14031202 Circuit Theory Chapter 18 Two-Port Circuits 14031202 Circuit Theory Chapter 18 Two-Port Circuits Prof. Adnan Gutub Full Credit of theses slides are given to Prof Imran Tasadduq whom generously shared them for academic benefit 1 Chapter Objectives 1 Be able to calculate two-port parameters with any methods: • Circuit analysis; • Measurements made on a circuit; • Converting from another set of two-port parameters using Table 18.1. 2 Be able to analyze a terminated two-port circuit to find currents, voltages, impedances, and ratios of interest using Table 18.2. 3 Know how to analyze a cascade interconnection of two-port circuits. 2 Instructor: Prof. Adnan Gutub 1 14031202 Circuit Theory Chapter 18 Two-Port Circuits Course Learning Outcome (CLO) No. 3 Chapters/Topics covering CLO 3 the CLO Chapter 18 The ability to analyze basic Two-port circuits, phasors and time- two port circuits. domain representations, impedance, terminal equations 3 Two-Port Building Block 4 Instructor: Prof. Adnan Gutub 2 14031202 Circuit Theory Chapter 18 Two-Port Circuits Why Two-Port Circuits? • Some cases, only focus on two pairs of terminals at input & output • These ”pairs” are called ports, hence, we have two ports: – input port – output port • Only interested in terminal variables: i1, v 1, i 2, v 2. • Not interested in calculating currents & voltages inside the circuit. 5 Example of a two-port Circuit Source Load (e.g. (e.g. CD speaker) player) 6 Instructor: Prof. Adnan Gutub 3 14031202 Circuit Theory Chapter 18 Two-Port Circuits Restrictions • No energy is stored within the circuit • No independent sources within the circuit • Current into port must equal current out of port • All connections are to either input port or output port • No connections are allowed between the ports – not allowed connections Between: a & c , a & d or b & c , b & d 7 Terminal Equations • Four variables considered: I1, V 1, I 2, V 2 s-domain Two-port Basic Building Block • Only two are independent → If two variables known, other two can be found • Two simultaneous (synchronized) equations are needed to describe two- port network 8 Instructor: Prof. Adnan Gutub 4 14031202 Circuit Theory Chapter 18 Two-Port Circuits Terminal Equations V1 = z 11 I1 + z 12 I2 V2 = z 21 I1 + z 22 I2 [Z] is called “Impedance Matrix” 9 Impedance • Impedance is denoted by “Z” and is given by: • Z = R + jX • R is the resistance and X is called “reactance” • Reactance is the opposition (or resistance) to current due to inductance or capacitance or both. • Units of Z, R and X are Ω (Ohms) 10 Instructor: Prof. Adnan Gutub 5 14031202 Circuit Theory Chapter 18 Two-Port Circuits Finding parameters when V and I are known • Circuit is “black box”, 2 test experiments to find parameters: (1) Open port 2, apply current I1 to port 1, measure input voltage V1 and output voltage V2. (2) Open port 1, apply current I2 to port 2, measure terminal voltages V1 and V2. 11 Two Port Networks V1 = z 11 I1 + z 12 I2 Z parameters: V2 = z 21 I1 + z 22 I2 V z is the impedance seen looking into port 1 z = 1 11 11 I I = 0 when port 2 is open. 1 2 V z is a transfer impedance. It is the ratio of the 1 12 z = voltage at port 1 to the current at port 2 when 12 I I = 0 2 1 port 1 is open. V z = 2 z21 is a transfer impedance. It is the ratio of the 21 I I = 0 1 2 voltage at port 2 to the current at port 1 when port 2 is open. V z 2 = z22 is the impedance seen looking into port 2 22 I I = 0 2 1 when port 1 is open. * notes 12 Instructor: Prof. Adnan Gutub 6 14031202 Circuit Theory Chapter 18 Two-Port Circuits Answers: Example 18.1 V1 = z 11 I1 + z 12 I2 V = z I + z I • Find the z parameters for 2 21 1 22 2 the circuit shown For I 2 = 0 z11 = V 1 / I 1 = 10 Ω z21 = V 2 / I 1 → V 2 = 15V1 /(5+20) = 0.75V1 → z21 = V2 / I 1 = 0.75V1 / I 1 = 7.5 Ω For I 1 = 0 z22 = V 2 / I 2 = 9.375 Ω z12 = V 1 / I 2 → V 1 = 20V2 /(5+20) = 0.8V2 → z 12 = V 1 / I 2 = 7.5 Ω 13 Practice Problem • Find z parameters for the circuit shown Answers: • z11 = 18 Ω • z12 = z 21 = 6 Ω • z22 = 9 Ω 14 Instructor: Prof. Adnan Gutub 7 14031202 Circuit Theory Chapter 18 Two-Port Circuits Answers: V2 = (V 1×12)/(12+1) = 12V1/13 I1 = V 1/13; Find z V2/I 1 = (12V1/13)/(V 1/13) Parameters? V2/I 1 = 12 V1 = (V 2×12)/(12+4) = 12V2/16 V1 = z 11 I1 + z 12 I2 I2 = V 2/16; V /I = (12V /16)/(V /16) V2 = z 21 I1 + z 22 I2 1 2 2 2 V1/I 2 = 12 15 Find z parameters? Answers: For I 2 = 0 z11 = V 1 / I 1 = 4 Ω z21 = V2 / I 1 = 1 Ω For I 1 = 0 z22 = V 2 / I 2 = 5 / 3 = 1.67 Ω z12 = V 1 / I 2 = 1 Ω V1 = z 11 I1 + z 12 I2 V2 = z 21 I1 + z 22 I2 16 Instructor: Prof. Adnan Gutub 8 14031202 Circuit Theory Chapter 18 Two-Port Circuits Six possible sets of terminal equations V1 z11 z12 I1 = × ⇒ []Z : impedance V2 z21 z22 I 2 I1 y11 y12 V1 1- = × ⇒ [][]Y = Z : admittance I 2 y21 y22 V2 transmission V1 a11 − a12 V2 V b − b V = × ⇒ []A 2 = 11 12 × 1 ⇒ B = A 1- I a − a I [][] 1 21 22 2 I2 b21 − b22 I1 hybrid V1 h11 h12 I1 I1 g11 g12 V1 1- = × ⇒ []H = × ⇒ [][]G = H I 2 h21 h22 V2 V2 g21 g22 I 2 17 Six possible sets of terminal equations V = z I + z I Impedance 1 11 1 12 2 Admittance I1 = y 11 V1 + y 12 V2 Z parameters Y parameters V2 = z 21 I1 + z 22 I2 I2 = y 21 V1 + y 22 V2 V = a V – a I V = b V – b I Transmission 1 11 2 12 2 Transmission 2 11 1 12 1 A parameters B parameters I1 = a21 V2 – a22 I2 I2 = b 21 V1 – b22 I1 V = h I + h V Hybrid I1 = g 11 V1 + g 12 I2 Hybrid 1 11 1 12 2 G parameters H parameters I = h I + h V V2 = g 21 V1 + g 22 I2 2 21 1 22 2 18 Instructor: Prof. Adnan Gutub 9 14031202 Circuit Theory Chapter 18 Two-Port Circuits Just intro: Difference between s-domain and Time-domain • What is the difference between i and I ? • What are phasors? • What advantages of phasors instead of time-domain representation? 19 From Example 18.2/p. 705 Given V, I as phasors below, Find a parameters V1 = a11 V2 – a12 I2 I1 = a21 V2 – a22 I2 20 Instructor: Prof. Adnan Gutub 10 14031202 Circuit Theory Chapter 18 Two-Port Circuits Assessment Problem 18.3 Answers: 21 Assessment Problem 18.1 Answers: For V 2 = 0 y = I / V = 0.25 S • Find the y parameters? 11 1 1 y21 = I2 / V 1 = -0.2 S For V 1 = 0 y22 = I 2 / V 2 = 4/15 = 0.267 S y12 = I 1 / V 2 = -1/5 = -0.2 S I1 = y 11 V1 + y 12 V2 I2 = y 21 V1 + y 22 V2 22 Instructor: Prof. Adnan Gutub 11 14031202 Circuit Theory Chapter 18 Two-Port Circuits Assessment Problem 18.2 • Find g and h parameters ?? Answers: 23 P 8.13 (p.699) Find g parameters? 24 Instructor: Prof. Adnan Gutub 12 14031202 Circuit Theory Chapter 18 Two-Port Circuits Relations among the 6 matrixes If we know one matrix, we can derive all the others analytically (Table 18.1). [Y]=[Z] -1, [B]=[A]-1, [G]=[H]-1 E.g. −1 z11 z12 y11 y12 1 y22 − y12 1 a11 ∆a = = = z21 z22 y21 y22 ∆y − y21 y11 a21 1 a22 where ∆y ≡ det[Y] = y11 y22 - y12 y21 ∆a ≡ det[A] = a11 a22 - a12 a21 25 Copyright ©2011, ©2008, ©2005 by Pearson Education, Inc. Electric Circuits, Ninth Edition Upper Saddle River, New Jersey 07458 James W. Nilsson • Susan A. Riedel All rights reserved. Instructor: Prof. Adnan Gutub 13 14031202 Circuit Theory Chapter 18 Two-Port Circuits Copyright ©2011, ©2008, ©2005 by Pearson Education, Inc. Electric Circuits, Ninth Edition Upper Saddle River, New Jersey 07458 James W. Nilsson • Susan A. Riedel All rights reserved. Problem 18.4 Answers: • The z parameters were calculated as: z 11 = 13, z21 = 12, z 22 = 16, z 12 = 12. • Use above results to find the y parameters? 29 Instructor: Prof. Adnan Gutub 14 14031202 Circuit Theory Chapter 18 Two-Port Circuits Example 18.3 Answers: From Short-circuit test: V1 = h 11 I1 + h 12 V2 30 I2 = h 21 I1 + h 22 V2 Cont. Solution Example 18.3 • We found h11 and h21 using these equations by putting V 2 = 0 • To find h12 and h22 , we need port 1 to be open. But this information has not been given in the question • We need to use Table 18.1, find another set of parameters and then use conversion formulas • We can use only those parameters that do not require port 1 to be opened or short circuited • “a” parameters can be computed without opening or short circuiting port 1 31 Instructor: Prof.