Transmission Line Termination Reflections. Current Waves.
The TLs we’ve considered so far were semi-infinitely long. This is useful when trying to understand basic behavior such as V and I wave propagation, for example. Obviously semi-infinite TLs are not very practical. As an example of a practical application, interconnected ICs on a PCB provide “termination” of TLs. We will see in this lecture that terminations of a TL generally will cause the voltage and current waves to (partially) reflect from the load. Consider a TL with resistive termination and the same step input voltage from the last example:
As we saw in the last example, a step voltage is launched at time
t = 0 with amplitude (V0 ./ Rc (Rc + Rs) . Here, the disturbance will propagate to the right along the TL until it encounters RL.
When the disturbance reaches the load, there will be created a new voltage disturbance that propagates to the left (-z direction). This is called the reflected wave.
Voltage Reflection Coefficient
What is the amplitude of the reflected voltage wave? To answer this very important question we begin with the general TL equations from
At the load z = L these two equations evaluate to:
However, at the load we also have the resistor’s lumped element terminal relationship
VL( t)= RL IL(t)
We can see from the figure above that
VL( t)=V( z = L, t) IL(t)= I( z=t)
(Note one subtle point here. Both RHSs above are the sum of +z and -z propagating waves! At z = L, both of these waves sum to give the terminal voltage VL or the terminal current IL .) Now, substitute
Dividing by V+( + (t+ + L / u)) gives
Solving this equation for the ratio V- - /V+ + results in
This ratio is incredibly important to our work with TLs. Consequently, we will define this ratio by the symbol as
which, we can deduce, is the voltage reflection coefficient at the load position z = L. In terms of
.Some general characteristics of the voltage reflection
o Special cases for tL
In this last case, there is no reflection of the wave from the load!
. Note from (8) that the voltage reflection coefficient is only a function of the load resistance and the characteristic resistance of the TL. tL is not dependent on the amplitude of the incident voltage. Therefore, whenever a voltage V + is incident on the load, a reflected voltage V - equal to
is launched that propagates in the -z direction.
. This reflected voltage adds to the incident voltage to produce the total voltage anywhere on the TL.
. When this reflected voltage disturbance reaches the source, there will be another reflection of the voltage wave
where Rs is the source resistance.
. This process of the voltage wave reflecting off the source, propagating down the TL, then reflecting from the load, then propagating down the TL, then reflecting from the source, etc. repeats indefinitely.
Example N13.1. A previously discharged TL shown below is excited by a DC source of 30 V at time t = 0. Sketch the voltage along the TL at times 1 ms, 2.5 ms, 4.5 ms, and 6.5 ms.
The voltage reflection coefficients are:
. At the load, from (8):
. At the source, from :
The time to transit the TL is
See the sketches on the next page for the solution:
. 2 The wave reflects off the load with tL =1/3 . The wave front then propagates to the left. (Can think of this as originating at z = 2L, if you wish.)
. The total voltage equals the sum of the incident and all reflected waves.
. 3 The wave front reflects off the source. Complete reflection with tL=-1.
. 4 The wave front propagates down the TL to the right and then reflects again off the load (tL=1/3).
See VisualEM Example 7.2 Worksheet [set nskip=1, W=20 ms (=10tL)]:
. Animation in time.
. Compare solution on next page at t = 1 ms and t = 2.5 ms.
. Voltage is converging to V( (z,t) ) » 30 V. Expected?
. V( (z,t) )» 30 V everywhere on the TL for t >> tL . This is the “steady state.” Do we expect a nonzero steady state voltage? Yes, since the source remains 30 V for t > 0. For t >>tL , have
In steady state, this is just a simple resistive circuit, maybe two parallel wires interconnecting a battery and a resistor. . In steady state on the TL, notice that
Why is this? Because
So far we’ve only considered the propagation and reflection of voltage waves on a TL. We know, however, that with a voltage wave there must also be an associated current wave: We cannot have one without the other.
Consider once again the TL of Fig. 1 excited by a unit-step voltage source. The current on the TL is expressed by (2). When the source initially turns on at t = 0, we can construct an equivalent circuit at the input to the TL in order to determine the amplitude of the initial current wave amplitude I + . In the case of Vs( t)=V0 u( t) the equivalent circuit is
From this circuit, we can determine that
A current wave with this amplitude I + in (10) then propagates down the TL in Fig. 1 until it encounters the resistive load. Like the voltage wave, this current wave is partially reflected by the load and the remaining energy is “transmitted” into the load resistance RL. We can determine this current reflection amplitude by beginning with (6)
As we learned in Lecture 12, the amplitudes of V+ + and V -- are related to I+ + and I- - as
Substituting these relationships into (11), we find that
This result in (14) tells us that the current reflection coefficient at the load is the negative of the load voltage reflection coefficient. It turns out that this is a general result: Current reflection coefficients are the negative of the corresponding voltage reflection coefficients