Resonance in transmission lines

• ergospherical
In summary, the speaker has calculated the relationships between input impedance, line impedance, and terminal impedance for a mismatched transmission line. They are now being asked to consider the case where the source, an oscillator, is in resonance with the line. This could mean that the oscillator is driving the line at its resonant frequency or that the two are resonating together at a different frequency. The speaker suggests looking at the relationship between source impedance and load impedance in this case. They also mention the importance of Q in determining the oscillator frequency and the possibility of multiple resonances in a complicated network. The speaker then shares their own solution, which involves using the results from the previous analysis and solving for the input impedance to determine the conditions for resonance.
ergospherical
I've calculated the relationships between the input impedance, line impedance and terminal impedance for a mismatched transmission line (parallel-wire type and of length ##l##). I'm now asked to consider the case where "the source [an oscillator] is in resonance with the line". What exactly does this condition mean (in terms of equations, preferably)?

ergospherical said:
I'm now asked to consider the case where "the source [an oscillator] is in resonance with the line".
There are a couple of possibilities with the question.

A fixed frequency oscillator could drive a line near the line's resonant frequency, so the Q of the line would then be important, and there will be multiple harmonics of line length.

Or, the combined oscillator and line could resonate together at some other frequency.
https://en.wikipedia.org/wiki/Lecher_line

DaveE and ergospherical
So in either of the two scenarios @Baluncore suggested, you'll want to look at the relationship between the source impedance and the load impedance. As in the model below. In the first case ZS must be negligible w.r.t. ZL (or you have to account for it somehow). In the second case the implication is that the resonance is for the combination.

If they haven't told you what the source impedance is, I would assume the first case, otherwise it could be anything.

PS: Also note that you could have more than one resonance if the total network is complicated.

ergospherical
At resonance the line presents a purely resistive load to the oscillator.
If however, the oscillator is not isolated from the line by a buffer amplifier or such like, then the two resonant circuits will be coupled together. In such a case, the impedances (or admittances, depending on the circuit) of the two circuits will be added together and in general the frequency of the oscillator will change. In practice, whichever circuit has the higher Q will largely determine the oscillator freqeucncy.

ergospherical
tech99 said:
If however, the oscillator is not isolated from the line by a buffer amplifier or such like, then the two resonant circuits will be coupled together.
Yes. This is the key point and requires an analysis of the source impedance vs. the downstream stuff.

tech99 said:
In practice, whichever circuit has the higher Q will largely determine the oscillator freqeucncy.
Yes, if there are two (or more) separate networks, or resonances. But this refers to the largest response. A lower frequency resonance can be excited by higher frequencies and will also up on the spectrum analyzer. Higher frequency resonances can also show up, excited by noise or transients.

tech99 said:
At resonance the line presents a purely resistive load to the oscillator.
Thanks, yes this is how I finally arrived at the solution. It's given that an open-ended, quarter-wavelength line of length ##2a = 0.25 \ \mathrm{m}## is at resonance. When a capacitance ##C## is then connected across the end of the line, resonance occurs when the line length has been reduced to ##a##. I used the results from the previous analysis to write down\begin{align*}
\dfrac{Z_I}{Z} &= \dfrac{Z_0 \cos{ka} + iZ\sin{ka}}{Z \cos{ka} + iZ_0 \sin{ka}} \\ \\

&= \dfrac{i (Z^2 - Z_0^2) \sin{ka} \cos{ka} + Z_0 Z }{Z^2 \cos^2{ka} + Z_0^2 \sin^2{ka}}
\end{align*}where ##Z_0## is the terminal impedance, ##Z## the (real) line impedance and ##Z_I## the input impedance. The initial quarter-wavelength line condition implies that ##k = 2\pi \ \mathrm{m}^{-1}##, which means ##ka = \pi/4## and ##\sin{ka} \cos{ka} = 1/2##. Given that ##Z_0 = 1/i\omega C##, then ##Z_I## is real if\begin{align*}
i (Z^2 - Z_0^2) \sin{ka} \cos{ka} + Z_0 Z = \dfrac{i}{2} \left(Z^2 + \dfrac{1}{\omega^2 C^2} \right) - \dfrac{iZ}{\omega C}
\end{align*}is real, which is only the case if it is zero. It remains to solve\begin{align*}
Z^2 - \dfrac{2}{\omega C} Z + \dfrac{1}{\omega^2 C^2} = 0
\end{align*}Since ##\omega = ck = 2\pi c##, this becomes
\begin{align*}
Z^2 - \dfrac{1}{\pi c C} Z + \dfrac{1}{4\pi^2 c^2 C^2} = 0
\end{align*}Inserting the value for ##C## gives, to my surprise, the correct answer.

DaveE and hutchphd

1. What is resonance in transmission lines?

Resonance in transmission lines is a phenomenon where the voltage and current in a transmission line become in phase with each other, resulting in a higher amplitude of the voltage and current. This occurs when the frequency of the input signal matches the natural frequency of the transmission line.

2. How does resonance affect the performance of transmission lines?

Resonance in transmission lines can cause an increase in the voltage and current, which can lead to overheating and damage to the line. It can also cause distortion and loss of signal quality, affecting the performance of the transmission line.

3. What factors can cause resonance in transmission lines?

The length and characteristic impedance of the transmission line, as well as the frequency and amplitude of the input signal, can all contribute to the occurrence of resonance in transmission lines. Other factors such as the type of material used and the presence of any discontinuities in the line can also play a role.

4. How can resonance in transmission lines be prevented?

To prevent resonance in transmission lines, it is important to carefully design and select the appropriate length and characteristic impedance for the line. Additionally, using proper termination and avoiding any discontinuities can help minimize the chances of resonance occurring.

5. What are the consequences of ignoring resonance in transmission lines?

If resonance in transmission lines is ignored, it can lead to damage to the line and other connected components. It can also result in signal distortion and loss of data, affecting the overall performance of the system. In extreme cases, it can even cause a complete breakdown of the transmission line.

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