Generalized Reflection Coefficient. Crank Diagram. VSWR.
As we saw in the previous lecture, for a lossless TL with an arbitrary load and this chosen coordinate system
the voltage and current on the TL can be written as
And
In these expressions
is the voltage reflection coefficient at the load. We can “generalize” the concept of voltage reflection coefficient to be the ratio of the (complex) amplitudes of the -z and +z traveling voltage waves at any point along the TL. That is, we define the generalized reflection coefficient for a lossless TL as
Now, substituting
It is worthwhile to memorize (3)–(5). These equations are the foundation upon which you can understand and solve sinusoidal steady state TL problems.
Crank Diagram
Taking the magnitude of
|V(Z)| = |V_{0}| | 1+T(z)|
In the “complex T( z)” plane, the quantity |1+T(z)| can graphically be interpreted as
As we move along the TL from the load in the -z direction:
1. From (3) we see that |T(z)|=|T_{L} L ^{e}j2b z|=|T_{L}, which is constant. Hence, T(z)) traces a circle in this plane.
2. There is CW rotation, rather than CCW, because of the factor e j2b z in (3) and our movement in the –z direction when moving towards the source. Both of these facts are illustrated in the crank diagram shown above.
Also notice from the crank diagram that we obtain the same T(z) value every 2b z = 2p or b z =p rad movement along the TL. Consequently, from (6) we will measure the same |V(z)| every b z =p rad movement along any (lossless) TL. This makes sense since we’re only looking at the magnitude of the voltage. Why is |V(z)| important? Because this quantity is “easy” to measure accurately. For example, using a square law detector.
Voltage Standing Wave Ratio
As we’ve seen repeatedly in our studies of TLs, there is generally some amount of reflection of voltage and current waves from loads attached to a TL. To help quantify the amount of interference that exists on a TL, we define the voltage standing wave ratio (VSWR) as
where | V( z)|_{max} and |V(z)|_{min} are the maximum and minimum voltage magnitudes, respectively, found anywhere on a long TL. Using the crank diagram above, we can easily determine expressions for these quantities. Specifically, we can see that
Substituting these into the definition of VSWR in
From this expression, we can definitely see that VSWR is intimately related to the amount of reflection at the load (through T_{L} ) and the subsequent interference on the TL. Special cases:
1. If Z_{L}=0 (short circuit load) then T_{L}=1 Consequently,
|T_{L}|Þ1 VSWR = ¥
2. If Z _{L}= (open circuit load) then T_{L}=1 Consequently,
|T_{L}|¥1 VSWR = ¥
3. If Z_{L}= Z_{0} (matched load) then 0 T_{L}=1 Consequently,
|T_{L}|Þ0 VSWR = 1
Regardless of the load, 1£ VSWR £ ¥.
Example N19.1: For the TL shown below, determine the VSWR on the TL, the time averaged power delivered to the load, and the voltage at the load.
For this TL:
Therefore,
To determine the time averaged power delivered to the load, we’ll compute the time averaged power at the TL input. Because the TL is lossless, these two quantities will be the same. We can construct an equivalent lumped element circuit at the TL input as:
From (6) in the previous lecture
With
Then
Very curious result! Z_{L} is complex, but Z_{in} is purely real. This is an example that TLs act as impedance transformers. Referring to the equivalent circuit above,
Since this is a lossless TL, all the time averaged power at the input to the TL will be delivered to the load. Therefore,
Or
In this example, the time averaged power delivered to the load
To determine the voltage at the load, we begin with (4)
V (z)=V^{+}_{0+e}^{-jb z}[1+T(z)]=V^{+}_{0+e}^{-j z} [1+T _{0+e}^{-j2b z}]
The only unknown quantity in this equation is o V +. We can determine this complex constant by applying the boundary condition at the source location on the TL. In particular, at the input to the TL
Solving this equation for V_{0}^{+} we find
Substituting this result back into
Or
With VisualEM
1. Confirm calculations in this example using the “Section 7.3/Problem 7.3.1” worksheet. This is a useful TL calculator.
2. See the plot of the voltage magnitude and phase. VSWR: what does it mean? Also see the time domain amplitude of oscillations and the time delay with respect to the source.
3. To better understand this TL behavior, also see the “Example 7.5” worksheet. Enter the revised numbers for this example:
. Animate the voltage on the TL,
. This total voltage is the sum of waves traveling in opposite directions on the TL. See the second animation in this worksheet:
. The maximum amplitude occurs when the +z and –z waves add “in phase.”
.The minimum amplitude occurs when the +z and –z waves add “out of phase.”