Input Impedance of transmission line


Input Impedance of Transmission Lines Excitation and the Source Conditions.

Keeping with standard circuits concepts, we can define the TL impedance at any position z as simply the ratio

The total voltage and current are themselves the sum of “+” and “-” propagating waves.
It is helpful to have an analytical expression for the input impedance of an arbitrarily terminated TL.

Towards this objective, we saw in the previous two lectures that the voltage and current everywhere on a homogeneous TL are

We can readily construct according to (1) an input impedance expression for a TL of length L by dividing (2) and (3) for some arbitrary load reflection coefficient TL at z = 0:

Such that

Substituting for TL and simplifying gives

This is the input impedance for a lossless TL of length L and characteristic impedance Z0 with an arbitrary load ZL. It is easy to see from (6) that one function of a TL can be as an impedance transformer.
Input Impedance for Special Loads
Three special cases for the input impedance in (6) are:
1. With an open circuit load ( ZL=¥), (6) yields
Zin = - jZ0 cot ((bL))
In other words, the input impedance is purely reactive
Zin= jXin where ( ) Xin= -Z0 cot ((bL)
A plot of this input reactance is:

Notice that any value of inductive or capacitive reactance can be achieved at the input of the TL. All we need to do is adjust the electrical length ((bL) of the TL to change this value of input reactance. The electrical length can be changed either by varying the signal frequency or the physical length of the TL. This is an example of the impedance transformation capability of TLs.
2. With a short circuit load ( Z0=0 ), (6) yields
Zin=jZ0tan (b L) [beta]
This input impedance is also purely reactive
Zin= jXin where ( ) in 0 Xin= Z0 tan (b L)
A plot of this input reactance is:

Again, we see here (as we did earlier with the open circuit load) that any value of reactive impedance can be realized at the input to the TL simply by adjusting the electrical length of the TL.
3. With the resistive load ZL= Z 0, (6) yields Zin= Z0 [beta]
The input impedance is Z0 regardless of the length of the TL.
Again, note that both input impedances (7) and (9) are purely reactive, which is expected since neither type can dissipate energy, assuming lossless TLs.
Excitation of Transmission Lines
For the sinusoidal excitation of TLs,

the total voltage at z = -L can be found using the input impedance of the TL and this equivalent circuit at the input:

By voltage division, we can compute Vin from this equivalent circuit at the TL input as

This process is illustrated in the following example.
Example N18.1: Determine an expression for the voltage at the input to the following TL (open circuit load) assuming Rs = Z0:

To calculate the input voltage Vin , we’ll first determine the input impedance Zin . To repeat, this is the effective impedance seen at the TL input terminals z = -L seen looking towards the load. For an open circuit load, we have from (7) that

An equivalent circuit can now be constructed at the input to the
TL by using Rs (= Z0) and Zin as

From voltage division

which is the desired quantity.
This circuit voltage Vin is also the voltage on the TL at z = -L.
That is, from (4) above with T L = - 1+ V( z= -L)2V0+cos(-bL)
Since ( ) V in=V( z= -L ), we can equate these two voltages giving

More often than not, expressions of this type are used to Determine V0+ + in terms of Vs and Rs