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Two-Port Networks ELL 100 - Introduction to Electrical Engineering LECTURE 20: TWO-PORT NETWORKS 1 Two-port networks • A port of a network consists of two terminals — one for the entry of current and other for its exit. • The currents in the two terminals of a port are equal and opposite. • ‘Porte’ in French means ‘door’. IIT Delhi 2 1-port 2-port 2-port 2-port 1-port IIT Delhi 3 Single port networks Thevenin and Norton Equivalent Circuits IIT Delhi 4 Single port networks I1 + One port V1 - network Zin Yin Input impedance Input admittance V I Z 1 Y 1 in I in 1 V1 IIT Delhi 5 Single port networks Example 1: Determine the input impedance of the circuit below I1 I1 Z2 Z3 Zin IIT Delhi 6 Single port networks Example 1: Determine the input impedance of the circuit below I 1 Vin Iin I1 I1 Z2 I1 Z2 => VZIin(1 ) 2 in Z3 Zin => ZZin (1 ) 2 IIT Delhi 7 Single port networks Example 2: Determine the output impedance of the circuit below I1 Iout + Vout Z1 I1 Z3 Zout - IIT Delhi 8 Single port networks Example 2: Determine the output impedance of the circuit below I1 Iout + Vout IIIout 11 (1 ) Vout Z1 Z1 I1 V Z Z3 Zout out 1 - => Zout Iout 1 IIT Delhi 9 Two-port networks I1 I2 + + Two port V1 V2 - network - Note the sign conventions for voltage polarity and current direction IIT Delhi 10 Two-port networks Various two-port descriptions: i g(,) v v Admittance or 1 1 1 2 iv g() model i2 g 2(,) v 1 v 2 Port current Port voltage v r(,) i i vi r() or 1 1 1 2 Impedance model v2 r 2(,) i 1 i 2 v1 h 1(,) i 1 v 2 Hybrid model i2 h 2(,) i 1 v 2 IIT Delhi 11 Two-port networks Example: Transistor Amplifier in Common-Emitter (CE) Fixed-Bias Configuration IIT Delhi 12 Two-port networks Small signal (AC) equivalent circuit IIT Delhi 13 Two-port networks Equivalent circuit models for transistors Hybrid equivalent model re equivalent model IIT Delhi 14 Y-Parameters The admittance or Y-parameters of a two port network is defined by I1 y11 y12 V1 I 2 y21 y22 V2 In scalar form I1 I2 I1 y 11 V 1 y 12 V 2 + + Input Output V1 The Network V2 I2 y 21 V 1 y 22 V 2 _ Port Port _ IIT Delhi 15 Y-Parameters I y 1 y11 is the input admittance seen looking into port 1 11 V when port 2 is shorted. 1 V 0 2 I y12 is a transfer admittance. It is the ratio of the current y 1 induced at port 1 to the voltage applied at port 2 with 12 V V 0 port 1 shorted. 2 1 I y21 is a transfer admittance. It is the ratio of the current y 2 21 V induced at port 2 to the voltage applied at port 1 with 1 port 2 shorted. I y is the input admittance seen looking into port 2 y 2 22 22 V 2 when port 1 is shorted. 16 Condition for Reciprocity A network is said to be reciprocal, if '' yy II12 12 21 IIT Delhi 17 Condition for Symmetry A two-port network is said to be symmetrical if its ports can be interchanged without changing the port voltages and currents. For Symmetric II12 Network or yy11 22 VV12 VV2100 IIT Delhi 18 Y-Parameters Numerical 1: Determine the admittance parameters for the shown circuit. Is the network reciprocal or symmetrical? I1 I2 Y2 + + 0.5V1 V1 Y1 Y3 V2 - - IIT Delhi 19 I1 I2 IYVYVVYYVYV111212 ()() 12122Y 2 + 0.5V + IVYVYVVYVYYV0.5 ( ) (0.5 ) ( 1 ) 2V1 132221Y1 Y3 21232 V2 - - IV11 YYY1 2 2 y11 Y 1 Y 2, y 12 Y 2 IV22 0.5 YYY2 2 3 y210.5 Y 2 , y 22 Y 2 Y 3 20 Y-Parameters Numerical 2: Compute the Y-parameters of the shown circuit. Is the network reciprocal or symmetrical? ˆ I1 1W I1 I2 1:a + + + ˆ V1 1W 1W V1 V2 - - - IIT Delhi 21 I 1W 1 ˆ I IVVVVVVV ( ˆˆ )1 2 2 I1 2 1 1 1 1 1 1 1a 2 1:a + + + 1ˆ 1 ˆ ˆ 1 2 IIVVVVV2 1 V 1 () 1 1 1 Vˆ 2 V a1 a1W a 1Wa2 1 2 1 - - 2- IV a 11 IV 12 Network is reciprocal 22 a a2 IIT Delhi 22 Y-Parameters Numerical 3: Find the Y-parameters of the circuit shown below. Is the network reciprocal or symmetrical? 5Ω I1 I2 + + V1 20Ω 15Ω V2 _ _ IIT Delhi 23 I 5 Ω I 1 2 VI 20.......( 1 ) + 1 a Ia 5 V1 20 Ω II .......(2 ) a 25 1 _ sub (1) (2) I 1 YS 1 11 V 4 V1 5I2 1 I2 1 Y21 S V1 5 24 24 V 15I .......(3) I1 5 Ω I2 2 x + I 5 x I x I 2.......(4) 15 Ω V2 25 _ sub (3) (4) I 2 4 Y22 S V2 15 V2 5I1 I1 1 1 1 Y S 12 Network is 4 5 V2 5 Y S 1 4 reciprocal 5 15 25 25 Y-Parameters Numerical 4: Find the Y-parameters of the circuit shown. Is the network reciprocal or symmetrical? I 10 Ω 1 2 Ω j4 Ω I2 + + + V1 10I2 V2 _ -j20 Ω _ _ IIT Delhi 2626 I 2Ω 10Ω 1 j4Ω I2 + + 10I V 2 1 _ _ I2 0 V1 (2 j4)I1 I1 1 Y11 (0.1 - j0.2) S V1 2 j4 I2 Y21 0 S V1 IIT Delhi 27 I1 2 Ω j4 Ω 10 Ω I2 + + 10I 2 _ V -j20 Ω 2 _ I Y 2 (0.05 j0.025) S 22 V -10I 2 I 2 ........(1) 1 2 j4 sub (2) (1) I1 V2 V2 - 10I2 Y12 (-0.1 j0.075) S I2 V - j20 10 2 In matrix form; 1 1 0.1 j0.2 0.1 j0.075 2I2 V2 .......(2) 10 - j20 Y S 0 0.05 j0.025 28 Terminated two-port networks I1 I2 + + Two port Y V1 V2 L - I1networyk11 y12 -V1 I 2 y21 y22 V2 I2 YLV2 y21V1 y22V2 0 y21V1 (y22 YL )V2 I1 y11 y 12 V 1 0 y21 y 22 YL V 2 IIT Delhi 29 Terminated two-port networks I1 y12 0 y22 YL (y22 YL )I1 From Cramer’s rule, V1 y11 y12 y11(y22 YL ) y12 y21 y21 y22 YL y12 y21 => The input admittance Yin y11 (y22 YL ) Also, y21 V 1 () y 22 YL V 2 Vy => Voltage Gain 2 21 y VV 21 V1 y 22 YL 21yY 22 L 30 Terminated two-port networks Rs I1 I2 + + y yV12 2 yV21 1 + 11 y22 vs V1 V2 YL - - - Yin Terminated two-port Y-parameter model IIT Delhi 31 Z-Parameters The impedance or Z-parameters of a two port network is defined by V1 z 11 z 12 I 1 V2 z 21 z 22 I 2 In scalar form I1 I2 + + V1 z 11 I 1 z 12 I 2 Input Output V V _1 Port The Network Port _2 V2 z 21 I 1 z 22 I 2 IIT Delhi 32 Z-Parameters V z 1 z11 is the input impedance seen looking into port 1 11 I I 0 when port 2 is open. 1 2 V z12 is a transfer impedance. It is the ratio of the z 1 12 I voltage induced at port 1 to the current applied at port I 0 2 1 2 with port 1 open. V z21 is a transfer impedance. It is the ratio of the z 2 21 I voltage induced at port 2 to the current applied at port 1 1 with port 2 open. V z 2 z22 is the input admittance seen looking into port 2 22 I when port 1 is open. 2 33 Conditions for Reciprocity and Symmetry A network is said to be reciprocal if zz12 21 A network is said to be symmetrical if VV12 or zz11 22 II12 II2100 34 Z-Parameters Numerical 1: Compute the Z-parameters of the given circuit. Is the network reciprocal or symmetrical? R I1 2 I2 + + R V1 R1 3 V I3 2 - - 35 VRIRI1 1 1 1 3 R VRIRI2 3 2 3 3I1 2 I2 + + 0 RIRIRRRI ( ) R 1 1 3V 21 1 2R1 3 3 3 V I3 2 R1 - R3 - III3 1 2 RRRRRR1 2 3 1 2 3 36 2 R1 RR13 VRII1() 1 1 2 RRRRRR1 2 3 1 2 3 RRRRR1() 2 3 1 3 II12 RRRRRR1 2 3 1 2 3 R I1 2 I2 + + RRRR 2 V1 R1 VIRI1 3 3 () V 3 2I3 1 32 2 - RRRRRR1 2 3- 1 2 3 RRRRR1 3 3() 1 2 II12 RRRRRR1 2 3 1 2 3 IIT Delhi 37 R I1 2 I2 + R1(R2 R3 ) R1R3 + Network isV z11 Rz12 R R RR3 R R R 1 1 I 1 2 3 1V2 2 3 reciprocal 3 R R R (R R ) - z21 z21 1 3 3 - 1 2 R1 R2 R3 R1 R2 R3 38 Z-Parameters Numerical 2: Given the following circuit, determine the Z-parameters. I I 1 8 W 10 W 2 + + V1 20W 20 W V2 _ _ IIT Delhi 39 z11 = 8 + 20||30 = 20 W z22 = 20||30 = 12 W I I 1 8 W 10 W 2 + + V 20I 8I z 1 V 2 20 8I z 2 8 W 12 I V1 1 20W 2 2012W V2 2 I 0 20 30 I2 1 _ _ Network is reciprocal (contains z z 8 W only passive elements) 21 12 40 I I 1 8 W 10 W 2 The +Z parameters can be expressed in matrix form as+ V1 V 20W20 8 I20W V2 1 1 _ _ V2 8 12I 2 IIT Delhi 41 Z-Parameters Numerical 3: Compute the Z-parameters of the given circuit.
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