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.

## Best Answer

Suppose we view the short transmission line as a two port network.

The input voltage and current are the same as the output voltage and current (magnitude and phase relationship).

The input voltage (Vin) and current (Iin) will be determined by the termination load (ZL) and the source impedance (Zs). It doesn't 'see' the lumped LCR components of the transmission wire (or at most it sees the resistance of the wire), just the terminating load (ZL).

Its as if the transmission wire doesn't exist and you directly connect the load impedance to the source.

The output voltage (Vout) and current (Iout) are the values that would be obtained connecting the output impedance directly to the source.

Its when we increase the length of the transmission wire(in relationship to the wavelength of the signal) we start to see thatVout and Iout are no longer equal to Vin and Iinso the original simplification no longer stands.Voltage and current change phase and reflection becomes more important on an unbalanced line.

We use the short wire approximation all the time without even realising it. Any cable connecting any form of AC is subject to the same basic physics. Its only at very high frequencies do the physical lengths of the cable start to become significant in terms of transmission line theory and we need to consider termination impedances (source and load).

As Andy correctly points out there are always reflections on a mismatched line but in the case of the short wire they are barely measureable.