IMPEDANCE MATCHING IN HIGH FREQUENCY LINES UNIT - III

Impedance Matching 10/7/20152 Maximum power is delivered when the load is matched the line and the power loss in the feed line is minimized Impedance matching sensitive receiver components improves the signal to noise ratio of the system Impedance matching in a power distribution network will reduce amplitude and phase errors Complexity Bandwidth Implementation Adjustability

Half and Quarter wave transmission lines The relationship of the input impedance at the input of the half-wave transmission line with its terminating impedance is got by letting L = wavelength/2 in the impedance equation. Z input = Z L The relationship of the input impedance at the input of the quarter-wave transmission line with its terminating impedance is got by letting L =wavelength/4 in the impedance equation. Z input = (Z input Z output ) 0.5

Series Stub Input impedance=1/S Voltage minimum

Single Stub Tunning 10/7/20155ELCT564 Shunt Stub Series Stub G=Y 0 =1/Z 0

Single Shunt Stub Tuner Design Procedure 10/7/20156ELCT Locate normalized load impedance and draw VSWR circle (normalized load admittance point is 180 o from the normalized impedance point). 2. From the normalized load admittance point, rotate CW (toward generator) on the VSWR circle until it intersects the r = 1 circle. This rotation distance is the length d of the terminated section of t-tline. The nomalized admittance at this point is 1 + jb. 3. Beginning at the stub end (rightmost Smith chart point is the admittance of a short-circuit, leftmost Smith chart point is the admittance of an open-circuit), rotate CW (toward generator) until the point at 0 - jb is reached. This rotation distance is the stub length l.

Smith Chart Impedances, voltages, currents, etc. all repeat every half wavelength The magnitude of the reflection coefficient, the standing wave ratio (SWR) do not change, so they characterize the voltage & current patterns on the line If the load impedance is normalized by the characteristic impedance of the line, the voltages, currents, impedances, etc. all still have the same properties, but the results can be generalized to any line with the same normalized impedances

Smith Chart The Smith Chart is a clever tool for analyzing transmission lines The outside of the chart shows location on the line in wavelengths The combination of intersecting circles inside the chart allow us to locate the normalized impedance and then to find the impedance anywhere on the line

Smith Chart Real Impedance Axis Imaginary Impedance Axis

Smith Chart Constant Imaginary Impedance Lines Constant Real Impedance Circles Impedance Z=R+jX =100+j50 Normalized z=2+j for Z o =50

Smith Chart Impedance divided by line impedance (50 Ohms) Z1 = j50 Z2 = 75 -j100 Z3 = j200 Z4 = 150 Z5 = infinity (an open circuit) Z6 = 0 (a short circuit) Z7 = 50 Z8 = 184 -j900 Then, normalize and plot. The points are plotted as follows: z1 = 2 + j z2 = 1.5 -j2 z3 = j4 z4 = 3 z5 = infinity z6 = 0 z7 = 1 z8 = j18S

Smith Chart Thus, the first step in analyzing a transmission line is to locate the normalized load impedance on the chart Next, a circle is drawn that represents the reflection coefficient or SWR. The center of the circle is the center of the chart. The circle passes through the normalized load impedance Any point on the line is found on this circle. Rotate clockwise to move toward the generator (away from the load) The distance moved on the line is indicated on the outside of the chart in wavelengths

Toward Generator Away From Generator Constant Reflection Coefficient Circle Scale in Wavelengths Full Circle is One Half Wavelength Since Everything Repeats

Single-Stub Matching Load impedance Input admittance=S

Single Stub Tuning Single-stub tuning circuits. (a) Shunt stub. (b) Series stub.

2 adjustable parameters d: from the load to the stub position. B or X provided by the shunt or series stub. For the shunt-stub case, Select d so that Y seen looking into the line at d from the load is Y 0 +jB Then the stub susceptance is chosen as –jB. For the series-stub case, Select d so that Z seen looking into the line at d from the load is Z 0 +jX Then the stub reactance is chosen as –jX.

Shunt Stubs Single-Stub Shunt Tuning Z L =60-j80.

(b) The two shunt-stub tuning solutions. (c) Reflection coefficient magnitudes versus frequency for the tuning circuits of (b).

To derive formulas for d and l, let Z L = 1/Y L = R L + jX L. Now d is chosen so that G = Y 0 =1/Z 0,

If R L = Z 0, then tanβd = -X L /2Z 0. 2 principal solutions are To find the required stub length, B S = -B. for open stub for short stub

Series Stubs Single Stub Series Tuning Z L = 100+j80 (a) Smith chart for the series-stub tuners.

(b) The two series- stub tuning solutions. (c) Reflection coefficient magnitudes versus frequency for the tuning circuits of (b).

To derive formulas for d and l, let Y L = 1/Z L = G L + jB L. Now d is chosen so that R = Z 0 =1/Y 0,

If G L = Y 0, then tanβd = -B L /2Y 0. 2 principal solutions are To find the required stub length, X S = -X. for short stub for open stub

Analytic Solution To the left of the first stub in Fig. 5.7b, Y 1 = G L + j(B L +B 1 ) where Y L = G L + jB L To the right of the 2nd stub, At this point, Re{Y 2 } = Y 0

Since G L is real, After d has been fixed, the 1 st stub susceptance can be determined as The 2 nd stub susceptance can be found from the negative of the imaginary part of (5.18)

B 2 = The open-circuited stub length is The short-circuited stub length is

For a load impedance ZL=60-j80Ω, design two single-stub (short circuit) shunt tunning networks to matching this load to a 50 Ω line. Assuming that the load is matched at 2GHz and that load consists of a resistor and capacitor in series.

Single Stub Tunning 10/7/201529ELCT564 y L =0.3+j0.4 d1= =0.110λ d2= =0.260λ y1=1+j1.47 y2=1-j1.47 l1=0.095λ l1=0.405λ

Single Stub Tunning 10/7/201530ELCT564

Results

For a load impedance ZL=25-j50Ω, design two single-stub (short circuit) shunt tunning networks to matching this load to a 50 Ω line.

Single Stub tunning 10/7/201533ELCT564 y L =0.4+j0.8 d1= =0.063λ d2= =0.260λ y1=1+j1.67 y2=1-j1.6 l1=0.09λ l1=0.41λ

Single Series Stub Tuner Design Procedure 10/7/201534ELCT Locate normalized load impedance and draw VSWR circle 2. From the normalized load impedance point, rotate CW (toward generator) on the VSWR circle until it intersects the r = 1 circle. This rotation distance is the length d of the terminated section of t-tline. The nomalized impedance at this point is 1 + jx. 3. Beginning at the stub end (leftmost Smith chart point is the impedance of a short-circuit, rightmost Smith chart point is the impedance of an open-circuit), rotate CW (toward generator) until the point at 0 ! jx is reached. This rotation distance is the stub length l.

For a load impedance ZL=100+j80Ω, design single series open-circuit stub tunning networks to matching this load to a 50 Ω line. Assuming that the load is matched at 2GHz and that load consists of a resistor and inductor in series.

Single Stub Tunning 10/7/201536ELCT564 z L =2+j1.6 d1= =0.120λ d2= =0.463λ z1=1-j1.33 z2=1+j1.33 l1=0.397λ l1=0.103λ

Single Stub Tunning 10/7/201537ELCT564

Single Stub Tunning 10/7/201538ELCT564

Double Stub Matching Network jB 1 jB 2 YLYL a b ba

Double-Stub Tuning If an adjustable tuner was desired, single-tuner would probably pose some difficulty. Smith Chart Solution y L  add jb 1 (on the rotated 1+jb circle)  rotate by d thru SWR circle (WTG)  y 1  add jb 2  Matched Avoid the forbidden region.

Double Stub Tunning 10/7/ ELCT564 The susceptance of the first stub, b1, moves the load admittance to y1, which lies on the rotated 1+jb circle; the amount of rotation is de wavelengths toward the load. Then transforming y1 toward the generator through a length d of line to get point y2, which is on the 1+jb circle. The second stub then adds a susceptance b2.

Design a double-stub shunt tuner to match a load impedance Z L =60-j80 Ω to a 50 Ω line. The stubs are to be open-circuited stubs and are spaced λ/8 apart. Assuming that this load consists of a series resistor and capacitor and that the match frequency is 2GHz, plot the reflection coefficient magnitude versus frequency from 1 to 3GHz.

Double Stub Tunning 10/7/ ELCT564 y L =0.3+j0.4 b 1 =1.314 b 1 ’ = y 2 =1-j3.38 l1=0.46λ l2=0.204λ

Double Stub Tunning 10/7/ ELCT564

Double-stub tuning. (a) Original circuit with the load an arbitrary distance from the first stub. (b) Equivalent- circuit with load at the first stub.

Smith chart diagram for the operation of a double-stub tuner.

Solution to Example 5.4. (a) Smith chart for the double- stub tuners. Z L = 60-j80 Open stubs, d = λ/8

(b) The two double-stub tuning solutions. (c) Reflection coefficient magnitudes versus frequency for the tuning circuits of (b).

0 r=1 x=1 x=-1 Real part of Refl. Coeff. P short circuit P open circuit r=0.5 Smith Chart YLYL

0 r=1 x=1 x=-1 Real part of Refl. Coeff. P short circuit P open circuit r=0.5 Smith Chart YLYL Rotate the the G=1 circle through an angle -  The intersection of G=1 and the G L circle determine the point P 2 P2P2 P3P3 G 1 =1

0 r=1 x=1 x=-1 Real part of Refl. Coeff. P short circuit P open circuit r=0.5 Smith Chart YLYL The shaded range is for the load impedance which cannot be matched when d=1/8 wavelength

10/7/201553ELCT564