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Sinusoidal Steady State Analysis of Transmission Lines

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

This 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 ZC, 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]).

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

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

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A Dash of Maxwell’s: A Maxwell’s Equations Primer – Part Two

Maxwell’s Equations are eloquently simple yet excruciatingly complex. Their first statement by James Clerk Maxwell in 1864 heralded the beginning of the age of radio and, one could argue, the age of modern electronics.

(1.1a)

 (1.1b)

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

Note: In [1] these constants were denoted as  and . 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  and . 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 and . 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

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(1.2a)

while the backward-traveling voltage wave is given by

(1.2b)

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

(1.3a)

Thus,

(1.3b)

From Eq. (1.3b) we obtain

(1.4)

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

(1.5a)

or

(1.5b)

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

(1.6a)

or

(1.6b)

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

(1.7a)

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

(1.7b)

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

(1.8a)

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

(1.8b)

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),

(1.9)

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

(1.10a)

(1.10b)

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

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

(1.11a)

(1.11b)

or

(1.11c)

(1.11d)

or

(1.11e)

(1.11f)

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

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

(1.12a)

(1.12b)

or

(1.12c)

(1.12d)

or

(1.12e)

(1.12f)

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

(1.13a)

(1.13b)

or

(1.13c)

(1.13d)

Next, let us turn our attention to the undetermined constants   and . 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

(1.14a)

(1.14b)

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

(1.15)

and thus

(1.16)

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

(1.17)

and thus

(1.18)

These two undetermined constants,    and , 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

(1.19a)

(1.19b)

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

(1.20)

and thus

(1.21)

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

(1.22)

and thus

(1.23)

Observation: To obtain the voltage or current at any location z, away from the source, we need the knowledge of the undetermined constants,  and , (or at least ). To obtain the undetermined constant,  and , 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, , shown in Figure 2, is defined as the ratio of the total voltage to the total current at that point.

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

(2.1)

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

(2.2a)

or

(2.2b)

or

(2.2c)

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

(2.3a)

or

(2.3b)

or

(2.3c)

Since the constants, and , 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,

(2.4)

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

(2.5)

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

(2.6a)

or

(2.6b)

or

(2.6c)

Now,

(2.7a)

(2.7b)

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

(2.8a)

or, using the Euler’s formulas

(2.8b)

leading to

(2.9a)

or equivalently,

(2.9b)

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

(3.1a)

(3.1b)

Now, from the knowledge of  and we can determine the constants and from

(3.2a)

(3.2b)

or

(3.2c)

(3.2d)

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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