Question 2.4: IMPEDANCE MEASUREMENT WITH A SLOTTED LINE The following two-...
IMPEDANCE MEASUREMENT WITH A SLOTTED LINE The following two-step procedure has been carried out with a 50 Ω coaxial slotted line to determine an unknown load impedance:
1. A short circuit is placed at the load plane, resulting in a standing wave on the line with infinite SWR and sharply defined voltage minima, as shown in Figure 2.14a. On the arbitrarily positioned scale on the slotted line, voltage minima are recorded at z=0.2 cm,2.2 cm,4.2 cm.
2. The short circuit is removed and replaced with the unknown load. The standing wave ratio is measured as SWR = 1.5, and voltage minima, which are not as sharply defined as those in step 1, are recorded at
z = 0.72 cm, 2.72 cm, 4.72 cm,
as shown in Figure 2.14b. Find the load impedance.
Knowing that voltage minima repeat every\ \frac{\lambda}{2} , we have from the data of step 1 that \ \lambda=4.0 cm. In addition, because the reflection coefficient and input impedance also repeat every \ \frac{\lambda}{2}, we can consider the load terminals to be effectively located at any of the voltage minima locations listed in step 1. Thus, if we say the load is at 4.2 cm, then the data from step 2 show that the next voltage minimum away from the load occurs at 2.72 cm, giving \ l_{\min }=4.2-2.72=1.48 cm=0.37\lambda .
Applying (2.58)–(2.60) to these data gives
\ \left|\Gamma \right| =\frac{1.5-1}{1.5+1} =0.2,\ \theta =\pi +\frac{4\pi }{4.0} \left(1.48\right) =86.4° ,
so
\ \Gamma =0.2e^{ j86.4°}= 0.0126 + j0.1996.The load impedance is then
\ Z_{L}=50\left(\frac{1+\Gamma }{1-\Gamma } \right) =47.3 + j19.7 \Omega .
For the Smith chart version of the solution, we begin by drawing the SWR circle for SWR = 1.5, as shown in Figure 2.15; the unknown normalized load impedance must lie on this circle. The reference that we have is that the load is \ 0.37\lambda away from the first voltage minimum. On the Smith chart the position of a voltage minimum corresponds to the minimum impedance point (minimum voltage, maximum current), which is the horizontal axis (zero reactance) to the left of the origin. Thus, we begin at the voltage minimum point and move \ 0.37\lambda toward the load (counterclockwise), to the normalized load impedance point,
\ z_{L}=0.95+j0.4, as shown in Figure 2.15. The actual load impedance is then \ Z_{L}=47.5+j20\Omega , in close agreement with the above result using equations. Note that, in principle, voltage maxima locations could be used as well as voltage minima positions, but voltage minima are more sharply defined than voltage maxima and so usually result in greater accuracy.
