑ ∂ ∂x ∂ ∂y ∂ ∂z E x E y E z    1 -jωµ 0    x ŷ ẑ ∂ ∂x 0 0 0 0 E z    = 1 -jωµ 0 ŷ(- ∂ ∂x E z ) = 1 jωµ 0 ŷ ∂ ∂x E z
So,

H a∞ = lim R→∞ H = 1 jωµ 0 ŷ ∂ ∂x E a∞ z = 1 jωµ 0 ŷ(jk)E a∞ z = k ωµ 0 ŷE a∞ z = 1 η 0 ŷE a∞ z = J 2 (exp(+jkx))ŷ (82)

Superposition of retarded and advanced waves

According to the viewpoint of Wheeler Feynman absorber theory [?, ?],

current radiates half the retarded wave and half the advanced wave. The author agrees with this viewpoint, that is,

E ∞ = 1 2 (E r∞ + E a∞ ) ( 83
)
H ∞ = 1 2 (H r∞ + H a∞ ) (84)
When the wavelength is very large, i.e

kr = 2π λ r → 0 (85)
there is,

lim kr→0 E ∞ = lim kr→0 1 2 (-η 0 J 2 exp(-jkx)ẑ + η 0 J 2 exp(+jkx)ẑ) = 1 2 (-η 0 J 2 + η 0 J 2 )ẑ = 0 (86) lim kr→0 H ∞ = lim kr→0 1 2 ( J 2 (exp(-jkx))ŷ + J 2 (exp(+jkx))ŷ) = 1 2 ( J 2 + J 2 )ŷ = J 2 ŷ ( 87
)
From above, we can see that the electric field is completely cancelled and the magnetic field is superimposed. This seems to be a bit problematic. This makes the half retarded and half advanced scheme proposed by Wheeler

Feynman cannot be relaized when

r λ (88)
The condition we proposed 1.2 cannot be met.

Correction of magnetic field

The previous calculations E r∞ , H r∞ are the retarded electric and magnetic fields of infinite plate currents calculated according to Maxwell's electromagnetic field theory. This electric field and magnetic field are in phase.

Such calculations can be found in FeynmanLectures on Physics [?]. Feynman specifically brought up this issue for discussion, and it seems that it is very entangled with this issue. According to Maxwell's theory,

E r∞ = -η 0 J 2 exp(-jkx)ẑ ( 89
)
H r∞ = lim R→∞ H = J 2 (exp(-jkx))ŷ (90)
Due to the fact that the plane wave generated by the flat plate current is itself a far-field, radiating electromagnetic field. According to the author's theory, a correction should be made to the magnetic field,

E cr∞ = E r∞ = -η 0 J 2 exp(-jkx)ẑ (91)
H cr∞ = (-j)H r∞ = (-j) J 2 (exp(-jkx))ŷ (92)
The superscript c represents corrected, the superscript r represents retarded, the superscript a represents advanced, and the superscript ∞ represents the electric and magnetic fields generated by infinite plate currents.

We calculate the electric and magnetic fields in the x-direction of the flat plate current.

E ca∞ = E a∞ = -η 0 J 2 exp(+jkx)ẑ ( 93
)
H a∞ = - J 2 (exp(+jkx))ŷ ( 94
)
For the advanced wave correction factor, it is not (-j) but (j), therefore there is,

H ca∞ = (j)H ca∞ = -j J 2 (exp(+jkx))ŷ (95)
This way,

E c∞ = 1 2 (E cr∞ + E ca∞ ) ( 96
)
H c∞ = 1 2 (H cr∞ + H ca∞ ) ( 97
)
E c∞ = 1 2 (-η 0 J 2 exp(-jkx)ẑ -η 0 J 2 exp(+jkx)ẑ) (98)
H c∞ = 1 2 (-j J 2 exp(-jkx)ŷ -j J 2 exp(+jkx)ŷ) (99) lim kr→0 E c∞ = -η 0 J 2 ẑ ( 100
)
lim kr→0 H c∞ = -j J 2 (101)
Consider that we can modify a phase factor for both the electric and magnetic fields above, such as

exp(jψ) = j (102)
After this correction, the phase difference between the electric and magnetic fields of the plane wave will not change. What we really care about is the phase difference between the magnetic and electromagnetic fields. So we can also write it as,

lim kr→0 E c∞ = jη 0 J 2 (-ẑ) (103) lim kr→0 H c∞ = j(-j) J 2 = J 2 (104)
We can see that according to the author's revised plane wave, the retarded electric field and the advanced electric field remain superimposed at kr → 0, and the retarded magnetic field and the advanced magnetic field also overlap, rather than cancel. In this way, we have implemented the theory of half retarded and half advanced proposed by Wheeler Feynman

[?, ?].

In addition, according to the author's electromagnetic theory, the electric and magnetic fields of plane waves maintain a 90 degree phase difference. This is consistent with the quasi-static situation. Under quasi-static conditions, electromagnetism also lags behind the magnetic field by a phase of j = exp(j π 2 ). In this way, half retarded waves and half advanced waves can degenerate into quasi-static electromagnetic fields when the condition r λ is met.

Simple circuit

The following figure 2 is a simple circuit, for which we can consider the problem under magnetic quasi-static conditions.

Consider a simple circuit composed of a power supply and a resistor, for this circuit. Obviously, we are considering that the voltage between the two wires is U So the electrostatic field is

E s = U L z (-ẑ) = -ẑ U L z (105)
That is to say, the electric field should be calculated based on voltage.

Alternatively, the electric field is calculated by the electric potential φ. We are applying

E s = -∇φ (106)
U = φ up -φ down ( 107
)
The above is the electric field. The magnetic field is calculated by the anpere circuital theorem. For upper current conductors,

H up J 2 ŷ ( 108
)
H down J 2 ŷ ( 109
)
The symbol " " in the above equation is because the width of the wire is finite, so the approximate value obtained above is.

H = H up + H down (110) H 2 J 2 ŷ = J ŷ = I L y ŷ (111)
So the wire power

W = U I * W = U U * R (112)
On the other hand, Poynting vector,

S = E × H * = -ẑ U L z × I * L y ŷ = x U I * L z L y (113)
We can calculate the power based on the Poynting vector, and the output power is,

W = ¨σ S • xdσ = ¨σ U I * L z L y dσ = U I * L z L y L z L y = U I * (114)
The two methods for calculating power are consistent. From the above simple circuit method, we can see that the magnetic field of the circuit should be calculated according to the current through the ampere circuital law. The electrostatic field is calculated based on the potential -∇φ. Our question is, does this circuit have an induced electric field? Of course, if we assume that the current is alternating current. So the magnetic field will induce an induced electric field. We know,

˛C E i • dl = -jωµ 0 " Γ H • ndΓ ∼ -jI (115)
It can be seen that the induced electric field near the power source is

E i ∼ -jI ẑ = jI(-ẑ)
The Poynting vector can be calculated from the induced electric field, which is

E i × H * ∼ j (116)
The Poynting vector of an induced electric field is a pure imaginary number and does not transfer energy. We will calculate the transmission line later. Transmission lines are very similar to simple circuit. The transmission line should consider the transmission of waves. But there are some properties that should be preserved. For example, a magnetic field can be calculated from current. The Poynting vector composed of induced electric and magnetic fields is a pure imaginary number. Energy should be transmitted through the Poynting vector composed of a static electric field and a magnetic field. That is to say, the transmission line should meet the conditions in the wave field (88) and can degenerate to the simple circuit mentioned above.

Review of Using Circuit Methods to Study Transmission Line Problems

Few people use Maxwell's equations to solve transmission line problems.

In most cases, the circuit method is used. This chapter reviews this method.

We are considering a small section of the transmission line, ∆x. Assuming the impedance per unit length of this circuit is

Z = R + jωL (117)
The conductivity per unit length is,

Y = g + jωC (118)
Further consider the input voltage and current as,

U in = U (x) (119)
I in = I(x) (120)
The output voltage is

U out = U (x + ∆x) (121)
I out = I(x + ∆x) (122)
Using the Kirchhoff voltage theorem,

Z Y U(x) U(x+Delta x) Delta x I(x) I(x+Delta x)
图 3: The voltage drop on the transmission line is U (x), and the current is

I(x) U (x) = IZ∆x + U (x + ∆x) (123)
or

-IZ∆x = U (x + ∆x) -U (x) (124)
or

-IZ = U (x + ∆x) -U (x) ∆x (125)
or

-IZ = ∂ ∂x U (x) (126)
According to Kirchhoff's current theorem,

I(x) -I(x + ∆x) = U (x)Y ∆x (127) or I(x + ∆x) -I(x) ∆x = -U (x)Y (128) or ∂ ∂x I = -U (x)Y (129)
Consider (125) Consider

I = - 1 Z ∂ ∂x U (x) (130) Substitution (129) ∂ ∂x (- 1 Z 1 ∂ ∂x U (x)) = -U (x)Y (131) or ∂ 2 ∂x 2 U (x) = ZY U (x) (132)
If R and g are both small quantities, they can be ignored,

ZY = (R + jωL)(g + jωC) jωLjωC = -ω 2 LC (133) Hence, ∂ 2 ∂x 2 U (x) = -ω 2 LCU (x) (134)
or

k = ω 2 LC (135) ∂ 2 ∂x 2 U (x) + k 2 U (x) (136)
The solution is

U (x) = U 0 exp(-jkx) (137)
Similarly, it can be concluded that,

I(x) = I 0 exp(-jkx) (138)
Consider,

I(x) = - 1 Z ∂ ∂x U (x) = - 1 Z (-jk)U (x) = - 1 jωL (-jk)U (x) = 1 ωL kU (x) = 1 ωL ω √ LCU (x) = C L U (x) (139)
or

I(x) = C L U (x) (140) U (x) = L C I(x) (141)
It can be seen that U (x) and I(x) are in phase.

Z 0 = L C (142)
The characteristic impedance of a transmission line. Where C is the capacitance per unit length. L is the inductance per unit length. Consider a special case where the radius of the wire is assumed to be a. The distance between two wires is d, so the capacitance per unit length is

C = 2π 0 ln d a (143)
The inductance per unit length is,

L = µ 0 2π ln d a (144)
The characteristic impedance is

Z 0 = µ0 2π ln d a 2π 0 ln d a = ln d a 2π µ 0 0 (145)
5 Using the electromagnetic field theory in today's textbook to calculate parallel transmission lines

The transmission line consists of an upper conductor and a lower conductor, forming a waveguide device. This device can confine waves within the transmission line. At the transmission line x=0, there is a current J in the direction of the current along the z-axis. This current generates a magnetic vector potential A.

Parallel transmission lines in electromagnetic theory textbooks

Our textbooks generally approach transmission line problems by considering TEM waves, where the electric field follows the z-axis direction and the magnetic field follows the y-axis direction,

E = E z (x, t)ẑ (146)
H = H y (x, t)ŷ (147) Satisfy Maxwell's equations

∇ × E = - ∂ ∂t µ 0 H (148) ∇ × H = + ∂ ∂t 0 E (149)
The above equation assumes that J and ρ are both 0. Therefore, φ is 0.

∇ × E =    x ŷ ẑ ∂ ∂x ∂ ∂y ∂ ∂z E x E y E z    =    x ŷ ẑ ∂ ∂x 0 0 0 0 E z    = -ŷ( ∂ ∂x E z ) (150) ∇ × H =    x ŷ ẑ ∂ ∂x ∂ ∂y ∂ ∂z H x H y H z    =    x ŷ ẑ ∂ ∂x 0 0 0 H y 0    = ẑ ∂ ∂x H y (151)
The Maxwell equation is transformed into (148,149),

∂ ∂x E z = ∂ ∂t µ 0 H y (152) ∂ ∂x H y = ∂ ∂t 0 E z (153) Or ∂ 2 ∂x 2 E z = ∂ ∂t µ 0 ∂ ∂x H y = µ 0 0 ∂ 2 ∂t 2 E z (154) Or ∂ 2 ∂x 2 E z = µ 0 0 ∂ 2 ∂t 2 E z (155) Or ∂ 2 ∂x 2 E z = µ 0 0 (jω) 2 E z (156) Or ∂ 2 ∂x 2 E z + k 2 E z = 0 ( 157
)
wherein

k 2 = µ 0 0 ω 2
The above equation is the Helmholtz equation, and its solution is,

E z = E 0 exp(-jkx) (158)
Similarly, for magnetic fields,

H y = H 0 exp(-jkx) (159) Consider (152), ∂ ∂x E z = ∂ ∂t µ 0 H y (160) Or -jkE z = jωµ 0 H y (161)
Or

-kE z = ωµ 0 H y (162)
Or

-E z = ωµ 0 k H y = ωµ 0 ω √ 0 µ 0 H y = µ 0 0 H y = η 0 H y ( 163
)
E z = -η 0 H y ( 164
)
This indicates that the electric field and magnetic field are in phase.

We assume that,

H = H 0 exp(-jkx) (165)
Obtain,

E = -η 0 H 0 exp(-jkx) (166)
The reference literature for solving transmission line problems using the field method can be seen from [?], and Liang Changhong particularly compared the circuit method with the field method,

∂ ∂x (-U (x)) = jωLI ∂ ∂x I = jωC(-U (x)) vs ∂ ∂x E z = jωµ 0 H y ∂ ∂x H y = jω 0 E z (167)
If we take a double wire transmission line as an example, assuming the radius of the wire is a and the distance between the two wires is d, the capacitance of the wire is

C = 2π 0 ln d a ( 168
)
Inductance is

L = µ 0 2π ln d a (169) Hence, ∂ ∂x (-U (x)) = jωLI ∂ ∂x I = jωC(-U (x)) = ∂ ∂x (-U (x)) = jω µ0 2π ln d a I ∂ ∂x I = jω 2π 0 ln d a (-U (x)) (170) or ∂ 2 ∂x 2 (-U (x)) = jω µ 0 2π ln d a ∂ ∂x I = jω µ 0 2π ln d a (jω 2π 0 ln d a (-U (x))) = -ω 2 µ 0 0 (-U (x)) (171) or ∂ 2 ∂x 2 U (x) + ω 2 µ 0 0 U (x) = 0 (172)
lets,

k 2 = ω 2 µ 0 0
we obtain,

U (x) = U 0 exp(-jkx) ( 173
)
This indicates that the field method can obtain the same results as the circuit method. If considering the following changes

I, -U (x), L, C ↔ H y , E z , µ 0 , 0 (174)
The left and right sides of the formula (167) are equivalent.

Problems with Methods in Parallel Transmission Line Textbooks

This method seems to have no problem calculating the electric and magnetic fields according to Maxwell's equations. Actually, it is an incorrect algorithm (the author believes it is incorrect). For transmission lines, the electric field consists of two parts: the static electric field and the induced electric field. This is already evident in the previous chapter 3. The transmission line should be able to degenerate into a simple circuit. Because there is voltage between the upper and lower wires. This voltage will generate an static electric field, which is generated by electric potential and is also in the same phase as the current, that is,

E s = -∇φ (175)
Additionally, at wire x = 0, there is a power source, or equivalent power source. This power supply has current. This current should contribute to the magnetic vector potential

J = J 0 ẑ (176)
We should calculate the induced electric field based on the magnetic vector potential A. This method is exactly the same as the one we use to solve antenna problems. Otherwise, one method is used for solving the antenna, and another method is used for calculating the transmission line.

That is in correct.

The author believes that electrostatic fields

E sz = - (φ up -φ down ) L z = - U L z (177)
E s It's an electrostatic field. φ up is the potential of the upper wire.

φ down is the potential of the lower wire.

U ≡ φ up -φ down U = U 0 exp(-jkx) ( 178
)
U 0 is the initial voltage, which is a constant.

E sz = - 1 L z U 0 exp(-jkx) (179)
E s = 1 L z U 0 exp(-jkx)(-ẑ) = E s0 exp(-jkx)(-ẑ) (180)
wherein,

E s0 = 1 L z U 0 (181)
At this point, we will find that the curl of this electrostatic field is not zero

∇ × E s = -ŷ( ∂ ∂x E z ) = -ŷ ∂ ∂x (E s0 exp(-jkx)(-1)) = ŷ ∂ ∂x (E s0 exp(-jkx) = -jkE s0 exp(-jkx)ŷ (182)
The author believes that this part of the electrostatic field cannot follow the Faraday formula

∇ × E s = -jω ∂ ∂t B (183)
Contribute to the magnetic field. Therefore, the above formula cannot be used to calculate the magnetic field. This is because Maxwell's original equation was

E = -∇φ - ∂ ∂t A (184)
In this equation, we can only use -∇φ to find the electrostatic field E s

.

E s = -∇φ (185)
We can only use vector potential A to calculate the induced electric field

E i = - ∂ ∂t A (186)
This formula is equivalent to Faraday's law

∇ × E i = - ∂ ∂t B = -jωB (187)
Therefore, we can only using the curl of induced electric field E i to calculate the magnetic field B. Alternatively, use the curl of vector potential A to find the magnetic field

B B = ∇ × A (188)
The curl of the electrostatic field in formula (182) is not 0, which is a very special case. In this case, write again

∇ × E = - ∂ ∂t B (189)
It's not correct because the above equation means,

∇ × E i + ∇ × E s = - ∂ ∂t B (190)
And at this point

∇ × E s = 0 (191)
The equation ( 184) is the Faraday's law in Maxwell's own Maxwell equation, while the equation ( 189) is a modified Maxwell's equation by Maxwell's descendants. At this point, it can be seen that Maxwell's own Maxwell's equation is more advanced than that of Maxwell's descendants.

Revise the above theory

The equation (146,147) actually means,

E i = E zi (x, t)ẑ (192)
H = H y (x, t)ŷ (193)
The subscript (i) means the induced electric field E i instead of the electric field E. The induced electric field and magnetic field satisfy the following relationship,

∇ × E i = - ∂ ∂t µ 0 H (194)
∇ × H = + ∂ ∂t 0 E i (195)
Comparing the above equation with (148,149) shows the difference between the two. The solution of the above equation

H y = H 0 exp(-jkx) (196)
E iz = -η 0 H 0 exp(-jkx) (197)
The above two formulas are the solution obtained based on the retarded wave that satisfies the Maxwell's equations. If written completely, it should be

H (r) yM axwell = H 0 exp(-jkx) (198)
E (r) izM axwell = -η 0 H 0 exp(-jkx) (199)
The superscript (r) represents retarded, and the subscript i represents induced. The subscript M axwell represents the value calculated according to Maxwell's electromagnetic theory. The magnetic field in the above equation is already correct, so we need to modify the induced electric field according to [START_REF] Shuang Ren Zhao | Definition, measurement and calibration of magnetic field of electromagnetic wave -correct the defects of maxwell's classical electromagnetic field theory[END_REF][START_REF] Shuang Ren Zhao | Energy conservation law and energy flow theorem for transformer, antenna and photon[END_REF])

E (r) i = jE (r) iM axwell = jη 0 H 0 exp(-jkx)(-ẑ) (200)
H (r) = H (r) M axwell = H 0 exp(-jkx)ŷ (201)
In this way, we obtain a magnetic field and an induced electric field.

We noticed that after the correction, the phase of the induced electric field is consistent with that of a simple circuit, as described in chapter 3.

The Poynting vector is calculated from electrostatic and magnetic fields,

S = E s × H * = E s0 exp(-jkx)(-ẑ) × (H 0 exp(-jkx)ŷ) * = E s0 H * 0 x ∼ x (202)
In the above equation,

E s ∼ U ∼ I ∼ H (203)
That is to say, electrostatic field E s The phase of is consistent with the phase of the voltage and the phase of the current, therefore it is also consistent with the phase of the magnetic field H. The above equation ( 202) is a real number, so this Poynting vector transmits active power. If we calculate based on the induced electric field,

E i × H * = jη 0 H 0 exp(-jkx)(-ẑ) × H 0 exp(-jkx)ŷ = jη 0 H 0 H * 0 x ∼ j x (204)
The above equation is a pure imaginary number, indicating that reactive power does not transfer energy. Therefore, the power corresponding to the induced electric field is reactive power. That is to say, the author believes that there is still an induced electric field within the transmission line. Comparing to the electrostatic field E s , the phase of the induced electric field E i lag by j = exp(j π 2 ).

6 Calculate the electric and magnetic fields according to the circuit method

We hope to solve this problem in multiple ways. Firstly, we consider that the voltage and current have been solved according to the circuit problem in Chapter 4. Assuming we have already calculated,

I = I 0 exp(-jkx) (205)
U = U 0 exp(-jkx) (206)
Now we need to calculate the electric and magnetic fields. among

U 0 I 0 = Z = L C (207)
is the impedance of the transmission line. Here we assume that there is no reflection.

Calculation of electrostatic field

E s = - U L z ẑ (208)
So the electrostatic field is,

E s = U L z (-ẑ) = U 0 exp(-jkx) L z (-ẑ) (209)
L z It is the height of the transmission line.

Calculation of magnetic field

Considering the contribution of the above current to the magnetic field, due to the infinite length of the wire, the vector potential is infinite. Consider parallel transmission lines being relatively wide. L y Compared to L z

Much larger, so we can approximately obtain

J x = I L y (210) H = 1 2 J x ŷ = 1 2 I L y ŷ ( 211
)
We know that the contribution of the lower wire to the magnetic field is the same as that of the upper wire, so the total magnetic field is,

H = I L y ŷ = 1 L y I 0 exp(-jkx)ŷ (212)

Calculating Power

The average power is

P = U I * = U 0 exp(-jkx)(I 0 exp(-jkx)) * = U 0 I * 0 = U 2 0 Z (213)
We will calculate the power based on the electric and magnetic fields

E s × H * = U 0 exp(-jkx) L z (-ẑ) × ( 1 L y I 0 exp(-jkx)ŷ) * (214) So S = E s × H * = U 0 I * 0 L z L y x (215)
P = " Γ S • n = L z L y U 0 I * 0 L z L y x • x = U 0 I * 0 (216)
Therefore, the power calculated using Poynting vector is consistent with the power directly calculated using voltage and current.

Calculation of induced electric field

Now we know that magnetic fields and induced electric fields can be calculated according to Faraday's law in Maxwell's equation,

∇ × E i = -jωµ 0 H (217) ∇ × E i =    x ŷ ẑ ∂ ∂x ∂ ∂y ∂ ∂z E ix E iy E iz    =    x ŷ ẑ ∂ ∂x 0 0 0 0 E iz    = -ŷ( ∂ ∂x E iz ) (218) -( ∂ ∂x E iz ) = -jωµ 0 H y (219) -(-jkE iz ) = -jωµ 0 H y (220) -(kE iz ) = ωµ 0 H y (221)
E iz = - ωµ 0 k H y = -η 0 H y (222)
This means that

E (r) izM axwell = -η 0 H (r) yM axwell (223)
wherein,

H (r) yM axwell = 1 L y I 0 exp(-jkx) (224)
E (r) izM axwell = -η 0 1 L y I 0 exp(-jkx) (225)
We know that the above magnetic field calculation is correct, so we need to make corrections to the induced electric field [START_REF] Shuang Ren Zhao | Definition, measurement and calibration of magnetic field of electromagnetic wave -correct the defects of maxwell's classical electromagnetic field theory[END_REF][START_REF] Shuang Ren Zhao | Energy conservation law and energy flow theorem for transformer, antenna and photon[END_REF])

E (r) iz = jE (r) izM axwell = -jη 0 1 L y I 0 exp(-jkx) (226)
H (r) y = H (r) yM axwell = 1 L y I 0 exp(-jkx) (227)
So we see that if we calculate the Poynting vector,

S = E (r) iz ẑ × H (r) y ŷ = (-jη 0 1 L y I 0 exp(-jkx))( 1 L y I 0 exp(-jkx)) * (-x) = jη 0 1 L y 1 L y I 0 I * 0 x (228)
The Poynting vector composed of induced electric and magnetic fields is a pure imaginary number. It is reactive power and does not transfer energy.

These results are consistent with the 5.3 section of the previous chapter.

Here, we calculate the magnetic field from the current and the induced electric field from the magnetic field, and the calculated induced electric field must be corrected. At this point, the induced electric field is corrected.

7 Calculate the transmission line based on the curl of the magnetic vector

In this chapter, we do not consider the contribution of the upper and lower wires to the magnetic vector potential. Two wires act as waveguides to confine waves in one-dimensional space. Electromagnetic waves propagate along the x-axis. It can be considered that the magnetic vector potential propagates along x.

Calculating vector potential

The transmission line problem is a one-dimensional problem, where electromagnetic waves can only propagate in the x direction. The vector potential generated by this problem should be consistent with the vector potential generated by an infinite plate. The vector potential of an infinite plate current is,

A ∼ ˆV J r exp(-jkx)dV (229) A ∼ J 0 ẑ exp(-jkx) (230)
Now we consider the magnetic vector potential of the current extending along the z-axis at the source x = 0, where the current is I 0 , the surface current density is,

J 0 = I 0 L y (231)
Hence,

J = J 0 ẑ (232)
Consider that the magnetic vector potential should propagate along the x-axis without decay. Therefore, the magnetic vector is,

A ∼ A 0 exp(-jkx)ẑ = J 0 ẑ exp(-jkx) (233)

Calculation of static electric field

There is a current on the transmission line and a voltage U between two wires. This voltage can of course form an electrostatic field, which is consistent with the previous calculation.

U = U 0 exp(-jkx) (241) E s = -U L z ẑ = U 0 L z exp(-jkx)(-ẑ) (242)

Correction of magnetic field

Since we calculate the induced electric field based on the magnetic vector A, the phase of the induced electric field calculated is correct, and the phase of the magnetic field calculated from the induced electric field must be corrected,

H real = (-j)H M axwell (243)
H M axwell = jJ 0 exp(-jkx)ŷ (244)

H real = (-j)jJ 0 exp(-jkx)ŷ = J 0 exp(-jkx)ŷ ( 245
)
H real is the true magnetic field, which is also the magnetic field obtained after correction according to the author's electromagnetic field theory. It can be seen that the corrected magnetic field and current are in phase.

Discussion

According to classic electromagnetic field theory textbooks such as literature [?, ?], the electromagnetic field satisfies the Maxwell equation,

∇ × E = - ∂ ∂t µ 0 H (246) ∇ × H = + ∂ ∂t 0 E (247)
The Maxwell equation above ignores the current J and charge density ρ. Due to ignoring ρ. Therefore, the electrostatic field generated by charges has also been ignored

E s = -∇φ (248)
In fact, the transmission line is composed of two conductors, and there is an electrostatic field between them, which is generated by -∇φ. This part of the field is in the same phase as the voltage and current on the transmission line. such as

φ = z U L z exp(-jkx) (249)
This indicates that the electric potential varies linearly with the variable z.

E zs = - ∂ ∂z φ = - ∂ ∂z (z U L z exp(-jkx)) = - U L z exp(-jkx) (250)
from ( 240) we can obtain,

E i = - ∂ ∂t A ∼ jη 0 J 0 exp(-jkx)(-ẑ) (251)
Therefore, according to the author's theory, there are actually two types of electric fields within the transmission line, the electrostatic field E s and induced electric field E i . According to general textbooks, there is only one type of electric field within the transmission line, whether it is an induced electric field or an electrostatic field. The induced electric field calculated by the author has a phase factor j. The induced electric field in general textbooks [?, ?] does not have this phase factor. Therefore, both methods can be verified through experiments.

Since the induced electric field can be regarded as a circuital field, we can measure it using a coil. When measuring with a coil, it is impossible for the electrostatic field to enter the measurement system.

Experiments and Measurements

Assuming we use an electromagnetic signal with a wavelength of 1 meter, the frequency of this meter wave is

f = c λ = 3 * 10 8 1 = 300M Hz (252)

Measurement of magnetic field

Magnetic field is the easiest to measure, using traditional current measurement equipment that actually uses Hall elements to measure the magnetic field around wires. Therefore, this device precisely provides the phase of the magnetic field. This phase is the phase of the current on the wire and also the phase of the magnetic field.

Measurement of electrostatic field

The phase of the electrostatic field can be obtained by measuring the voltage between two wires. This phase should be the same as the previously measured magnetic field phase.

Measurement of induced electric field

The induced electric field can be measured using coil type magnetic field measurement equipment. Because this method of measuring magnetic field actually calculates the magnetic field by measuring the induced electric field (induced electromotive force). Therefore, we can use it to directly measure the phase of the induced electric field. Compare the phase of the induced electric field with the phase of the current.

According to classical electromagnetic field theory, this part of the induced electric field is generally ignored in textbooks. Or do not mention this part of the induced electric field at all. Alternatively, it can be considered that induced electric fields and static electric fields are the same thing, which is an electric field. The author believes that there is a phase lag between the induced electric field and the magnetic field, which is the factor of (j). Therefore, induced electric fields and electrostatic fields are completely different. This is reflected in a simple circuit. The author believes that the transmission line problem must be able to degenerate into a simple circuit. Therefore, the induced electric field should lag behind the current by j = exp(j π 2 ).

8.4 Direct measurement of induced electric field using an electric field near field measuring pen

The electric field near-field measurement pen can be used to measure the electric field. This measurement should include both electrostatic and induced electric fields, which are the sum of electrostatic and induced electric fields. According to classical electromagnetic theory, there is only one type of electric field on a transmission line, which is in phase with the current.

According to the author's theory, there are electrostatic fields and induced electric fields on the transmission line, and the phase of the induced electric field is 90 degrees more than that of the magnetic field. Since the values of the two electric fields are basically the same, the phase factor will become 45 degrees. Namely Therefore, according to the author's method, it is calculated that the total electric field measured by the near-field measurement pen should have a phase lag of exp(j π 4 ). According to classical electromagnetic theory, this phase is 0.

E = E s + E i (258)

Conclusion

The author has made revisions to the classical electromagnetic field theory, which includes incorporating advanced waves into the theory. In order to ensure that the electromagnetic theory including advanced and retarded waves remains self consistent. The author has made corrections to the solution of the Maxwell equation, mainly by correcting the phase of the far-field magnetic field. The Poynting vector of the antenna in the far field after correction is reactive power. At the same time, the mutual energy flow becomes active power. This revised theory is difficult to be

  theory also applies the principle of action at a distance [?, ?, ?]. In addition, Stephenson also conducted in-depth research on the theory of advanced waves around 1980 [?]. Cramer further developed this theory into a transactional interpretation of quantum mechanics in 1986 [?, ?]. Welch proposed the electromagnetic field reciprocity theorem including advanced waves in 1960 [?].

  Wheeler and Feynman also attempted to solve this contradiction in the absorber theory. Their viewpoint is that the current radiates half of the retarded wave and half of the advanced wave. The retarded wave carries positive radiation energy, while the advanced wave radiates negative energy. Therefore, current does not radiate energy in average. The author agrees with Wheeler Feynman's viewpoint of half retarded wave and half advanced wave, but does not agree that the energy of the retarded and advanced waves emitted by the current can be offset. The author proposed the concept of time reversal wave in 2017 [?]. Time reversal waves can cause both retarded and advanced waves to collapse in the opposite direction. The reverse collapse cancels out the energy flow of the original retarded wave and advanced wave. This leaves only the mutual energy flow to transfer energy. The law of conservation of energy is guaranteed. That is to say, Article 2 has been guaranteed. But time reversal waves require adding new electromagnetic fields to the Maxwell's equations, which makes the problem more complex. And it cannot solve all the problems. Since 2017, the author has attempted to apply the theory proposed by the author in many examples. These examples include electromagnetic waves of double infinite plane currents. The energy flow problem from the primary to secondary of the transformer. The energy flow problem from the dipole transmitting antenna to the dipole receiving antenna.

  This article studies the transmission line problem. The transmission line problem is a very simple problem. The electromagnetic waves inside the transmission line are TEM waves, which should be easily solved by Maxwell's classical electromagnetic theory. However, we have discovered a strange phenomenon. Few people use Maxwell's equations to solve transmission line problems, usually using circuit methods. The author reviews circuit methods for solving transmission line problems. Review the plane wave problem. Review simple electromagnetic field transmission problems in transmission lines. The dilemma of using Maxwell's classical electromagnetic theory to solve transmission line problems is studied. Then, using the electromagnetic field theory proposed by the author to calculate the transmission line problem, the author found that the results are consistent with the circuit method. The author's revision of Maxwell's electromagnetic theory has not been widely accepted, although several papers have been published. If the transmission line problem can be perfectly solved according to the author's theory, but there are difficulties in solving the classical electromagnetic theory using Maxwell's equations, this itself supports the author's electromagnetic field theory. On the other hand, any measurement within the transmission line is very easy to perform. If experimental verification is required, it can also be done very easily on the transmission line.

图 1 :

 1 Plane electromagnetic waves are generated by an infinite flat plate current.

图 2 :

 2 Simple circuit. It is composed of a power supply and a resistor, connected by a wide wire as shown in the figure. Line width L y , line height L z , with current I 0 . The voltage drop on the line is U .

图 4 :

 4 Parallel transmission line, line width L y , line height L z , with current I 0 . The voltage drop on the line is U .

A transmission line of 5 11 = 24 . 16 × 10 - 12 = 24 7 *

 51124161012247 meters is sufficient. It is already 5 times the wavelength, which can be considered long enough. For the convenience of measurement, double parallel wire transmission lines are selected. Assuming the radius of two wires a is one millimeter, a = 0.001M etre (253) Assuming the distance d between the wires is one centimeter, × 10 -12 = 2.416 * 10 -ln 10 = 4.60517 * 10 -7 = 0.460517 * 10 -6 = 0.460517µH(256)The characteristic impedance of the transmission line is, 136 Ω resistor should be connected at the end of the transmission line.

E

  = ηH 0 ŷ + jηH 0 ŷ = ηH 0 (1 + j) = ηH 0 exp(j

Calculation of induced electric field

The induced electric field is,

The induced electric field calculated from the magnetic vector potential does not need to be corrected.

Calculation of magnetic field from vector potential

The magnetic field can have the curl calculation of vector potential,

So, according to Maxwell's electromagnetic theory

We know that the induced electric field can be obtained from the Maxwell equation E i in the (-ẑ) direction, the magnitude includes a spatial impedance η 0 . phase that is completely consistent with H, that is,

The above equation is based on Maxwell's electromagnetic theory to obtain the induced electric field, which is in phase with the magnetic field. Numerically, there is an additional impedance factor η 0 . Our research is only interested in phase, and the above equation is consistent with the formula (234).

verified through measurement. For example, for transmitting and receiving antennas, it is necessary to measure the far-field of the transmitting antenna.

However, the far-field magnetic field of the transmitting antenna is very weak and difficult to measure. Both electric and magnetic fields are easy to measure on transmission lines. Therefore, this article chooses to further calculate the transmission line using the author's electromagnetic theory.

The author also reviewed the calculation of transmission lines based on classical electromagnetic theory. Comparing the calculation results of the two, according to the author's electromagnetic theory, there is not only an electrostatic field but also an induced electric field on the transmission line. If we discover an induced electric field through measurement and the induced electric field lags behind the current by a factor j = exp(j π 2 ), it indicates that the author's electromagnetic theory is correct. The author is preparing to complete this experiment in the next step. This experiment is relatively easy, and the author encourages readers to do it themselves too.

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