Problems Two

Question 3.

A coaxial cable feeds a load (2+j1)Zo. The line is lossless and has characteristic impedance Zo. Define, and find values for, the following:-

The Voltage Standing Wave Ratio (VSWR) on the line.

• The VSWR is a property of the standing wave pattern on the line. It is a dimensionless number, equal to the ratio of voltage at a standing wave pattern maximum to the voltage at a standing wave pattern minimum, 1/4 of a transmission line wavelength away along the line.
• Numerically, the VSWR = (1+|s|)/(1-|s|) where |s| is the modulus of the complex reflection coefficient or scattering parameter somewhere along the line. Strictly speaking, the VSWR is only defined precisely in terms of the wave pattern on lossless lines. However, the definition in terms of |s| extends the concept of the VSWR to lines having attenuation or loss, and in this case the value of the VSWR depends on where it is measured along the line.
• In this case, s = (ZL-Zo)/(ZL+Zo) = (1+j)/(3+j) and |s| = sqrt(2/10) = 0.4472 so the VSWR = 2.618

The return loss, in deciBels.

• The return loss is defined as the number of dB by which the reflected power from a load is less than the incident power. Again, it is a function of position along the line only on lossy lines.
• In this case, the return loss = -20 log10(|s|) = 6.99dB

The position (in numbers of guide wavelengths from the load) of the first standing wave voltage minimum.

• The position of the first standing wave minimum is measured from the position of the load, and it is where a probe measuring the total voltage on the line would pick up the smallest signal, as the probe was run back along the line from the load. It is also the place where the scattering parameter s has phase angle 180 degrees modulo 360 degrees.
• In this case, the phase of the scattering parameter s at the load is arctan(1/1)-arctan(1/3) = 45.00-18.43 degrees = 26.57 degrees. At the first VSWR minimum the phase of s is -180 degrees. A change in the s parameter phase angle of -360 degrees represents a translation of 0.5 of a line wavelength away from the load and towards the generator. In this case the phase shift is -206.57 degrees or 0.2869 wavelengths from the load to the first SW minimum.

The value of impedance (ratio of total line voltage to total line current) at the first standing wave voltage minimum.

• The definition is inherent in this part of the question. Here, the voltage and currents at the standing wave minimum are those that would be measured if we could include a current meter in one of the wires of the transmission line, and measure the voltage between the wires of the transmission line, both measurements being taken at the standing wave minimum.
• At the first standing wave minimum, the total line voltage = (V+)[1-|s|] and the total line current = (I+)[1+|s|]. The ratio of these is therefore Zo([1-|s|]/[1+|s|]), since V+ = ZoI+
• In our case using the results above, the impedance = Zo/(VSWR) = Zo/2.618 = 0.3820*Zo

The value of impedance at the first standing wave voltage maximum.

• By direct analogy to the above part of the question, the value of impedance at the first standing wave maximum is Zo*VSWR = 2.618*Zo

The scattering parameter (complex reflection coefficient) at the load.

• The scattering parameter at the load is the complex ratio (backward wave voltage)/(forward wave voltage), at the load, where the voltages are regarded as complex phasors describing the oscillating voltages.
• In this case, as before, s = (ZL-Zo)/(ZL+Zo) = (2+j1-1)/(2+j1+1) = (1+j)/(3+j) = 0.4472*exp(j26.57) = 0.4 + j0.2

The scattering parameter at the first standing wave voltage minimum.

• The scattering parameter varies only in phase along a lossless line. Therefore its magnitude is 0.4472 and the phase angle is -180 degrees at the first standing wave minimum. In real and imaginary parts it is -.4472+j0

The input impedance if the line length is 1.73 wavelengths. and Zo is 75 ohms.

• The input impedance is the ratio of line voltage to current at the input, or generator end, of the line.
• In our case Zin/Zo = (1+s)/(1-s) where s is taken at the line input. Again, |s| doesn't vary along the line so it is still 0.4472 at the line input. The line length is 1.73 wavelengths = 3*0.5+.23 wavelengths. Each 0.5 wavelength represents a change in phase of the s parameter of -360 degrees and can be ignored. Thus we are left with a phase shift input -> load -> input of -360*.23*2 degrees = -165.60 degrees. At the input therefore the scattering parameter has phase 26.57-165.60 = -139.03 degrees.
• Converting to real and imaginary parts, at the input the scattering parameter s = -0.3377-j0.2932 so Zin = Zo(0.6623-j0.2932)/(1.3377+j0.2932) with Zo=75 ohms. The trusty calculator finds Zin = 31.99 -j23.45 ohms

The scattering parameter at the input to the transmission line.

• See the previous section

Question 4.

Define the term "scattering matrix" for a 2-port microwave network. Explain carefully the output and input variables related by the scattering parameters. Explain why the scattering parameters have dimensionless complex character. Explain clearly the meaning of the term "reference plane" with respect to which the scattering parameters are defined and measured. Explain what difference in s parameters occurs if the reference plane is moved further from the 2-port. Explain why scattering parameters are usually given in the form "modulus and phase angle".

• Most of the section above is dealt with in the notes on scattering parameters .

• If the reference plane is moved the s parameters change phase but not modulus. Thus it is best to express them in this format.

• The round trip distance is 2/5 wavelength, or 360*2/5 degrees which is 144 degrees. The phase retards as we move with the wave. One pass of the line is therefore -72 degrees, and the line is lossless. Thus s12=s21 = 1 angle -72 degrees. s11=s22 = 0 because there are no inherent reflections within the length of line.
• In terms of real and imaginary parts, s12=s21= 0.309-j0.951

This transmission line is connected to an antenna which has scattering parameter 0.1 angle 0 degrees. Determine the fraction of the forward wave power which is radiated. Design a matching network to be placed at the input to the transmission line so that the generator sees a perfect match at its terminals.

• The reflected wave power is 0.1*0.1 = 1% of the incident power. Thus 0.99 of the incident power is radiated. (99%).

  One might think this is a good enough match. However, at the input of the transmission line the reflection coefficient (1-port s parameter) is 0.1 angle -144 degrees using the results of the last section. This is 0.2 wavelengths towards the generator from the antenna normalised impedance of

  
                (1+0.1)/(1-0.1) = 1.2222 + j0
  
  which gives us an input impedance of (read graphically)
  
                  0.84  - j0.1
  
  from the SMITH plot 0.2 wavelengths towards the generator
  
  or we can CALCULATE the reflection coefficient gamma
  
         (0.1 angle -144 degrees) = -0.0809 - j0.0588 
  
  which gives us an input impedance of (calculated)
  
         (1 - 0.0809 - j0.0588)/(1 + 0.0809 + j0.0588)
  
  which evaluates to be
         
                0.8449 - j0.1004 
  
  which is satisfactorily in agreement with the SMITH plot.
  
  

• We therefore need a single stub match to the impedance
  

               0.8449 - j0.1004
  

• A stub of series impedance -j0.2 placed 0.186 wavelengths further towards the generator does this. The stub length for an impedance -j0.2 is 0.468 wavelengths from a short.