Sinusoidal Steady State Analysis of Transmission Lines

EMC concepts explained

Part II: Voltage, Current, and Input Impedance Calculations – Circuit Model 1

his is the second of the three tutorial articles devoted to the frequency-domain analysis of a lossless transmission line. In the previous article, [1], the general solution for the voltage and current in sinusoidal steady state was derived and the concept of the input impedance to the line was presented. This article shows numerous methods of calculating the voltage, current, and input impedance at various locations on the transmission line, using the Circuit Model 1, [1], described next.

1. Voltage and Current at Any Location z Away from the Source

Consider a lossless transmission line with the characteristic impedance Z_c, driven by the source located at z = 0 and terminated by the load located at z = L, as shown in Figure 1. (This circuit was referred to as Circuit Model 1, in [1]).

The voltage and current at any location z away from the source were derived in [1] as

where β is the phase constant of the sinusoidal voltage source and the V_z+ and V_z– are yet to be determined constants.

Note: In [1] these constants were denoted as V+ and V-. Here, we use a different notation to distinguish between the constants for two different circuit models. Using Model 1, shown in Figure 1, we move from the source at z = 0 to the load at z = L, and use constants V_z+ and V_z–. In Model 2, discussed in the next article, we move from the load at d = 0 to the source at d = L, and use constants V_d+ and V_z–. These two sets of constants are different.

The solutions in Eqns. (1.1) consist of the forward- and backward-traveling waves. The forward-traveling voltage wave is described by

while the backward-traveling voltage wave is given by

Using these two waves, we define the voltage reflection coefficient at any location z, as the ratio of the backward-propagating wave to the forward-propagating wave

Figure 1: Transmission line circuit with the source located at z = 0 and the load at z = L

Thus,

From Eq. (1.3b) we obtain

Utilizing Eq. (1.4) in Eq. (1.1a) gives

Utilizing Eq. (1.4) in Eq. (1.1b) gives

Equations (1.5b) and (1.6b) express voltage and current at any location z, away from the source, in terms of the unknown constant V_z+ and the voltage reflection coefficient Γ at any location z away from the source.

Let us return to this reflection coefficient, given by Eq. (1.3b). Letting z = L, we obtain the voltage reflection coefficient at the load

Note that the load reflection coefficient can always be obtained directly from the knowledge of the load and the characteristic impedance of the line as

Let us return again to the reflection coefficient given by Eq. (1.3b).

Thus, the voltage reflection coefficient at any location z, away from the source, can be expressed in terms of the load reflection coefficient as

Equation (1.8b) can be used to determine the voltage reflection coefficient at the input to the line, i.e., at z = 0, (we will need it shortly),

Utilizing Eq. (1.8b) in Eqns. (1.5b) and (1.6b) gives

Equations (1.10) express voltage and current at any location z, away from the source, in terms of the unknown constant V_z+ , and the load reflection coefficient.

In summary, the voltage and current at any location z, away from the source, can be obtained from

The last set of equations is perhaps the most convenient since the load reflection coefficient, Γ_L, can be obtained directly from Eq. (1.7b) and the only unknown in this set is the constant V_z+.

The three sets of equations (1.11) can be used to determine the voltage and current at the input to the line, and at the load.

Letting z = 0, in Eqns. (1.11) we obtain the voltage and current at the input to the line as

Letting z = L, in Eqns. (1.11) we obtain the voltage and current at the load as

Next, let us turn our attention to the undetermined constants V_z+ and V_z–. These constants can be determined from the knowledge of the voltage and current at the input to the line.

Eqns. (1.12a) and (1.12b) can be rewritten as

Adding Eqns. (1.14a) and (1.14b) gives

and thus

Subtracting Eq. (1.14b) from Eq. (1.14a) gives

and thus

These two undetermined constants, V_z+ and V_z–, can alternatively be obtained from the knowledge of the voltage and current at the load.

Eqns. (1.13a) and (1.13b) can be rewritten as

Adding Eqns. (1.19a) and (1.19b) gives

and thus

Subtracting Eq. (1.19b) from Eq. (1.19a) gives

and thus

Observation: To obtain the voltage or current at any location z, away from the source, we need the knowledge of the undetermined constants, V_z+ and V_z–, (or at least V_z+). To obtain the undetermined constant, V_z+ and V_z–, we need the knowledge of the voltage and current at the input to the line, or at the load. We resolve this stalemate by introducing the concept of the input impedance to the line.

2. Input Impedance to the Line at any Location z away from the Source

At any location z, away from the source, the input impedance to the line, Z_in, shown in Figure 2, is defined as the ratio of the total voltage to the total current at that point.

Since the total voltage and current at any location z away from the source can be obtained from the three different sets of Eqns. (1.11), it follows that the input impedance to the line, at any location z away from the source can be obtained from

Letting z = 0, in Eqns. (2.2) we obtain the input impedance to the line at the input to the line as

Figure 2: Input impedance to the line at any location z away from the source

Since the constants, V_z+ and V_z–, are still unknown, in the calculations of the input impedance to the line at the input to the line, we are left with the remaining two equations, (2.3b) and (2.3c).

Since,

at this point, we effectively have just one equation (2.3c) to determine the input impedance to the line at the input to the line. Towards this end, we first determine the load reflection coefficient from

and then use Eq. (2.3c) or Eq. (2.9b), derived next, to calculate the input impedance to the line at the input to the line.

There is one more useful set of formulas for obtaining the input impedance to the line at the input to the line. Using Eq. (2.5) in Eq. (2.3c) we get

Now,

Utilizing Eqns. (2.7) in Eq. (2.6c) we get

or, using the Euler’s formulas

leading to

or equivalently,

3. Voltage and Current at the Input to the Line

At the input to the line, we have a situation depicted in Figure 3.

Figure 3: Equivalent circuit at the location z = 0

It is apparent the voltage and current at the input to the line can be now obtained from

Now, from the knowledge of V and I we can determine the constants V_z+ and V_z– from

At this point we can obtain the voltage, current, or impedance at any location z away from the source using the previously derived equations.

In the next article, we will analyze the circuit where we move from the load is located at d = 0 towards the source located at d = L (Model 2). Such a circuit is shown in Figure 4.

Figure 4: Transmission line circuit with the load located at d = 0 and the source at d = L

References

Adamczyk, B., “Sinusoidal Steady State Analysis of Transmission Lines – Part I: Transmission Line Model, Equations and Their Solutions, and the Concept of the Input Impedance to the Line,” In Compliance Magazine, January 2023.

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Dr. Bogdan Adamczyk is professor and director of the EMC Center at Grand Valley State University (http://www.gvsu.edu/emccenter) where he regularly teaches EMC certificate courses for industry. He is an iNARTE certified EMC Master Design Engineer. Prof. Adamczyk is the author of the textbook “Foundations of Electromagnetic Compatibility with Practical Applications” (Wiley, 2017) and the upcoming textbook “Principles of Electromagnetic Compatibility with Laboratory Exercises” (Wiley 2022). He can be reached at adamczyb@gvsu.edu.

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