4.8: Exercises

1. In Section 4.3.1, the \(S\) parameters of a reciprocal error network were determined by applying three loads—\(Z_{1},\: Z_{2},\) and \(Z_{3}\)—and measuring the respective input reflection coefficients. If \(Z_{1}\) is a matched load, \(Z_{2}\) is a short circuit, and \(Z_{3}\) is an open, the \(S\) parameters of Equations (4.3.1), (4.3.2), and (4.3.3) are found. Use SFG theory to derive these results.
2. The \(S\) parameters of a line with a physical length of \(20\text{ cm}\) was measured at \(1\text{ GHz}\) in a \(Z_{\text{ref}} = 50\:\Omega\) system and found to be \(S_{11} = S_{22} = 0.1\) and \(S_{21} = S_{12} = −0.9\). What are the characteristic impedance, attenuation constant, and phase constant of the line at \(1\text{ GHz}\). It is known that the line is less than a wavelength long.
3. At \(10\text{ GHz}\) the propagation constant of a line is \(\gamma = 4.6 +\jmath 400\) and the characteristic impedance is \(Z_{0} = 60 −\jmath 0.5\). What are the \(R,\: L,\: G\) and \(C\) parameters of the line?
4. At \(100\text{ GHz}\), the propagation constant of a line is \(\gamma = 30 +\jmath 600\) and the characteristic impedance is \(Z_{0} = 27 +\jmath 0.7\). What are the \(R,\: L,\: G\) and \(C\) parameters of the line?
5. At \(1\text{ GHz}\), the propagation constant of a line is \(\gamma = 2.5 +\jmath 36\) and the characteristic impedance is \(Z_{0} = 105\jmath\). What are the \(R,\: L,\: G\) and \(C\) parameters of the line?
6. The \(S\) parameters of a line with a physical length of \(2\text{ mm}\) was measured at \(10\text{ GHz}\) in a \(Z_{\text{ref}} = 50\:\Omega\) system and found to be \(S_{11} = S_{22} = 0.1 −\jmath 0.001\) and \(S_{21} = S_{12} = −0.7 + \jmath 0.3\). It is known that the line is less than a wavelength long. For the line find the following at \(10\text{ GHz}\):
   1. Characteristic impedance.
   2. Why is it important to know the approximate length of the line in terms of wavelengths?
   3. Complex propagation constant.
   4. Attenuation constant, .
   5. \(R,\: L,\: G,\) and \(C\) parameters.
7. Repeat 6 but now the line is between one and two wavelengths long.
8. The properties of a \(5\text{ mm}\) long microstrip line on an unknown substrate are to be determined by terminating the line in a known impedance and measuring \(\Gamma_{\text{in}}\), the reflection coefficient at the input of the line. At \(10\text{ GHz}\) the load has a reflection coefficient \(\Gamma_{L} = 0.9\angle 0^{\circ}\) and \(\Gamma_{\text{in}} = 0.9\angle 170^{\circ}\). When the frequency is swept, on a Smith chart \(\Gamma_{\text{in}}\) traces out a circle centered at the origin. All measurements are referenced to \(50\:\Omega\). It is known that the substrate is not magnetic and so the relative permeability of the substrate is one.
   1. At \(10\text{ GHz}\) what is the electrical length of the line in degrees? (Assume that the line is less than a half-wavelength long.)
   2. What is the electrical length of the line in fractions of a wavelength?
   3. Since the line is \(5\text{ mm}\) long, what is the guide wavelength, \(\lambda_{g}\), of the line?
   4. What is the free space wavelength, \(\lambda_{0}\)?
   5. What is the relationship between \(\lambda_{g},\:\lambda_{0}\), and the line’s effective relative permittivity \(\varepsilon_{e}\)?
   6. What is \(\varepsilon_{e}\)?
   7. What is the characteristic impedance of the line?
   8. What is the loss of the line in terms of \(\text{dB}\) per meter?
   9. If there was no substrate, i.e. \(\epsilon_{r} =\epsilon_{0}\), what would the electrical length of the line be in terms of \(\lambda_{0}\)?
9. A long slightly lossy line has a frequency-independent input reflection coefficient located at the point \(\Gamma_{\text{in}} = 0.8\) on a Smith chart. What is the characteristic impedance of the line?
10. A long slightly lossy line has a frequency-independent input reflection coefficient located at the point \(\Gamma_{\text{in}} = −0.7\) on a Smith chart. What is the characteristic impedance of the line?
11. Port \(\mathsf{2}\) of a transmission line with characteristic impedance \(Z_{01} = 75\:\Omega\) is terminated in \(75\:\Omega\) and the input reflection coefficient \(\Gamma_{\text{in}}\) at Port \(\mathsf{1}\) is measured and plotted on a \(50\:\Omega\) Smith chart. As the frequency is varied \(\Gamma_{\text{in}}\) traces out a circle. What is the center and radius of that circle.
12. Port \(\mathsf{2}\) of a transmission line with characteristic impedance \(Z_{01} = 75\:\Omega\) is left open and the input reflection coefficient \(\Gamma_{\text{in}}\) at Port \(\mathsf{1}\) is measured and plotted on a \(50\:\Omega\) Smith chart. As the frequency is varied \(\Gamma_{\text{in}}\) traces out a circle. What is the center (use polar coordinates) and radius of that circle.
13. The input reflection coefficient \(\Gamma_{\text{in}}\) of a transmission line with unknown characteristic impedance \(Z_{01}\) and is measured using a VNA in a \(50\:\Omega\) system but the load terminating the line is unknown. On a \(50\:\Omega\) Smith chart the locus of \(\Gamma_{\text{in}}\) with respect to frequency is a circle centered at \(0.7\) on the horizontal axis of the Smith chart with a radius of \(0.3\). What is \(Z_{01}\)?
14. The input reflection coefficient \(\Gamma_{\text{in}}\) of a transmission line with unknown characteristic impedance \(Z_{01}\) and is measured using a VNA in a \(50\:\Omega\) system but the load terminating the line is unknown. On a \(50\:\Omega\) Smith chart the locus of \(\Gamma_{\text{in}}\) with respect to frequency is a circle centered at \(1.2\) on the horizontal axis of the Smith chart with a radius of \(0.25\). What is \(Z_{01}\)?

4.8.1 Exercises by Section

\(†\)challenging

\(§4.3\: 1†\)

\(§4.4\: 2, 3†, 4, 5, 6, 7, 8\)

\(§4.5\: 9, 10, 11, 12, 13, 14\)

4.8.2 Answers to Selected Exercises

4. \(\begin{array}{cc}{R=476.0\:\Omega\text{/m}}&{G=21.11\text{ mS/m}}\\{L=381.9\text{ nH/m}}&{C=106.1\text{ pF/m}}\end{array}\)