Questions about Circuit Models of Transmission Lines

In summary, Circuit Models of Transmission Lines are mathematical representations used to analyze and design electrical circuits that involve transmission lines. These models take into account parameters such as resistance, inductance, capacitance, and conductance to accurately simulate the behavior of electric signals as they travel through the transmission line. They can also be used to predict the effects of impedance mismatch and signal distortion on the circuit. Different types of circuit models, such as lumped element models and distributed models, are used depending on the complexity of the circuit. Understanding these models is crucial for the design and optimization of high-frequency circuits, such as those used in telecommunications and radio frequency systems.
  • #1
stephen8686
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TL;DR Summary
Can someone explain lumped element circuit model of transmission lines please?
This is for my graduate EM Theory class. My background is physics/optics, so I was able to understand when we solved maxwells eqs. for waves propagating between parallel conducting plates, but that lead into the lumped element circuit model of transmission lines which I don't understand. I've attached the figure we used. My question is why does the circuit look this way? Do the two horizontal wires represent the two parallel plates, or is the top one the transmission line and the bottom one ground? And how do we know that the RLC components should be arranged in this way instead of some other way?
 

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  • #2
The model you show is of a transmission line over a ground plane.
The line has inductance per unit length, and capacitance to ground per unit length.
Drawn as a ladder network, it makes a low-pass filter, with a very high cutoff frequency.
https://en.wikipedia.org/wiki/Transmission_line#Telegrapher's_equations

From the lumped model, you can determine the characteristic impedance and the velocity factor. Your lumped model has no resistance, so it is an ideal lossless line.
 
  • #3
Baluncore said:
The model you show is of a transmission line over a ground plane.
The line has inductance per unit length, and capacitance to ground per unit length.
Drawn as a ladder network, it makes a low-pass filter, with a very high cutoff frequency.
https://en.wikipedia.org/wiki/Transmission_line#Telegrapher's_equations

From the lumped model, you can determine the characteristic impedance and the velocity factor. Your lumped model has no resistance, so it is an ideal lossless line.
So the top wire represents the transmission line and the bottom is ground? And why would a transmission line act as a lo pass filter?
 
  • #4
stephen8686 said:
And why would a transmission line act as a lo pass filter?
Because a transmission line has a cutoff frequency, by virtue of the fact that it has series inductance, with parallel capacitance, distributed over its length.
 
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  • #5
The lumped element circuit has a cut off frequency but a real line is continuous and does not. We need more LC sections to raise the cut off frequency of this equivalent circuit.
 
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  • #6
Leon Brillouin wrote a beautiful little book called Wave Propagation in Periodic Structures that covers transmission lines from a physics perspective before moving on to waves (acoustic and optical) in crystals. With your background, it should be an enlightening read.
 
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  • #7
stephen8686 said:
So the top wire represents the transmission line and the bottom is ground?
What you show is the model for an unbalanced transmission like, like coax cable. The lumped model for balanced transmission lines (like twisted pair or twin-lead cable) would have inductors in the lower/return leg as well.

The Characteristic Impedance ##Z_0## of a Transmission Line includes the inductance per unit length and the capacitance per unit length, so for a lossless TL that is all you need to include in the lumpted model. Then all you have to decide is how fine you want that model to be per unit length (per meter, for example), based on the frequencies that you want to model propagating down that TL.

A better model includes the loss terms, because if you are trying to match a transmit impedance and a termination impedance to the TL, the loss terms and their effect on ##Z_0(f)## can be significant. In the twisted pair IoT world that I work in, this affects communication datarates in the 100kHz range, but not in the 10MHz range, for example.

1655073433170.png

https://en.wikipedia.org/wiki/Characteristic_impedance
 

Related to Questions about Circuit Models of Transmission Lines

1. What is a circuit model of a transmission line?

A circuit model of a transmission line is a simplified representation of a physical transmission line that allows for analysis and design of the line's behavior. It consists of lumped elements such as resistors, inductors, and capacitors that represent the distributed parameters of the transmission line.

2. What are the types of circuit models for transmission lines?

There are two main types of circuit models for transmission lines: lumped element models and distributed element models. Lumped element models use discrete components to represent the transmission line, while distributed element models divide the line into smaller segments and use distributed parameters to model its behavior.

3. What is characteristic impedance in a circuit model of a transmission line?

Characteristic impedance is the ratio of voltage to current in a transmission line, and it is represented by the symbol Z0. In a circuit model, it is typically represented by a resistor and is a measure of the line's ability to carry electrical signals without distortion.

4. How do you calculate the characteristic impedance of a transmission line?

The characteristic impedance of a transmission line can be calculated using the formula Z0 = √(L/C), where L is the inductance per unit length and C is the capacitance per unit length of the line. It can also be calculated using the velocity of propagation (v) and the line's distributed parameters: Z0 = √(L/C) = v/(C*L).

5. What is the purpose of a circuit model in transmission line analysis?

The purpose of a circuit model in transmission line analysis is to simplify the complex behavior of a physical transmission line into a more manageable and understandable form. It allows for the prediction of the line's performance and the design of circuits that use transmission lines, such as antennas and filters.

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