Representing a Transmission Line as a 2-Port Network

In this blog post I will derive equations representing a transmission line of characteristic impedance Zo and of length 'l' as a two-port network.  As a result, there will be a total of four different pairs of equations, each pair representing the two-port network in a different way.

These pairs of equation will be:

1.  V and I at Port 2, in terms of V and I at Port 1, for a lossy transmission line.

2.  V and I at Port 1, in terms of V and I at Port 2, for a lossy transmission line.

3.  V and I at Port 2, in terms of V and I at Port 1, for a lossless transmission line.

4. V and I at Port 1, in terms of V and I at Port 2, for a lossless transmission line.

(These last two sets of equations are special cases of the first two sets).

The image below shows these voltages and currents for both the transmission line and its two-port model.

The first two sets of equations that will be derived are summarized by the two matrix equations, below:

To derive the two equations in the first set of these equations, let's start with some fundamental definitions...

 

Some fundamentals:

Assume that the transmission line has on it both a forward voltage wave (traveling from left to right) and an independent reverse voltage wave (traveling from right to left).  At any point on the transmission line these forward and reverse voltage waves create forward and reverse current waves:

If we were to measure the voltage at any point on the line, the voltage we would measure at that point will be the sum of the forward voltage and the reverse voltage at that point.  

Note that the forward voltage and the reverse voltage each has its own amplitude and phase at that point.

Similarly, the current at that point will be the difference between the forward current and the reverse current (because the currents are flowing in opposite directions). 

Recognize that for either the forward wave or the reflected wave, their voltage and currents follow ohms law, independently of the other wave (because of the Principle of Superposition).  E.g. I⁺ = V⁺/Zo and I^-=V^-/Zo, where Zo is the transmission line's characteristic impedance.

So we can write equations for the Total Voltage and Current at each end of the transmission line in terms of the forward and reverse voltages at those points:

We can also express the forward and reverse voltages at either end of the transmission line as functions of the Total Voltage and Current at either end:

I will use equations 1, 2, 3, and 4 (in the figure, above) to derive the two-port network equations, but before I go any further in this derivation, let me first define a transmission line's "Propagation Constant."

Sidebar on the exponential terms e^-γl and e^γl :

Assuming a sinusoidal signal on the transmission line, the exponential term e^-γl  (or e^γl) represents the effect on a sine wave's amplitude and phase, due to the transmission line's Propagation Constant, γ, as that wave travels along the transmission line.

The other variable in the exponent is 'l', which represents the length that the wave has traveled (or will travel) along the transmission line.

The Propagation Constant is a complex number: γ = α + jβ, where α represents a "real" change in signal amplitude (either positive or negative, i.e. attenuation or gain), and jβ represents a phase shift of the sinusoidal signal, either a positive phase shift or a negative phase shift.  

Both of these values are in terms of unit-length, and so when substituted into the exponential, multiplying α by the length of the line gives overall attenuation (or gain, depending upon reference point), and multiplying jβ by length gives phase-shift for the wave's journey along a defined length of line.

If we were to take a snapshot in time and simultaneously compare the Forward Voltage V⁺ at point 1 with the Forward Voltage V⁺ at point 2, we would see that the voltage at point 2 has a negative phase shift compared to the phase of  V⁺ at point 1, because the voltage at point 2 represents an earlier version of  V⁺ that had been at point 1 at an earlier point in time, compared to the V⁺ that is currently at Point 1. 

We would also see that the amplitude of V⁺ at point 2 has been attenuated relative to its original amplitude at point 1 due to transmission line loss as it traveled from left to right along the line.   Thus, γ's "α" term (in the Propagation Constant's equation  γ = α + jβ) is negative to represent this loss at point 2 compared to the amplitude at point 1.

The figure below demonstrates both the attenuation and the phase shift at point 2, compared to point 1, for the Forward moving voltage wave:

With the same "snapshot in time," we can also simultaneously compare the Reverse Voltage V^- at point 1 with the Reverse Voltage V^- at point 2.  

We would see that the voltage at point 2 has a positive phase shift compared to the phase of  V^- at point 1, because the voltage at point 2 represents a later version of V^-  that has yet to arrive at point 1, compared to the phase of V^- that is currently at Point 1. 

We would also see that the amplitude of V^- at point 2 has not yet been attenuated compared to the amplitude of V^- at point 1, because it has not yet traveled from right-to-left along the line, and thus it has not yet been attenuated by line loss.  Therefore, γ's "α" term (in the Propagation Constant equation  γ = α + jβ) is positive for V^- at point 2 , not negative, because the amplitude at point 2 is larger compared to the amplitude at point 1.

The figure below demonstrates both the amplitude and phase at point 2, compared to point 1, for the Reverse moving voltage wave.:

With this information and the equations, above, we can derive network equations that represent a lossy transmission line of length 'l'.

Lossy Transmission Line Equations for V₂ and I₂ in terms of V₁ and I₁:

From the previous two figures that describe phase-shift and attenuation of V⁺ and V^- at Point 2 compared to our reference at Point 1, we can create the identities represented by equations 5 and 6 in the figure, below, and, using these, represent V₂ as a function of Point 1's V⁺ and V^- :

Similarly, we can represent the current I₂ at Point 2 as function of Point 1's V⁺ and V^- :

Next we substitute equations 1 and 2 (introduced earlier, above) into the V⁺ and V^- terms of these last two equations, rearrange, and get the following two equations:

Equations 9 and 10 represent V₂ and I₂ in terms of V₁ and I₁, and in themselves they satisfactorily represent the lossy transmission line as a two-port network (from the perspective of calculating V₂ and I₂ from V₁ and I₁).

But we can also represent these equations using Hyperbolic functions:

The matrix form of these last two equations is:

Lossy Transmission Line Equations for V₁ and I₁ in terms of V₂ and I₂:

In the same way that I defined V₂ and I₂ in terms of V₁ and I₁, I can define V₁ and I₁ in terms of V₂ and I₂.  (In fact, this is the typical form in which equations representing the transmission line as a two-port network appear).

This time, we begin by considering the Forward Voltage V⁺ and the Reverse Voltage V^-  from Point 2 (rather than from Point 1, as we did for the first derivations).

This will give us V⁺ and V^- at Point 1 in terms of  V⁺ and V^- at Point 2, as illustrated, below.

First, V⁺ at Point 1 in terms of  V⁺ at Point 2:  

Next, V^- at Point 1 in terms of  V^- at Point 2:

We can use these equations to represent the total Voltage at Port 1 in terms of V⁺ and V^- at Point 2:

Repeat for the total Current at Port 1:

Then, substituting into these two equations equations 3 and 4 (presented earlier) and rearrange.  We arrive at two equations expressing V₂ and I₂ in terms of V₁ and I₁:

The above form is a sufficient representation of these two equations, but often they will be represented in terms of Hyperbolic functions:

The matrix form of these last two equations is:

The above 2x2 matrix is in the form of an ABCD matrix (note that I2's direction is defined as exiting the network, not entering it).

Two-Port Network Equations for Lossless Lines:

Sometimes it is convenient to assume a transmission line is lossless.  What do I mean by lossless?

Recall the equation for the Propagation Constant: γ = α + jβ. If the line is lossless, α = 0 and thus e^-γl becomes e^-jβl. 

Similarly, e^γl becomes e^jβl. 

In either case, because α = 0, there is no attenuation as a wave travels along the transmission line, only phase shift.  With jβ and -jβ in the exponents of equations 9, 10, 17, and 18, we can use Euler's Formula and substitute cos and sin for the exponential terms in these four equations.

You can see equations 23 and 24 used in C. L. Ruthroff's article, "Some Broad-Band Transformers," in the August, 1959 issue of the "Proceedings of the I.R.E."

In matrix form, these equations are:

Note that these equations can also be expressed in terms of the Hyperbolic functions cosh and sinh in lieu of cos and sin.

Given γl = jβl, and using cos-cosh identities: cosh(γl) = cosh(jβl) = cos(j*jβl) = cos(-βl) = cos(βl).

The sinh term is similar:  sinh(γl) = sinh(jβl) = -j*sin(j*jβl) = -j*sin(-βl) = j*sin(βl).

Resources:

For further reading...

"Some Broad-band Transformers", C.L. Ruthroff, Proceedings of the IRE, August, 1959

"New Method of Impedance Matching in Radio-Frequency Circuits", G. Guanella, The Brown Boveri Review, September, 1944

"Transmission Line Transformers," Chapter Six of Radio Frequency Circuit Design, W. Alan Davis, Krishna Agarwal, John Wiley & Sonse, Inc. 2001

"A novel topology of Broad-band Coaxial impedance transformer", Centurelli, Piatella, Tommasino, Trifiletti, Proceedings of the 40th European Microwave Conference"

Phasors and Transmission Lines


And viewing...

Transmission Line YouTube Series, Professor Gregory D. Durgin, Georgia Tech.  Starting with this video:  https://www.youtube.com/watch?v=7Oz1sazpekM


Standard Caveat:

As always, I might have made a mistake in my equations, assumptions, drawings, or interpretations.  If you see anything you believe to be in error or if anything is confusing, please feel free to contact me or comment below.

And so I should add -- this information is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.