14031202 Circuit Theory Two-Port Circuits

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

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.

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

Two-Port Building Block

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.

Example of a two-port Circuit

Source Load (e.g. (e.g. CD speaker) player)

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

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”

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)

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.

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

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

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?

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:

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:

P 8.13 (p.699) Find g parameters?

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

Instructor: Prof. Adnan Gutub 13 14031202 Circuit Theory Chapter 18 Two-Port Circuits

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?

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. Adnan Gutub 15 14031202 Circuit Theory Chapter 18 Two-Port Circuits

Cont. Solution Example 18.3

AP 18.4 (p.686) Answers: The following measurements First calculate the b-parameters: made on two-port resistive circuit: With port 1 open:

V2 = 15 V, V 1 = 10 V, I2 = 30 A With port 1 short-circuited:

V2 = 10 V, I2 = 4 A, I1 = -5 A

Calculate the z parameters ?

Hint: you need to find b parameters first. 33

Instructor: Prof. Adnan Gutub 16 14031202 Circuit Theory Chapter 18 Two-Port Circuits

Reciprocal Two-Port Circuit Reciprocal : if interchange of ideal voltage source at one port with ideal ammeter at other port produces same ammeter reading.

Symmetric Two-Port Circuits

Reciprocal two-port circuit is symmetric if its ports can be interchanged without disturbing values of the terminal currents and voltages.

Instructor: Prof. Adnan Gutub 17 14031202 Circuit Theory Chapter 18 Two-Port Circuits

Figure 18.6 Four examples of symmetric two-port circuits. (a) A symmetric tee. (b) A symmetric pi. (c) A symmetric bridged tee. (d) A symmetric lattice.

Assessment Problem 18.5 Answers:

z11 = z 22 = V 1 / I 1 = 95/5 = 19 Ω

11.52 = 19 I1 - z12 (2.72) 0 = z 12 I1 - 19 (2.72)

Solving for z 12

2 ( z 12 ) + (72/17) z 12 - (6137 /17) = 0

V1 = z 11 I1 + z 12 I2 → z 12 = z 21 = 17 Ω

V2 = z 21 I1 + z 22 I2

Instructor: Prof. Adnan Gutub 18 14031202 Circuit Theory Chapter 18 Two-Port Circuits

Figure 18.7 A terminated two-port model.

• Zg ‰ Internal impedance of the source

• Vg ‰ Internal voltage of the source

38 • ZL ‰ Load impedance

Six Important Properties of Terminated Two-Port Circuit

1. Input impedance Zin = V 1/I 1 or Yin = I 1/V 1

2. Output current I2

3. Thevenin voltage and impedance ( VTh , ZTh ) respect to port 2

4. Current gain I2/I 1

5. Voltage gain V2/V 1

6. Voltage gain V2/V g

Instructor: Prof. Adnan Gutub 19 14031202 Circuit Theory Chapter 18 Two-Port Circuits

Instructor: Prof. Adnan Gutub 20 14031202 Circuit Theory Chapter 18 Two-Port Circuits

Interconnected Two-Port Circuits • Design of large systems is: • Simplified by first designing subsections (usually modeled by two-port circuits) • Then, interconnecting these units to complete the system. a. Cascade: Better use [A] b. Series: [Z ] c. Parallel: [Y] d. Series-parallel: [H].

e. Parallel-series: [G]. 43

Instructor: Prof. Adnan Gutub 21