Microwave Measurements & Scattering Matrix PDF

## **UNIT 4** - Scattering Matrix and Microwave Measurements ### **Syllabus** * **SCATTERING MATRIX** * Scattering Matrix Properties * Directional Couplers - 2 Hole, Bethe Hole * \[S] matrix of Magic Tee and Circulator * **MICROWAVE MEASUREMENTS** * Description of Microwa...

## **UNIT 4** - Scattering Matrix and Microwave Measurements ### **Syllabus** * **SCATTERING MATRIX** * Scattering Matrix Properties * Directional Couplers - 2 Hole, Bethe Hole * \[S] matrix of Magic Tee and Circulator * **MICROWAVE MEASUREMENTS** * Description of Microwave Bench - Different Blocks and their Features, Errors and Precautions * Measurement of Attenuation, Frequency * Standing Wave Measurements * Measurement of Low and High VSWR, Cavity Q, Impedance Measurements ### **LEARNING OBJECTIVES** * Concept of scattering matrix and its properties. * Basic idea and types of directional couplers. * Construction and working of two hole directional coupler and Bethe hole type directional coupler. * S-matrix of various waveguide components such as directional coupler, E-plane tee, H-plane tee, magic tee, gyrator, isolator and circulator * Description of microwave bench setup with explanation of its various blocks and its features. * Basic idea about errors in microwave measurements and precautions to avoid them. * Measurement of various parameters like attenuation, frequency, low & high VSWR, cavity Q and impedance. ### **INTRODUCTION** In the previous unit, we studied about the construction and working of commonly used waveguide components such as directional couplers, waveguide tees (like E- plane tee, H-plane tee and magic tee) and ferrite devices (i.e., gyrator, isolator and circulator). In this unit, we discuss the mathematical analysis of these components using Scattering parameters, also known as S-parameters, are the most common network parameters used to describe the performance of the microwave devices or circuits or networks in the frequency domain. In the field of microwave engineering, we often come across the need of measuring different values while using in different applications such as power, attenuation, phase shift, VSWR, impedance and Q factor etc. for the effective usage. In this unit, we will discuss about the different measurement techniques. ### **PART - A SHORT QUESTIONS WITH SOLUTIONS** **Q1. Define S-matrix and mention the use of scattering parameters in microwaves.** **(or)** **State the significance of S-parameters at high frequencies.** **(or)** **Define scattering matrix.** **(or)** **Explain the significance of scattering matrix.** **(or)** **Why the s-parameters are used in microwaves?** **Ans:** **S-matrix** The S-matrix (or) a scattering matrix is a matrix which is used to represent all inputs which are applied to the ports of a given network in a matrix form. The elements in this matrix are known as 'scattering coefficients' (or) 'scattering parameters'. **Significance of S-matrix or S-parameters** S-parameters can be used at any frequencies, but at high frequencies such as RF and microwave frequencies powers associated with travelling waves must be calculated. S-parameters at high frequencies make these calculations simple and accurate. S-parameters are used at high frequencies to describe the behaviour of complex networks. **Q2. Why Z and Y parameters are not measured at microwave frequencies?** **(or)** **Explain why h, Y and Z parameters cannot be measured at microwave frequencies.** **Ans:** Z, Y and h-parameters are used in the analysis of low frequency circuits. Because at low frequencies, the total voltages and total currents are measured in the network analysis. But at microwave frequencies, travelling waves come into the analysis. For this, the microwave powers associated with travelling waves must be calculated. Due to this reason s-parameters are used at microwave frequencies instead of Z, Y and h-parameters. **Q3. What are the properties of S-matrix?** **Ans:** The properties of S-matrix include, 1. Scattering matrix is always a square matrix i.e., the order of S-matrix is n x n. 2. The diagonal elements of an ideal N-port network are zero, if all the ports matched perfectly. 3. A reciprocal microwave device holds symmetric property. i.e., * S = S* 4. In matrix notation, it can be expressed as, * \[S] = [S]'] 5. For a lossless network, S-matrix exhibits unitary property, i.e., * \[S] \[S]* = [1] 6. If the ports of a network (2-port) shifted by a phase of o₁ and o₂ then S-matrix of device after shift are evaluated as, * \[S'] = * [e-j1 0] * [0 e-j2] \[S] [e-j1 0] * [0 e-j2] This is known as phase shift property and it is applicable for a microwave junction having number of ports. **Q4. List the advantages of S-matrix.** **Ans:** 1. The check unitary property of \[S] is used to the power of balance of lossless structure. 2. The \[S]-parameters change only in phase if reference planes are changed. 3. The \[S] matrix describes the properties of the component (as, if it were a black box). 4. The properties of a \[S] matrix gives us a clear idea of what can be achieved and what cannot be achieved. **Q5. What are the different types of directional couplers?** **Ans:** 1. Two-hole directional coupler. 2. Four-hole directional coupler. 3. Reverse-coupling directional coupler (Schwinger coupler). 4. Bethe-hole directional coupler. **Q6. Define coupling factor and directivity of a directional coupler.** **Ans:** * **Coupling Factor:** The factor which is a measure of the ratio of power levels in main and auxiliary waveguides of a two-hole directional coupler is known as coupling factor. * **Directivity:** The factor which is a measure of how well the forward travelling wave in the main waveguide couples only to a specific port of the auxiliary waveguide is known as directivity of the directional coupler. **Q27. A 20 dB coupler has a directivity of 30 dB. Calculate the value of isolation.** **Ans:** Given that, * For a coupler, * Coupling factor, C = 20 dB * Directivity, D = 30 dB * Isolation, I = ? The isolation can be defined as, * Isolation, I = Coupling factor (C) + Directivity (D) On substituting corresponding values in the above equation, * I = 20 + 30 = 50 dB. * Isolation, I = 50 dB **Q28. List out the different components used in a microwave bench.** **Ans:** The different components used in a microwave bench are, * Reflex klystron or gunn oscillator as a source * Isolator * Attenuator * Wave meter * Standarding wave detectors * Slotted waveguide * Tunable probe vernier scale * Rack and pinion arrangement * Waveguide detector. **Q29. What type of slot is used in microwave bench?** **Ans:** The slotted line is one of the basic instruments used in radio frequency test and measurement at microwave frequencies. It consists of a precision transmission line with a movable insulated probe inserted into a longitudinal slot cut into the line. The type of slot used in microwave bench is transmission line (waveguide) type. **Q30. Why Isolator is used in microwave measurements?** **(or)** **What is the need for an isolator in MW measurements and where it is placed?** **Ans:** Isolator is used in several microwave circuits such as microwave bench set up because it provides a very small amount of attenuation when transmitted from port (1) to port (2) and maximum amount of attenuation, when transmitted from port (2)to port (1). This feature is used in matching a source with a variable load. It is placed between variable attenuator and frequency meter. **Q31. How are microwave measurements different from low frequency measurements?** **Ans:** The differences between low frequency and microwave measurements are mentioned in table. | Low Frequency Measurement | Microwave Measurement | | ----------------------------- | ------------------------- | | In order to measure the power, initially the voltage | Power is measured directly without calculating the current and the voltage. | | and current must be calculated. | In case of power measurements, Ratio of powers | | Circuits are lumped elements. | Circuits are distributed. | | Exact values must be known. | Exact values are not necessary. | **Q32. What are the possible errors in high frequency measurements?** **Ans:** The different types of errors associated with high frequency measurements are, 1. Error due to hysterisis and nonlinearity of the modulator characteristics. 2. Mismatch error in the channel. 3. Error due to microwave frequency instability. 4. Error may be caused by vibrations of equipment and other external disturbances. 5. These measurements desire the manual interaction which, ultimately leads to an error. **Q33. What are the characteristics of detectors used in microwave measurements?** **Ans:** The characteristics of detectors used in microwave measurements are, 1. In order to detect low frequency square wave modulated microwave signal, a non-reciprocal detector diode must be mounted in the transmission line. * Examples of these diodes are point contact diodes or Schottky barrier diodes. 2. To detect the modulated signal, a crystal detector is connected at one end of the waveguide. 3. In order to avoid amplitude and phase variations and to get perfect detection, a tunable stub is used to match the detector to the microwave transmission system and that tunable stub may be, * Tunable waveguide detector * Tunable probe detector * Tunable coaxial detector. 4. Schottky detector can be used in which the output is proportional to the input power. **Q34. What is Q of a cavity resonator?** **Ans:** The quantity which is a measure of the frequency selectivity of a resonant or non-resonant circuit is known as quality factor 'Q'. Mathematically, * \[Q = 2π × \frac{Maximum energy stored}{Energy dissipated per cycle}\] * \[ i.e., Q = \frac{wW}{P}\] Where, * W - Maximum stored energy * P - Average power loss. **Q35. What is a VSWR meter and how will you determine the VSWR?** **Ans:** A VSWR meter is a high gain, high Q, low noise amplifier which is tuned normally to a fixed frequency modulated microwave signal. VSWR can be determined by direct method (slotted line) and double minimum method. **Q36. What are the different possible errors that will effect VSWR measurements?** **(or)** **Account for the different types of errors associated with the measurement of VSWR using a slotted line setup.** **Ans:** The different types of errors associated with the measurement of VSWR using a slotted line setup are, 1. Linearity error. 2. There is no accuracy in obtained results because of the limited dynamic range of detectors. 3. This process desires the manual interaction which, ultimately leads to an error. **Q37. What is a scattering matrix? Discuss the importance of S-parameters. List the properties of S-matrix.** **Ans:** The S-matrix (or scattering matrix) is a square matrix gives all possible relations between incident and reflected wave power of a microwave junction. The elements in this matrix are known as 'scattering coefficients' (or) 'scattering parameters'. It is denoted as, * \[ [S] \] * \[a] * [b]= [S] * \[a] * \[b] The scattering parameters are fixed properties of the linear circuits, which describe how the energy couples between a pair of ports or transmission lines connected to the circuit. Formally, S-parameters can be defined for linear electronic components. They are algebraically related to the impedance, admittance and also to transmission parameters. **Properties:** * Scattering matrix is always a square matrix i.e., the order of S-matrix is n x n. * The diagonal elements of an ideal N-port network are zero, if all the ports matched perfectly." * A reciprocal microwave device holds symmetric property. i.e., * S=S; ij * In matrix notation, it can be expressed as, * \[S] = [S]'] * For a lossless network, S-matrix exhibits unitary property. i.e., * \[S] \[S]* = [1] * If the ports of a network (2-port) shifted by a phase of o₁ and o₂ then S-matrix of device after shift are evaluated as, * \[S'] = * [e-j1 0] * [0 e-j2] \[S] [e-j1 0] * [0 e-j2] This is known as phase shift property and it is applicable for a microwave junction having number of ports. **Significance** The S-matrix is a square matrix, it gives the power relationships between the various input and output ports of a microwave junction. **Q38. State and derive the unitary property of S matrix.** **Ans:** Unitary property states that the sum of products of each term in S-matrix of a lossless network with its conjugate is unity * \[ i.e., Σ_{n=1}^N S_{n} S_{n} ] = 1\] where, * N - Number of ports. For a lossless network, the sum of power incident at input ports is equal to the total power reflected from output ports. It can be expressed as, * \[\Σ_{n=1}^N |a_n|^2 = \Σ_{n=1}^N |b_n|^2 = 1 \] Where, * a - Incident wave power * b - Reflected wave power. Since, \[b] = [S] \[a] * \[\Sigma_{n=1}^N |b_n|^2 = \Σ_{n=1}^N |S_{n}|^2 |a_n|^2\] On substituting the value of b₁ in equation (1),, * \[\Σ_{n=1}^N \Σ_{i=1}^N |S_{ni} |^2 |a_i|^2 = \Σ_{n=1}^N |a_n|^2 \] Assuming, only the ith port is excited and all remaining ports are terminated with matched loads. Then equation (2) becomes, * \[ ( |S_{i1}|^2 + |S_{i2}|^2 +....+ |S_{in}|^2) |a_i|^2 + S_{i1} |a_1|^2+....+ S_{i,i-1}|a_{i-1}|^2 + S_{i,i+1}|a_{i-1}|^2+.... + S_{in}|a_n|^2 = |a_i|^2 \] Here, the incident power a, is zero for all 'n' except for n = i. Which implies, * \[Σ_{n=1}^N |a_n|^2 = |a_i|^2 \] From equation (3),, * \[ \Σ_{n=1}^N |S_{ni}|^2 |a_i|^2 = |a_i|^2 \] * \[ \Sigma_{n=1}^N |S_{ni}|^2 = 1 \] Hence, for a lossless junction, * \[ \Σ_{n=1}^N S_{n} S_{n} ] = 1\] **Q39. Explain the working of a two-hole directional coupler with a neat diagram and derive the expression for the coupling and directivity of a two-hole directional coupler.** **Ans:** **Two-Hole Directional Coupler** A two-hole directional coupler with a travelling wave propagation consists of two waveguides namely, main waveguide and auxiliary waveguide with two tiny holes common between them as shown in figure. The spacing between the centers of two tiny holes must be, * \[L = (2n+1) \frac{λ_g}{4}\] Where, * λ_g – Guide wavelength When an incident power, P is fed at the port (1) of two-hole directional coupler, received power P₁ is obtained at port (2). The incident power coming out from holes A and B are out of phase by 180°, since the incident power travels a distance of (λ_g/4 + λ_g/4 ) when it comes back from hole B. So, the incident powers cancel each other and produces a zero power (P₃ = 0) at port (3). At hole B, both the incident powers are in phase so, they add up and produce a forward coupled power, P₄ at port (4). The magnitude of the power coming out of 2 holes depends upon the dimension of the two holes. The degree of coupling is determined by the size and location of the holes in the waveguide walls. It is very difficult to produce high degree directivity over a band of frequencies. **Coupling Coefficient** The factor which is a measure of the ratio of power levels in main and auxiliary waveguides of a two-hole directional coupler (shown in figure) is known as coupling factor. Mathematically, * \[Coupling factor = \frac{P_4}{P_1}\] * \[Coupling factor (dB) = 10 log_{10} (\frac{P_4}{P_1})\] Where, * P₁ - Power input to port (1) * P₄ - Power output from port (4) Coupling coefficient of an ideal two-hole directional coupler is in the range of 3 to 20 dB. **Directivity** The factor which is a measure of how well the forward travelling wave in the main waveguide couples only to a specific port of the auxiliary waveguide, is known as directivity of the directional coupler shown in figure. Mathematically, * \[Directivity, D = \frac{P_3}{P_1}\] * \[Directivity, D(dB) = 10 log_{10} ( \frac{P_3}{P_1})\] Where, * P₃ - Power output from port (3) Directivity can also be written as, * \[D = 10 log_{10} (\frac{P_4}{P_1}) + 10 log_{10} (\frac{P_1}{P_3})\] * \[D = 10 log_{10} (\frac{P_4}{P_1}) - 10 log_{10} (\frac{P_3}{P_1})\] * \[D = Isolation (dB) – coupling factor (dB) \] Since, Isolation (dB) = 10 log_{10} ( \frac{P_3}{P_1}) .. Directivity (dB) = Isolation (dB) - Coupling factor (dB) For an ideal directional coupler, directivity must be infinity. The typical value of the directivity is only 30 to 35 dB. **Insertion Loss** The amount of signal attenuation in the main guide is called 'insertion loss'. It is given by the ratio of power incident to the power transmitted in the main arm. i.e., I = \[\frac{Power in port 4}{Power in Port 1} \] (dB) The insertion loss of an ideal two-hole directional coupler is infinite. **Q40. Explain the principle of operation of Bethe hole directional coupler.** **Ans:** **Bethe-hole Coupler** Bethe-hole coupler or waveguide coupler contains two coupled waveguides arranged through a single hole as shown in figure. A hole is located either at the centre of common broad wall of inclined waveguides at an angle θ, or at an offset position 'd' of two parallel waveguides. The following considerations should be taken when the aperture is smaller than the wavelength. 1. It works as a electric dipole, which is perpendicular to the aperture, and having dipole moment directly proportional to the normal component of the electric field in the guide. 2. It also works as a magnetic dipole in the plane of aperture having dipole moment which is directly proportional to the tangential component of the exciting magnetic field at the aperture. 3. Due to radiations of electric and magnetic dipole coupling is available to the auxiliary guide. 4. Magnetic dipole radiates asymmetrically in longitudinal directions, while electric dipole radiates equally and longitudinally in both the directions. 5. The port 4 and port 3 powers can be controlled by adjusting the distance 'd' or by varying the angle between the two waveguides. Centre Hole: Generally, p₁ ≠ p₄ for centre coupling hole. Therefore, the coupling and directivity are defined as, * \[ C = -20 log_{10} (\frac{4β_o^2}{cosθ + K_o^2 }) dB] * \[D = 20 log_{10} ( \frac{2B^2 cosθ + K_o^2}{2B^2 cosθ - K_o^2 }) dB] \] Where, * r - Radius of hole * a × b - Cross-sectional waveguide dimensions * β_o = \frac{2π}{λ_o} * λ_o = \frac{c}{f_o} * K_o = \frac{2\pi}{λ_o} \frac{d}{2a} Here, the offset value 'd' is given by, * \[d = \frac{a}{√6} sin(\frac{πa}{λ_o}) \] While, Coupling 'C' = \[20 log_{10}(\frac{1+ x^2}{x})\] Directivity 'D' = \[20 log_{10} (\frac{ 1}{x})\] Where, * x = \frac{ 16π^2 r^2}{3aβλ_o} sin^2 (\frac{πd}{a}) **Q41. What are the characteristics of two hole direction coupler and derive the S-matrix of it.** **(or)** **Derive the S matrix of directional coupler and define all the parameters?** **Ans:** **Characteristics of Two Hole Direction Coupler** For answer refer Unit-4, Q20, (Refer Only Coupling coefficient and Directivity). **Derivation of S-matrix** The directional coupler contains four ports. So a 4 × 4 matrix is required to represent \[S] matrix. *  \[[S] = * \[S_{11} S_{12} S_{13} S_{14} \] * \[S_{21} S_{22} S_{23} S_{24} \] * \[S_{31} S_{32} S_{33} S_{34}\] * \[S_{41} S_{42} S_{43} S_{44}\]] The four ports of a directional coupler are perfectly matched to the junction. *  \[S_{11} = S_{22} = S_{33} = S_{44} = 0 \] From the symmetrical property, S = S* *  \[S_{12} = S_{21} S_{13} = S_{31}, S_{24} = S_{42}, S_{34} = S_{43} S_{23} = S_{32}, S_{41} = S_{14}\] By using the above property, *  \[S_{12} = S_{21} S_{13} = S_{31}, S_{24} = S_{42}, S_{34} = S_{43} S_{23} = S_{32}, S_{41} = S_{14}\] For ideal 3 dB directional coupler, the back power (P) is zero. i.e., P = 0 Hence, there is no coupling between the ports (1), (3) and the ports (2), (4). *  \[i.e., S_{13} = S_{31} = 0 and S_{24} = S_{42} = 0 \] Equation (1) can be written as, *  \[[S] = * \[0 S_{12} 0 S_{14} \] *  \[S_{12} 0 S_{23} 0 \] *  \[0 S_{23} 0 S_{34} \] *   \[S_{14} 0 S_{34} 0\] ] Using the relation \[S] \[S]* = I, *  \[[S] = *  \[0 S_{12} 0 S_{14} \] *  \[S_{12} 0 S_{23} 0 \] *  \[0 S_{23} 0 S_{34} \] *  \[S_{14} 0 S_{34} 0 \]] * \[ *  \[0 S_{12} 0 S_{14} \] *  \[S_{12} 0 S_{23} 0 \] *  \[0 S_{23} 0 S_{34} \] *  \[S_{14} 0 S_{34} 0 \]] = \[ *  \[1 0 0 0\] *  \[0 1 0 0 \] *  \[0 0 1 0 \] *  \[0 0 0 1 \]] *  \[ i.e., S_{12}S_{23} + S_{14} S_{34} = 0 \] *  \[S_{12}S_{14} + S_{23} S_{34} = 0 \] The above equation can also be written as, *  \[ S_{12}S_{23} = S_{14}.S_{34}\] *  \[S_{12}S_{14} = S_{23}S_{34}\] Similarly, \[S_{12}S_{14} = S_{23}S_{34}\] *  \[S_{12}S_{23} = S_{14}.S_{34}\] Dividing equation (3) by equation (4), *  \[ \frac{S_{12}S_{23}}{S_{12}S_{14}} = \frac{S_{14}||S_{34}}{ S_{23}|S_{34}} \] *  \[|S_{14}| = |S_{23}|\] Hence, \[|S_{14}| = |S_{23}|\] Similarly, \[|S_{12}| = |S_{34}|\] On assuming a positive and real variable 'p' such that, *  \[S_{12} = S_{34} = p [ real number, p ≠ 0] \] On assuming as positive and real variable 'q' such thats, *  \[S_{14} = S_{23}= jq [q ≠ 0] \] Since, \[S_{12}. S_{23} + S_{14}. S_{34} = 0 [·:· S_{12} = S_{34} = S_{34}\] *  \[p [S_{23} + S_{23}] = 0 \] Since, p ≠ 0, \[S_{23} =-S_{23}\] *  \[S_{23} = -ja [S_{23}=jq]\] Hence, the scattering parameters of a 3dB directional coupler are given by, *  \[[S] = *  \[0 S_{12} 0 S_{14} \] *  \[S_{12} 0 S_{23} 0 \] *  \[0 S_{23} 0 S_{34} \] *  \[S_{14} 0 S_{34} 0 \]] = \[ *  \[0 p 0 jq \] *  \[p 0 jq 0 \] *  \[0 jq 0 p\] *  \[jq 0 p 0 \] ] *  \[S_{12}=p and S_{14}=jq]\] **Q42. Explain the functional features of Directional coupler.** **(or)** **Discuss about directional coupler with neat diagram.** **(or)** **Write short notes on directional coupler.** **Ans:** Directional couplers are the microwave devices that are specially designed for power measurements. These can be built by connecting two or more waveguides together. Figure shows that a directional coupler is a four-port waveguide junction made up of two waveguides namely, main waveguide (1-2) and auxiliary waveguide (3-4). There is free transmission (without reflection) of power between port (1) and port (2) and no transmission of power between the pair of ports (1) and (3) or ports (2) and (4) respectively, when all the ports are terminated with their characteristic impedances. The structure of coupler determines the degree of coupling between port (1) and port (4) and between port (2) and port (3) respectively. When power is fed at port (1), the resulting powers are obtained at port (2) and port (4) respectively but, no power is obtained from port (3) because no coupling-exists between port (1) and port (3). Similarly, when the power is fed at port (2), the resulting power is obtained at port (1) and port (3) but, yields no power at port (4). Applications of directional coupler includes, 1. Power monitoring 2. Source leveling 3. Isolation of signal sources 4. Swept transmission and reflection measurements. **Q43. A microwave source with n number of ports is represented as shown in figure.** **Ans:** The scattering matrix is always a square matrix, it gives the power relationships between the various input and output ports of a microwave junction. **Q44. A 20 MW signal is fed to one of the collinear port1 of a lossless H-plane Tee. Calculate the power delivered through each port when other ports are match terminated.** **Ans:** For a lossless H-plane Tee, *  Power fed to the collinear port 1= 20 mW The other two ports are terminated with matched loads, *  Power delivered to port-1 = ? *  Power delivered to port-2 = ? *  Power delivered to port-3 = ? S-matrix for H-plane Tee is given as, *  \[S= *  \[1/√2 -1/√2 1/√2\] *  \[1/√2  1/√2  1/√2\] *  \[ 1/√2   1/√2 -1/√2 \]] But from the given data. *  Port-1 is fed with some power i.e., a₁ = 20 mW *  Port-2 and port-3 are terminated i.e., a₂ = a₃ = 0 Then, the power at ports are given as, * \[b₁ = \frac{1}{√2} α₁ + \frac{-1}{√2} α₂ + \frac{1}{√2} α₃\] * \[b₂ = \frac{1}{√2} α₁ + \frac{1}{√2} α₂ + \frac{1}{√2} α₃\] * \[b₃ = \frac{1}{√2} α₁ + \frac{1}{√2} α₂ + \frac{-1}{√2} α₃\] Consider, b₂ = \[ \frac{1}{√ 2} + \frac{1}{√ 2} α₂ + \frac{1}{√ 2} α₃ \] Since, a₂ = a₃ = 0 *  \[b₂ = \frac{1}{√2} α₁\]. then the power at port-1 is obtained as, *  \[ P₂ = |b₂|^2 = (\frac{1}{√2} α₁)^2 = \frac{1}{4} α₁^2\] *  \[...Power at port - 2 = 5m W \] Consider, b₃ = \[ \frac{1}{√2} + \frac{1}{√2} α₂ + \frac{-1}{√2} α₃ \], then the power at port-3 is obtained as, *  \[P3 = |b_3|^2 = (\frac{1}{√2} α_1)^2= \frac{1}{4} α_1^2\] *  \[... Power at port - 3 = 10m W \] **Q45. For the given scattering parameters for a two-port network calculate the equivalent impedance parameters if the characteristics impedance is 50 Ω.** **S₁₁ = 0.4 + j 0.7** **S12 = S21 = j 0.6** **S22 = 0.3-j 0.8.** **Ans:** Given that, * For a two-port network, * Scattering parameters, S₁₁ = 0.4 + j 0.7 *  S12=S21=j 0.6 *  S22 = 0.3-j 0.8 * Characteristic Impedance, Z = 50 Ω Equivalent Impedance parameters, Z11 Z12 Z21 Z22 = ? Impedance parameters interms of S parameters are obtained as follows, *  \[ Z_{11} = Z_{o} ( \frac{(1-S_{11})(1 -S_{22}) - S_{12}S{21}}{(1+ S_{11})(1 -S_{22}) - S_{12}S{21} } )\] *  \[ Z_{11} = 50 ( \frac{(1-0.4-j 0.7)(1 -0.3 + j 0.8) + (j 0.6)(j0.6)}{(1+ 0.4+j 0.7)(1 -0.3 + j 0.8) - (j 0.6)( j 0.6) })\] *  \[ Z_{11} = 50 ( \frac{(0.6-0.7)(0.7+0.8j) +0.36}{0.06 +1.61 j} )\] *  \[ Z_{11} = 50 (\frac{ 0.06 + 1.61 j }{1.34-0.01 j}) \] *  \[Z_{11} = 50 (0.035 +