# Question on TEM wave travel in parallel plate transmission line.

• yungman
In summary, the conversation discusses the propagation of TEM waves in a parallel plate transmission line. The existence of surface current and surface charge on the conductor plates does not affect the TEM wave traveling in the dielectric between the plates, which is considered "source free." However, in reality, there is a small energy loss due to the finite conductivity of the plates. The equations for the fields are incorrect and should include a dependence on the wave vector in the y direction. The modes for the TE_z solution are defined by the eigenmodes for a desired frequency. The distance between the plates must be much smaller than the wavelength for a pure TEM mode to be present. The case of an infinite parallel plate waveguide is not useful as it does not support
yungman
I want to confirm that the TEM wave is actually travel inside the dielectric between the two plates of the parallel plate transmission line. So even though surface current and surface charge appeared on the surface of the conductor plates as the consequence of the boundary conditions between the dielectric and conductor boundries, the TEM wave traveling in the dielectric between the plates are still consider as "source free" where

$$\tilde E_y=\hat y E_0 e^{-\delta z} \;\;\hbox { and } \;\; \tilde H_x = (-\hat x)\frac{E_0}{\eta} e^{-\delta z}$$

The thing that confuse me is the TEM wave do induce surface current and charges on the conducting plate. That should take energy to assemble the charges. So power should be loss along the line! But in reality we all know that it does not.

If we assume that the plates are perfect conductors and that the dielectric is lossless then no energy is lost as the wave propagates. With a PEC material, there is no energy loss in generating currents. But once we allow for the plates to have a finite conductivity then we introduce a small loss since we have to inject a certain amount of energy to keep the currents flowing.

The equations for your fields are incorrect though. The guided modes of a parallel plate waveguid bounce back and forth off of the plates. The angle of incidence upon the plates is determined by the frequency and mode of your wave. This is so that the wave will strike the plates in such a way as to satisfy the appropriate boundary conditions. So if your waveguide is setup so that the guided direction of propagation is along the z-axis and the plates are in the x-z plane, then your waves must have a dependence upon

$$\sim e^{ik_yy + ik_zz}$$

The waves are still traveling as TEM waves but we generally decompose the solutions into those that are TE and TM to the direction of guided propagation (z in this case). So your TE_z solution is something of the form

$$\mathbf{E}(\mathbf{r}) = E_0 \hat{x} e^{ik_yy + ik_zz}$$
$$\mathbf{H}(\mathbf{r}) = H_0 (\alpha\hat{y} + \beta\hat{z}) e^{ik_yy + ik_zz}$$

TM_z solutions are of the form,

$$\mathbf{E}(\mathbf{r}) = E_0 (\alpha\hat{y} + \beta\hat{z}) e^{ik_yy + ik_zz}$$
$$\mathbf{H}(\mathbf{r}) = H_0 \hat{x} e^{ik_yy + ik_zz}$$

Further, we note that we can have a solution that travels in the +y and +z direction and also in the -y and +z direction at the same location. That is, we can have both upward and downward bouncing waves. So we generally combine the two traveling wave components as a standing wave component where the standing wave is just the superposition of two traveling waves. So that means that the E field of the TE_z solution is expressed more compactly as,

$$\mathbf{E}(\mathbf{r}) = E_0 \hat{x} \sin (k_yy) e^{ik_zz}$$

Again, the selection of the wave vector k is dependent upon the mode and frequency of the solution (obviously in the above we note that at the locations of the plates (say y=0, y=a) the tangential electric field is zero). This, in essence, is an eigenvalue problem. So it's a simple matter to see that in our case that,

$$k_y = \frac{\pi m}{a}$$
$$k_z = \sqrt{ k_0^2 - \left( \frac{\pi m}{a} \right)^2 }$$

Thus the modes for our TE_z solution for a desired frequency are defined by the above eigenmodes for m=1,2,3, ...

EDIT: Ok, pfew... I think that's actually pretty much the treatment of the parallel plate waveguide in its entirety.

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I actually copy my equation from the book. The book specified that the distance between the plates are much smaller than the wave length to ensure it is pure TEM mode to simplify the situation. This is true in electronic circuit distribute element design, you want the dielectric thickness less than 1/10 of wave length and normally smaller than the width of the trace.

Are your equations without the same requirement so you have to worry about bouncing between plates. Therefore your propagation constant k has both y and z components?

Why there would be no energy spent to assemble the charges and surface current on the surfaces of the plates like we deal with the energy spent to assemble charges in electrostatic study? Is it because it is sinodal and the assemble and disassemble cancel out or what?

yungman said:
I actually copy my equation from the book. The book specified that the distance between the plates are much smaller than the wave length to ensure it is pure TEM mode to simplify the situation. This is true in electronic circuit distribute element design, you want the dielectric thickness less than 1/10 of wave length and normally smaller than the width of the trace.

Are your equations without the same requirement so you have to worry about bouncing between plates. Therefore your propagation constant k has both y and z components?

Why there would be no energy spent to assemble the charges and surface current on the surfaces of the plates like we deal with the energy spent to assemble charges in electrostatic study? Is it because it is sinodal and the assemble and disassemble cancel out or what?

If that's the case then it is an evanescent mode, which bears with your original equations assuming that \delta is real (in which case the solution is decaying exponentially). We can see that they come out easily enough. Assuming that the width is 1/10th of a wavelength,

$$k_z = \sqrt{ k_0^2 - \left( \frac{\pi m}{a} \right)^2 } = \sqrt{ \left( \frac{2\pi}{\lambda} \right)^2 - 100 m^2 \left( \frac{\pi}{\lambda} \right)^2} = \frac{\pi}{\lambda} \sqrt{ 4-100m^2} = i\delta$$

But I expect that this is not the same as the case you are thinking about with electronic circuits as this is an infinite parallel plate waveguide. Such a configuration is not useful since it does not support propagation modes with such small separations. Those would be microstrip or striplines which have a different set of governing equations and behaviors. The dispersion relations for microstrip lines allow solutions down to DC. (EDIT: This does not work with a parallel plate waveguide since we cannot have a TEM_z solution that satisfies the boundary conditions. This is probably why the given expressions in your text do not have the sinusoidal portion. They are basically enforcing an unsupported mode.)

It doesn't take any energy because it's a PEC. The voltage difference between any two points on or inside the PEC is zero. So there is no work needed to be expended to move a charge about or on the PEC. So inducing currents on the PEC does not expend energy.

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What is PEC?

Perfect electrical conductor.

Born2bwire said:
Perfect electrical conductor.

Thanks for the help. I think the book use parallel plates as the first step to show how to calculate capacitance, inductance and impedance. I don't even see a lot of use with parallel plate tx line, later in the book, it just lead into microstrip and stripline which are the most common in circuit boards.

Another question is why there is not x component in the k, ie the propagation has no x component in your equations in #3? That is one question I don't have also. What prevent the EM wave from skipping out from the side of a true parallel plate tx line? Your equation still assume the wave propagate in y and z direction only.

Alan

yungman said:
Thanks for the help. I think the book use parallel plates as the first step to show how to calculate capacitance, inductance and impedance. I don't even see a lot of use with parallel plate tx line, later in the book, it just lead into microstrip and stripline which are the most common in circuit boards.

Another question is why there is not x component in the k, ie the propagation has no x component in your equations in #3? That is one question I don't have also. What prevent the EM wave from skipping out from the side of a true parallel plate tx line? Your equation still assume the wave propagate in y and z direction only.

Alan

The analysis of a parallel plate waveguide assumes that the plates are infinite. Since the problem is invariant in two directions, we can solve the problem as a 2D problem. The choice of coordinates for the 2D representation is arbitrary.

Born2bwire said:
The analysis of a parallel plate waveguide assumes that the plates are infinite. Since the problem is invariant in two directions, we can solve the problem as a 2D problem. The choice of coordinates for the 2D representation is arbitrary.

What is the reason the EM wave stay inside between the parallel plate even when the guide line turn corner?

I want to verify that the way to launch the TEM is from the voltage at the beginning of the line that cause the EM wave and then the EM wave travel down the guided structure.

Then the next question is why terminating the line electrically can terminate the EM wave? Like you said, it does not take energy to generate current and charges in PEC. At the end of the line, only the voltage or the current see the termination. But the EM wave is still there going. How dose the EM wave get terminated?

Alan

There is no corner here, it's an infinite set of parallel plates. If we have a spatially variant waveguide, say a microstrip line, then the discontinuity that is presented by the ending of the structure at a corner causes the wave to change direction. Anytime that we present a wave with a change in the impedance of its propagation it will undergo a reflection and transmission. Some energy maybe lost when the wave hits a corner but there are ways to minimize this from occurring. In addition, to help minimize the back reflection but instead strengthen the reflection around the corner we may apply other techniques (which is why there are specific design rules on how to place bends and corners using microstrips).

The EM wave gets "terminated" as you put it by being absorbed. This can be done by gradual loss of the wave by leaking out of the waveguide. It can be absorbed due to the lossy materials (finitely conductive metals, lossy dielectrics) that make up the waveguide. And it can be absorbed by the receiving object like an antenna or chip. Without some kind of mechanism to absorb or leak the energy of the wave the wave will always just travel down, back and forth, the waveguide.

The voltage and currents are consequences of the electromagnetic wave. When the wave impinges on a conductor, it induces currents. In addition, the electric field and magnetic fields induce a voltage difference across sections of the waveguide. When we want to excite an electromagnetic wave we can do this by replicating the currents or voltages that the wave would induce on the line by virtue of reciprocity.

## 1. What is a TEM wave?

A TEM (transverse electromagnetic) wave is an electromagnetic wave that has both electric and magnetic fields perpendicular to the direction of propagation. It does not have any longitudinal components and can only exist in material mediums with a zero conductivity or perfect conductors. TEM waves are commonly used in transmission lines and antennas.

## 2. How does a TEM wave travel in a parallel plate transmission line?

In a parallel plate transmission line, the TEM wave propagates in the space between the two parallel conductors. The electric and magnetic fields are perpendicular to each other and to the direction of propagation, and they are confined between the two plates due to the presence of the conductors. The electric field is strongest between the plates, while the magnetic field is strongest outside the plates.

## 3. What factors affect the propagation of a TEM wave in a parallel plate transmission line?

The propagation of a TEM wave in a parallel plate transmission line is affected by the distance between the plates, the conductivity of the material between the plates, and the frequency of the wave. The distance between the plates determines the characteristic impedance of the transmission line, while the conductivity of the material affects the attenuation of the wave. The frequency of the wave determines the wavelength and the phase velocity in the transmission line.

## 4. How is the velocity of a TEM wave in a parallel plate transmission line calculated?

The velocity of a TEM wave in a parallel plate transmission line is calculated using the formula v = 1/√(LC), where v is the phase velocity, L is the inductance per unit length, and C is the capacitance per unit length. The phase velocity is the speed at which the wave travels, and it is dependent on the characteristics of the transmission line.

## 5. Can a TEM wave travel in any type of transmission line?

No, a TEM wave can only travel in a transmission line that has two conductors with equal and opposite charges and with a dielectric material in between. This is known as a TEM mode transmission line. Other types of transmission lines, such as coaxial cables or waveguides, do not support TEM waves and instead support other modes of propagation.

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