Questions and confusions on transmission line theory and lumped element model

Question

I want to model a simple signal transmission system by considering it as travelling through a transmission line. Here is the system: The original signal generated by the transducer is 0 to 10V step function like signal with very low 3Hz fundamental frequency. So the signal has high harmonics which matters since it is squarewave-like step function signal. This is a general question which is related to my previous question: Passive low-pass filtering question for a transducer output

Basically I have confusion when to use which model. Transmission line theory or lumped model... So I want to model this in LTspice as a Lumped LC Model or something in that sort to see phenomena like ringing. To achieve an approximate model I will write about each sections from left to the right of the circuit:

Questions when trying to use lumped model:

1-) Rload in the above illustration represents a scope or very high input impedance of a measurng instrument which is known. Rout in the above figure is the output of a transducer which is unknown. How to measure that if it is not a function generator with a known output impedance? I will use the following method to find the output resistance: http://www.qsl.net/w/w2aew//youtube/How_to_measure_output_impedance.pdf Do I really need the output impedance or can I take it roughly some hundreds of ohms?

2-) Coaxial-cable in the above illustration is a standard 50 Ohm intrinsic impedance BNC cable. I know its length and lets say it is 20 meters long. Since we use lumped element model we will not use 50 Ohm right? And capacitance and the inductance will vary with length? In other words, how can I model this cable in LTspice?

A question when trying to use impedance matching in transmission line theory:

Secondly let's forget about the LTspice and lumped model and just assume we only want to achieve impedance matching. Here is the model considering impedances:

So now we have a transducer output impedance which is Rout=Zout, coaxial cable impedance which is 50 Ohm, and we have a load Rload=Zload which is lets say 100M. So in this case to achieve impedance matching throughout the line I need a 50 Ohm resistor in parallel with Rload just before Rload to make the Rload 50 Ohm. And I also need to know the Rout and I need to add a series or parallel resistor to it such that its equivalent or Thevenin would be 50 Ohm. Is this method/model right? If it is than I will have the following issue:

The 50 Ohm parallel resistor will load the transducer and I will have more error right? It seems to me for sending data this might not be problem but in this case the signal's voltage level is important. What would you suggest in this case?

Answer 1

Coaxial-cable in the above illustration is a standard 50 Ohm intrinsic impedance BNC cable. I know its length and lets say it is 20 meters long. Since we use lumped element model we will not use 50 Ohm right? And capacitance and the inductance will vary with length? In other words, how can I model this cable in LTspice?

First, when we talk about transmission lines, we talk about characteristic impedance. "Intrinsic impedance" is not a term that has any specific meaning in the area of transmission lines.

A lumped element model of a transmission line with 50 ohms characteristic impedance does not involve a 50 ohm resistive element in series. Characteristic impedance describes the ratio between voltage and current in the travelling wave that can propagate along the line. It doesn't cause any power loss like a series resistance would.

It might involve a series of capacitive and inductive elements in a pi or T section arrangement. A pi-section model of a lossless unbalanced line would look like this:

^{simulate this circuit – Schematic created using CircuitLab}

C1 and C6 would have half the value of C3, C4, and C5, because the intermediate capacitors actually each represent the shunt legs of two pi sections in parallel. The total capacitance should add up to the line's capacitance per unit length times its length. The total inducance should add up to the line's inductance per unit length times its length.

Obviously this model will fail when the frequency gets too high, because the first and last capacitance elements will effectively short out signals approaching in the forward and reverse directions. By increasing the number of sections you can reduce the capacitance per section in the model, and so increase the frequency where this issue occurs.

So now we have a transducer output impedance which is Rout=Zout, coaxial cable impedance which is 50 Ohm, and we have a load Rload=Zload which is lets say 100M.

The load impedance is not particularly realistic. Typical scope inputs are 1 or 10 Megohms. Scopes designed for measuring reasonably high frequencies will usually have an option to program an input impedance of 50 ohms.

So in this case to achieve impedance matching throughout the line I need a 50 Ohm resistor in parallel with Rload just before Rload to make the Rload 50 Ohm.

Yes, if your scope doesn't have a 50 ohm input impedance option, and reflections become an issue, you can add a 50-ohm parallel resistance at the input to reduce these reflections. It will also reduce the signal seen by the scope.

And I also need to know the Rout and I need to add a series or parallel resistor to it such that its equivalent or Thevenin would be 50 Ohm.

You don't strictly need to match both ends of the transmission line. If you match one end very well it will eliminate reflections that reach that end, so you won't see ringing from multiple reflections.

The 50 Ohm parallel resistor will load the transducer and I will have more error right? It seems to me for sending data this might not be problem but in this case the signal's voltage level is important. What would you suggest in this case?

You could either provide a buffer amplifier at the transducer to produce a signal with low output impedance.

You could move the scope closer to the transducer so that the line can be shorter and impedance matching not needed.

You could provide an RC filter at the transducer output to reduce the edge transition speed so that less high frequency signal is present and impedance matching is not needed.