# Electromagnetic Field Theory: A Problem Solving Approach Part 65

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50 Vosinct Zo = 50 z= -I Z, = 50(1 + jl z=0 Point i= I Z,(z =-) l+j 2-j (1 -j) 4+.2j I IjZo/Vo 0.447 0.316 0.632 0.707 26.60 -18.4* -18.4" 8.1" Figure 8-20 (a) The load impedance at z = 0 reflected back to the source is found using the (b) Smith chart for various line lengths. Once this impedance is known the source current is found by solving the simple series circuit in (c). 615 616 8-4-4 Guided Electromagnetic Waves Standing Wave Parameters The impedance and reflection coefficient are not easily directly measured at microwave frequencies. In practice, one slides an ac voltmeter across a slotted transmission line and measures the magnitude of the peak or rms voltage and not its phase angle. From (6) the magnitude of the voltage and current at any position z is j (z) = IVl I +r(z)l (23) I f(z)1 = Yol V+1 I - r(z)I From (23), the variations of the voltage and current magnitudes can be drawn by a simple construction in the r plane, as shown in Figure 8-21. Note again that I V+J is just a real number independent of z and that Ir(z)l 5 1 for a passive termination. We plot II + r(z)l and II - F(z)I since these terms are proportional to the voltage and current magnitudes, respectively. The following properties from this con- r(z= 0) Towards generator (z < 0) = rL e +2i Figure 8-21 The voltage and current magnitudes along a transmission line are respectively proportional to the lengths of the vectors I1 +F(z)| and I I- (z)J in the complex r plane. Arbitrary Impedance Terminations 617 struction are apparent: (i) The magnitude of the current is smallest and the voltage magnitude largest when F(z)= 1 at point A and vice versa when r(z)= -1 at point B. (ii) The voltage and current are in phase at the points of maximum or minimum magnitude of either at points A or B. (iii) A rotation of r(z) by an angle ir corresponds to a change of A/4 in z, thus any voltage (or current) maximum is separated by A/4 from its nearest minima on either side. as in By plotting the lengths of the phasors I 1 ± F(z)I, Figure 8-22, we obtain a plot of what is called the standing wave pattern on the line. Observe that the curves are not sinusoidal. The minima are sharper than the maxima so the minima are usually located in position more precisely by measurement than the maxima. From Figures 8-21 and 8-22, the ratio of the maximum voltage magnitude to the minimum voltage magnitude is defined as the voltage standing wave ratio, or VSWR for short: I (z)m= 1+IL_ = VSWR iN(z), min 1- rL (24) The VSWR is measured by simply recording the largest and smallest readings of a sliding voltmeter. Once the VSWR is measured, the reflection coefficient magnitude can be calculated from (24) as VSWR- 1 IrLI=VSWR VSWR + 1 The angle 4 (25) of the reflection coefficient rL =IIrL eIw (26) can also be determined from these standing wave measurements. According to Figure 8-21, r(z) must swing clockwise through an angle 0 + ir as we move from the load at z = 0 toward the generator to the first voltage minimum at B. The shortest distance din, that we must move to reach the first voltage minimum is given by 2kdmin = + r (27) 1 (28) or ir =4 A 618 Guided Electromagnetic Waves Voltage Current r, = 0 VSWR = 1. ,r= 0.5e Figure 8-22 the VSWR. 11 + PI(2)1 /d4 Voltage and current standing wave patterns plotted for various values of A measurement of dmin, as well as a determination of the wavelength (the distance between successive minima or maxima is A/2) yields the complex reflection coefficient of the load using (25) and (28). Once we know the complex reflection coefficient we can calculate the load impedance ArbitraryImpedance Transformations 619 from (7). These standing wave measurements are sufficient to determine the terminating load impedance ZL. These measurement properties of the load reflection coefficient and its relation to the load impedance are of great importance at high frequencies where the absolute measurement of voltage or current may be difficult. Some special cases of interest are: (i) (ii) Matched line-If FL =0, then VSWR= 1. The voltage magnitude is constant everywhere on the line. Short or open circuited line-If I rLI = 1, then VSWR= oo. The minimum voltage on the line is zero. The peak normalized voltage Ii(z)/V+I is 1+ I LI while the minimum normalized voltage is 1-I r I. (iv) The normalized voltage at z =0 is I + r. I while the normalized current Ii(z)/ Yo V+ at z = 0 is )I -LI. (v) If the load impedance is real (ZL = RL), then (4) shows us that rL is real. Then evaluating (7) at z = 0, where F(z = 0) = L, we see that when ZL > Zo that VSWR = ZS.Zo while if ZL