TWO-PORT NETWORKS In many situations one is not interested in the internal organization of a network. A description relating input and output variables may be sufficient A two-port model is a description of a network that relates voltages and currents at two pairs of terminals LEARNING GOALS Study the basic types of two-port models Admittance parameters Impedance parameters Hybrid parameters Transmission parameters Understand how to convert one model into another ADMITTANCE PARAMETERS The network contains NO independent sources The admittance parameters describe the currents in terms of the voltages y21 determines the current I1 y11V1 y12V2 The first subindex identifies the output port. The second the input port. flowing into port 2 when the I 2 y21V1 y22V2 port is short - circuited and a voltage is applied to port 1 The computation of the parameters follows directly from the definition y11 I1 V1 V y12 I2 V1 V y22 2 0 y21 2 0 I1 V2 V 0 1 I2 V2 V 0 1 LEARNING EXAMPLE Find the admittance parameters for the network I1 y11V1 y12V2 I 2 y21V1 y22V2 Circuit used to determine y11, y21 I2 Circuit used to determine y12 , y22 1 3 I1 (1 )V1 y11 [ S ] 2 2 1 1 1 I2 I1 I 2 V1 y21 [ S ] 1 2 2 2 5 1 1 I 2 V2 y22 [ S ] 6 2 3 3 3 5 1 I1 I2 V2 y12 [ S ] 23 5 6 2 Next we show one use of this model An application of the admittance parameters Determine the current through the 4 Ohm resistor I1 y11V1 y12V2 I 2 y21V1 y22V2 3 1 I1 V1 V2 2 2 1 5 I 2 V1 V2 2 6 1 I V2 I1 2 A, V2 4 I 2 2 4 The model plus the conditions at the ports are sufficient to determine the other variables. 3 1 2 V1 V2 2 2 1 5 1 0 V1 V2 2 6 4 13 V2 6 8 V2 [V ] 11 2 I 2 [ A] 11 V1 LEARNING EXTENSION I1 Find the admittance (Y) parameters I2 V1 V2 I1 I1 y11V1 y12V2 I 2 y21V1 y22V2 I2 1 1 3 )V1 V1 21 42 42 42 I2 I1 21 42 I1 ( V1 I1 1 [S] 14 1 y21 [ S ] 21 y11 I2 V2 2 1 I 2 V2 21 21 10.5 I1 I2 21 10.5 1 y22 [ S ] 7 1 y12 [ S ] 21 Use the admittance (Y) parameters to find the current Io LEARNING EXTENSION IO I1 10 A V1 I2 I1 y11V1 y12V2 5 V2 I 2 y21V1 y22V2 1 y22 [ S ] 7 1 y12 [ S ] 21 Conditions at I/O ports I1 10 A 1 I 2 V2 5 Io I2 Replace in model 1 1 V1 (5 I o ) 14 21 1 1 I o V1 (5 I o ) 21 7 10 Solve for variable of interest 1 [S] 14 1 y21 [ S ] 21 y11 Io 420 [ A] 98 IMPEDANCE PARAMETERS The network contains NO independent sources V1 z11I1 z12 I 2 V2 z21 I1 z22 I 2 The ‘z parameters’ can be derived in a manner similar to the Y parameters z11 V1 I1 I z12 V1 I2 z21 2 0 V2 I1 z22 I1 0 I 2 0 V2 I2 I1 0 LEARNING EXAMPLE Find the Z parameters V1 z11I1 z12 I 2 V2 z21 I1 z22 I 2 z11 z12 Write the loop equations V1 2 I1 j 4( I1 I 2 ) V1 I1 I V1 I2 z21 2 0 z22 I1 0 V2 j 2 I 2 j 4( I 2 I1 ) rearranging V1 (2 j 4) I1 j 4 I 2 V2 j 4 I1 j 2 I 2 z11 2 j 4 z12 j 4 z21 j 4 z22 j 2 V2 I1 I 2 0 V2 I2 I1 0 LEARNING EXAMPLE Use the Z parameters to find the current through the 4 Ohm resistor V1 z11I1 z12 I 2 V2 z21 I1 z22 I 2 Output port constraint V2 4I 2 Input port constraint V1 120 (1) I1 V1 (2 j 4) I1 j 4 I 2 V2 j 4 I1 j 2 I 2 0 j 4 I1 (4 j 2) I 2 (3 j 4) 12 (3 j 4) I1 j 4 I 2 j4 48 j (16 (4 j 2)(3 j 4)) I 2 I 2 1.61137.73 LEARNING EXTENSION I1 I2 V1 V2 Find the Z parameters. Find the current on a 4 Ohm load with a 24V input source V1 12 I1 6( I1 I 2 ) V2 3 I 2 6( I1 I 2 ) z11 18, z12 6 z21 6, z22 9 I1 V1 I2 24V V2 V1 18 I1 6 I 2 V2 6 I1 9 I 2 output port constraint: V2 4I 2 input port constraint: V1 24[V ] 24 18 I1 6 I 2 0 6 I1 13 I 2 (3) 24 (39 6) I 2 I2 24 [ A] 33 4 HYBRID PARAMETERS The network contains NO independent sources V1 h11 I1 h12V2 I 2 h21 I1 h22V2 h11 V1 I1 V h21 V1 V2 h22 2 0 h12 I1 0 I2 I1 V h11 short - circuit input impedance I2 V2 h21 short - circuit forward current gain 2 0 I1 0 h12 open - circuit reverse voltage gain h22 open - circuit output admittance These parameters are very common in modeling transistors Find the hybrid parameters for this circuit LEARNING EXAMPLE Non-inverting amplifier V1 h11 I1 h12V2 Equivalent linear circuit I 2 h21 I1 h22V2 I R2 I DS V1 ( Ri R1 || R2 ) I1 h11 Ri I 2 I R 2 I DS R1R2 R1 R2 R1 ARi I1 I1 R1 R2 Ro AR R1 h21 i R R R o 1 2 V1 R1 R1 V2 h12 R1 R2 R1 R2 Vi 0 I 2 h22 V2 Ro || ( R1 R2 ) Ro R1 R2 Ro ( R1 R2 ) LEARNING EXTENSION I1 Find the hybrid parameters for the network I2 V1 V2 V1 h11 I1 h12V2 I 2 h21 I1 h22V2 I2 I1 V1 V1 I2 V2 0 V1 (12 (6 || 3)) I1 h11 14 6 2 I2 I1 h21 3 6 3 I1 0 V2 V1 6 2 V2 h12 3 6 3 I2 V2 1 h22 [ S ] 9 9 Determine the input impedance of the two-port LEARNING EXTENSION I1 V1 h11 I1 h12V2 I2 V1 I 2 h21 I1 h22V2 V2 V1 I1 4 output port constraint: V2 4I 2 V1 h11 I1 h12 (4 I 2 ) 2 3 1 h22 [ S ] 9 h11 14, h12 2 h21 3 Rin Rin 14 I 2 h21 I1 h22 (4 I 2 ) I 2 4h12 h21 I1 V1 h11 1 4h22 4(2 / 3)(2 / 3) 16 14 15.23 1 4(1 / 9) 13 Verification Rin 12 6 || 7 12 42 13 h21 I1 1 4h22 TRANSMISSION PARAMETERS ABCD parameters The network contains NO independent sources V1 AV2 BI 2 I1 CV2 DI 2 A V1 V2 B C I 2 0 V1 I2 V 2 0 I1 V2 D A open circuit voltage ratio I 2 0 B negative short - circuit transfer impedance I1 I2 V 2 0 C open - circuit transfer admittance D negative short - circuit current ratio LEARNING EXAMPLE Determine the transmission parameters V1 AV2 BI 2 I1 CV2 DI 2 A V1 V2 B C I 2 0 V1 I2 V 2 0 when I 2 0 1 j V2 V A 1 j 1 1 1 j V2 1 I I1 1 j j V2 when V2 0 1 1 j I2 I1 I1 1 1 j 1 j I1 V2 D I 2 0 I1 I2 V 2 0 D 1 j 2 j 1 (1 j )I 2 V1 1 (1 || ) I1 j 1 j B 2 j LEARNING EXTENSION I1 V1 Determine the transmission parameters V1 AV2 BI 2 I2 I3 V2 when I 2 0 10.5 V1 A 3 10.5 21 42 V I 1 I3 I1 2 C 1 [ S ] 42 21 10.5 V2 7 V2 I1 CV2 DI 2 A V1 V2 B C I 2 0 V1 I2 V I1 V2 D 2 0 I 2 0 I1 I2 V 2 0 When V2 0 I2 42 3 I1 D 42 21 2 3 V1 (42 || 21) I1 14 I1 14 ( I 2 ) 2 B 21 PARAMETER CONVERSIONS If all parameters exist, they can be related by conventional algebraic manipulations. As an example consider the relationship between Z and Y parameters V1 z11 I1 z12 I 2 V2 z21 I1 z22 I 2 V1 z11 V z 2 21 y11 y 21 1 z12 I1 I1 z11 z22 I 2 I 2 z21 y12 z11 y22 z21 z12 z22 1 1 Z z12 V1 y11 z22 V2 y21 y12 V1 y22 V2 z22 z12 z 21 z11 with Z z11z22 z21z12 In the following conversion table, the symbol stands for the determinant of the corresponding matrix Z z11 z21 z12 y , Y 11 z22 y21 y12 h h A B , H 11 12 , T y22 h21 h22 C D INTERCONNECTION OF TWO-PORTS Interconnections permit the description of complex systems in terms of simpler components or subsystems The basic interconnections to be considered are: parallel, series and cascade PARALLEL: Voltages are the same. Current of interconnection is the sum of currents The rules used to derive models for interconnection assume that each subsystem behaves in the same manner before and after the interconnection SERIES: Currents are the same. Voltage of interconnection is the sum of voltages CASCADE: Output of first subsystem acts as input for the second Parallel Interconnection: Description Using Y Parameters Interconne ction descriptio n I1 y11 I y 2 21 I YV y12 V1 y22 V2 In a similar manner I1a V1a y11a y12a I a ,Va ,Ya I a YaVa I b YbVb I V y y 2a 2a 21a 22 b Interconnection constraints : I I a I b I YaVa YbVb (Ya Yb )V I1 I1a I1b , I 2 I 2a I 2b V Va Vb V1 V1a V1b , V2 V2a V2b Y Ya Yb Series interconnection using Z parameters SERIES: Currents are the same. Voltage of interconnection is the sum of voltages Description of each subsystem Va Z a I a , Vb Z b I b Interconnection constraints Ia Ib I V Za I Zb I ( Za Zb ) I V Va Vb Z Za Zb Cascade connection using transmission parameters CASCADE: Output of first subsystem acts as input for the second Interconnection constraints : I 2a I1b V2a V1b V1 V1a V2 V2b I1 I1a I 2 I 2b V1a Aa I C 1a a Ba V2a Da I 2a V1b Ab I C 1b b Bb V2b Db I 2b V1 Aa I C 1 a V1 AV2 BI 2 I1 CV2 DI 2 V1 A I C 1 B V2 D I 2 Matrix multiplication does not commute. Order of the interconnection is important Ba Ab Da Cb Bb V2 Db I 2 Find the Y parameters for the network LEARNING EXAMPLE j2 I1 V1 y 1 j 1 S, y j 1 11a 12 a 1 1 3 1 2 j 2 2 V2 j j 5 2 5 2 [S] 1 1 Y y21a j S ; y22a j S 1 1 2 1 2 2 j j 5 2 5 2 V1 V2 j 2 I1 I 2 I1 I1 V1 I2 I2 1 2 1 V2 V1 2 I1 I 2 V2 I1 3I 2 1 2 1 1 3 1 Yb 5 1 2 1 3 Find the Y parameters for the network using a direct approach I1 Vx I2 V1 V2 Vx Vx V1 Vx V2 2V V 0 Vx 1 2 1 1 2 5 V1 V x V1 V2 1 j2 V V x V2 V1 I2 2 2 j2 I1 Replace Vx and rearrange LEARNING EXAMPLE Find the Z parameters of the network Network A Use direct method, or given the Y parameters transform to Z … or decompose the network in a series connection of simpler networks 2 2 j 3 2 j Za 2 3 2 j 2 3 2 j 2 4 j 3 2 j 1 1 Zb 1 1 5 4 j 3 2 j Z Za Zb 5 2 j 3 2 j Network B 5 2 j 3 2 j 5 6 j 3 2 j LEARNING EXAMPLE Find the transmission parameters By splitting the 2-Ohm resistor, the network can be viewed as the cascade connection of two identical networks A B 1 j C D j 2 j 1 j 1 j j A B 1 j C D j 2 j 1 j 2 j 1 j (1 j )(2 j ) (2 j )(1 j ) A B (1 j ) 2 (2 j ) j C D 2 j (2 j ) (1 j ) j (1 j ) (1 j )( j ) A B 1 4 j 2 2 C D 2 2 j 2 4 6 j 2 2 1 4 j 2 2 LEARNING by APPLICATION Given the demand at the receiving end, determine the conditions on the sending end V1 AV2 BI 2 I1 CV2 DI 2 Transmission parameters are best suited for this application In the next slide we show how to determine the transmission parameters for the line. Here we assume them known and use them for analysis | V2 | VL 300kV (line voltage) V2 | VL | 0 P P 3 | VL || I L | pf | I || I | L 2 3VL pf P 1 I2 cos pf 3VL pf Conditions at the sending end V1 AV2 BI 2 Preceived I1 CV2 DI 2 Psent V1 pfsending Psending I1 Ptotal 3 |Vline || I line | cos f Qtotal 3 |Vline || I line | sin f Vline f Power factor angle I line Determining the transmission parameters for the line V1 AV2 BI 2 I1 CV2 DI 2 I2 0 A V2 0 B V1 R Z L ZC 0.95900.27 V2 ZC V1 I2 V1 ( R Z L ) I 2 B R Z L 100.0084.84 IT IT ZC V I1 2 R Z L 2 ZC ZC I2 0 C V2 0 D R Z L 2 ZC I1 C 975.1090.13S V2 Z C2 I1 R Z L ZC 0.95900.27 I2 ZC LEARNING EXAMPLE Determine the effect of the load on the voltage gain A 20,000, Ri 1M, Ro 500, R1 1k, R2 49k Hybrid parameters are computed in next slide Ideal gain 1 G R2 50 R1 h21 h11h22 h12 h21 Using the hybrid parameters V1 h11 I1 h12V2 I 2 h21 I1 h22V2 Eliminating I1 and solving for V2 V2 h21V1 h11I 2 h11h22 h12 h21 Constraint at output port : V2 RL I 2 Solving for V2 h21 V2 V h11 1 h11h22 h12 h21 RL h11 RL 49.88 1.247 1 RL Effect of load resistance Computing the hybrid parameters for non-inverting amplifier (repeat earlier example) Non-inverting amplifier V1 h11 I1 h12V2 Equivalent linear circuit I 2 h21 I1 h22V2 I R2 I DS V1 ( Ri R1 || R2 ) I1 h11 Ri I 2 I R 2 I DS R1R2 R1 R2 R1 ARi I1 I1 R1 R2 Ro AR R1 h21 i R R R o 1 2 V1 R1 R1 V2 h12 R1 R2 R1 R2 Vi 0 I 2 h22 V2 Ro || ( R1 R2 ) Ro R1 R2 Ro ( R1 R2 ) LEARNING BY DESIGN Gain required = 10,000 on a load of 1kOhm Amplifier parameters R Ideal gain 1 2 10,000 A 20,000 R1 Ri 1M R 9.999 M Ro 500 2 R1 1k h11 1.001M, h12 1.0 104 h21 4.0 107 , h22 1.0mS From the conversion table H h12 h21 h11h22 h A B 11 h21 h21 h21 C h22 h21 D V1 AV2 BI 2 I1 CV2 DI 2 eliminating I1 1 h21 V2 RL I 2 V2 1 6667 V1 A B 1 166.7 RL RL For the final solution we will need to cascade amplifiers. Hence the transmission parameters will prove very useful Analysis of solution: -Even with infinite load the maximum gain is only 6,667 Likely causes: -R2 is higher than input resistance Ri -Desired gain is comparable to the maximum gain, A, of the Op-Amp Proposed solution: -Cascade two stages, each with ideal gain of 100. This also lowers R2 to 99kOhm Analysis of proposed solution Since the two stages will be cascaded, the transmission parameters of the proposed solution will be V1 Aa I C 1 a Ba Ab Da Cb 1 A Aa Ab Ba Cb V2 V1 A B B Aa Bb Ba Db RL Bb V2 Aa Db I 2 Ca Ba Ab Da Cb Bb V2 Db I 2 Effect of load resistance. G=10,000 Identical stages Two-Ports