EE-223 Microwave Circuits (Fall 2014) Lecture 3

Dr. Atif Shamim EE Program King Abdullah University of Science and Technology (KAUST) 1

Terminated Lines

LOAD REFLECTION COEFFICIENT L:

Terminated Lines

Reflection Coefficient at Some Location z

VOLTAGE STANDING WAVE RATIO: (also known as SWR)

Problem

Problem: A 50 line is teminated in a 150 load. Find (a) L (b) VSWR.

Answer: , 3

When the load is mismatched, not all of the available power from the generator is delivered to the load. This loss is called Return Loss, and is defined in dBs as

RL= -20 log || dB

Input Impedance Zin

INPUT IMPEDANCE Zin

Applying wave equations (6.21) and (6.23), we now look into the line at z = -l to obtain:

Input Impedance Zin

Figure 6-8 (p. 277) The terminated T-line can be replaced by an equivalent lumped-element input impedance.

Use of (6.56) Replace: T-line plus load ZL by: Zin

Note: The input impedance, Zin , refers to the ratio of the total voltages and currents, while the characteristic impedance, Zo , refers to the ratio of the voltage to the current for only one of the waves, the incident one or the reflected one

Special Cases of Lossless Terminated Lines

Line terminated in a short circuit (ZL =0) For this case =-1, and Zin = j Zo tan l Line terminated in an open circuit (ZL =) For this case =1, and Zin = -j Zo cot l Terminated T-line with length l=/2 Zin = ZL (meaning that /2 does not alter impedance) Terminated T-line with length l=/4 Zin = Z02 /ZL (quarter wave transformer)

Problems

1) Find the input impedance of a lossless 50 T- line terminated into a 150 load for lengths

(a) /8 (b) /4 Answer: (a) 30- j40 , (b) 16.7

2) Find the input impedance for a 0.6 m long T- line

of characteristic impedance 75 + j 25 , having a propagation constant of 0.01 +j 0.1 m-1 and terminated into a load 25 j40 .

Answer: 23- j34

Complex Loads

In general, use s-domain impedance values of R, L, C to model complex loads

= 32nH

Problem

A /6 line with Zo =50 is terminated in a 25 resistance. Find the input impedance along with its equivalent lumped-element circuit at 1.0 GHz.

Answer: Zin= 57+ j 37 ; this resembles a 57 resistor in series with a 5.9 nH inductor.

Purely Reactive Load

PURELY REACTIVE LOAD ZL = jX

The Complete Circuit

The Complete Circuit

Figure 6-10 (p. 279) The circuit after adding a source, and the equivalent circuit.

Example 6.3

Example 6.3

Figure 6-11 (p. 21) Circuit for Example 6.3.

The Smith Chart

Figure 6-12 (p. 281) The Transmission Line Calculator, commonly referred to as the Smitha Chart.

Most commonly used graphical tool Polar plot of the voltage reflection coefficient (= || ej) || is plotted as radius from the center of the chart (|| 1) angle is measured from the right hand side of the horizontal diameter (-180o 180o) Real utility in its ability to transform from reflection coefficients to normalized impedances/admittances and vice versa

Constant Resistance and Reactance Circles

EE-223 Microwave Circuits(Fall 2014)Lecture 3Terminated LinesTerminated Lines Reflection Coefficient at Some Location zProblemInput Impedance ZinInput Impedance Zin Special Cases of Lossless Terminated LinesProblemsComplex LoadsProblemPurely Reactive LoadThe Complete Circuit The Complete Circuit Example 6.3Example 6.3Figure 6-11 (p. 21)Circuit for Example 6.3.The Smith ChartConstant Resistance and Reactance Circles