# 4.8: Exercises

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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

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

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