Two-Port Networks One-Port Networks • Definitions i 1 + • Impedance Parameters One-Port v • Admittance Parameters Network - • Hybrid Parameters i' 1 • Transmission Parameters • A pair of terminals at which a signal (voltage or current) may • Cascaded Two-Port Networks enter or leave is called a port • Examples • A network having only one such pair of terminals is called a one-port network • Applications • No connections may be made to any other nodes internal to the network • By KCL, we therefore have i 1 = i ′ 1 • We discussed in ECE 221 how one-port networks may be modeled by their Th´ evenin or Norton equivalents J. McNames Portland State University ECE 222 Two-Port Networks Ver. 1.11 1 J. McNames Portland State University ECE 222 Two-Port Networks Ver. 1.11 2 Two-Port Networks: Definitions & Requirements Two-Port Networks: Defining Equations I 1 ( s ) I 2 ( s ) i 1 i 2 + + Two-Port + + v 1 v 2 Two-Port Network V 1 ( s ) V 2 ( s ) Network - - i' 1 i' 2 - - • Two-port networks are used to describe the relationship between a • If the network contains dependent sources, one or more of the pair of terminals equivalent resistors may be negative • The analysis methods we will discuss require the following • Generally, the network is analyzed in the s domain conditions be met • Each two-port has exactly two governing equations that can be 1. Linearity written in terms of any pair of network variables 2. No independent sources inside the network • Like Th´ evenin and Norton equivalents of one-ports, once we know 3. No stored energy inside the network (zero initial conditions) a set of governing equations we no longer need to know what is 4. i 1 = i ′ 1 and i 2 = i ′ 2 inside the box J. McNames Portland State University ECE 222 Two-Port Networks Ver. 1.11 3 J. McNames Portland State University ECE 222 Two-Port Networks Ver. 1.11 4
Impedance Parameters Impedance Parameter Measurements + + + + Two-Port Two-Port I 1 ( s ) V 1 ( s ) V 2 ( s ) I 2 ( s ) I 1 ( s ) V 1 ( s ) V 2 ( s ) Network Network - - - - V 1 = z 11 I 1 + z 12 I 2 � � � � � � V 1 z 11 z 12 I 1 = = z 11 I 1 + z 12 I 2 V 1 V 2 = z 21 I 1 + z 22 I 2 V 2 z 21 z 22 I 2 V 2 = z 21 I 1 + z 22 I 2 • Suppose the currents and voltages can be measured If the right port is an open circuit ( I 2 = 0 ), then we can easily solve • Alternatively, if the circuit in the box is known, V 1 and V 2 can be for two of the impedance parameters: calculated based on circuit analysis � � z 11 = V 1 z 21 = V 2 • Relationship can be written in terms of the impedance parameters � � � � I 1 I 1 � � • We can also calculate the impedance parameters after making two I 2 =0 I 2 =0 sets of measurements J. McNames Portland State University ECE 222 Two-Port Networks Ver. 1.11 5 J. McNames Portland State University ECE 222 Two-Port Networks Ver. 1.11 6 Impedance Parameter Measurements Continued Impedance Parameter Measurements Summarized + + + + Two-Port Two-Port V 1 ( s ) V 2 ( s ) I 2 ( s ) I 1 ( s ) V 1 ( s ) V 2 ( s ) I 2 ( s ) Network Network - - - - � � z 11 = V 1 z 12 = V 1 V 1 = z 11 I 1 + z 12 I 2 � � � � I 1 I 2 � � V 2 = z 21 I 1 + z 22 I 2 I 2 =0 I 1 =0 � � z 21 = V 2 z 22 = V 2 � � If the left port is an open circuit ( I 1 = 0 ), then we can easily solve for � � I 1 I 2 � � I 2 =0 I 1 =0 the other two impedance parameters: � � z 12 = V 1 z 22 = V 2 � � � � I 2 I 2 � � I 1 =0 I 1 =0 J. McNames Portland State University ECE 222 Two-Port Networks Ver. 1.11 7 J. McNames Portland State University ECE 222 Two-Port Networks Ver. 1.11 8
Impedance Parameter Equivalent Example 1: Impedance Parameters 200 Ω I 1 ( s ) I 2 ( s ) 40 Ω z 11 z 22 I 1 + I 2 + + + 500 Ω 800 Ω V 1 ( s ) z 12 I 2 z 21 I 1 V 2 ( s ) V 1 V 2 - - 1 k Ω - - V 1 = z 11 I 1 + z 12 I 2 Find the z parameters of the circuit. = z 21 I 1 + z 22 I 2 V 2 • Once we know what the impedance parameters are, we can model the behavior of the two-port with an equivalent circuit. • Notice the similarity to Th´ evenin and Norton equivalents J. McNames Portland State University ECE 222 Two-Port Networks Ver. 1.11 9 J. McNames Portland State University ECE 222 Two-Port Networks Ver. 1.11 10 Example 1: Workspace Example 2: Parameter Conversion I 1 ( s ) I 2 ( s ) + + Two-Port V 1 ( s ) V 2 ( s ) Network - - V 1 = z 11 I 1 + z 12 I 2 V 2 = z 21 I 1 + z 22 I 2 In general, the two defining equations can be written in terms of any pair of variables. For example, rewrite the defining equations in terms of the voltages V 1 and V 2 . J. McNames Portland State University ECE 222 Two-Port Networks Ver. 1.11 11 J. McNames Portland State University ECE 222 Two-Port Networks Ver. 1.11 12
Example 2: Workspace Example 2: Workspace Continued J. McNames Portland State University ECE 222 Two-Port Networks Ver. 1.11 13 J. McNames Portland State University ECE 222 Two-Port Networks Ver. 1.11 14 Impedance & Admittance Parameters Hybrid Parameters I 1 ( s ) I 2 ( s ) I 1 ( s ) I 2 ( s ) + + + + Two-Port Two-Port V 1 ( s ) V 2 ( s ) V 1 ( s ) V 2 ( s ) Network Network - - - - Hybrid Parameters Impedance Parameters � h 11 � � I 1 V 1 = h 11 I 1 + h 12 V 2 V 1 = z 11 I 1 + z 12 I 2 � V 1 � � � � � � � � h 12 V 1 z 11 z 12 I 1 = = I 2 h 21 h 22 V 2 V 2 z 21 z 22 I 2 I 2 = h 21 I 1 + h 22 V 2 V 2 = z 21 I 1 + z 22 I 2 Inverse Hybrid Parameters Admittance Parameters � g 11 � � V 1 I 1 = y 11 V 1 + y 12 V 2 I 1 = g 11 V 1 + g 12 I 2 � � � � � � � I 1 � � I 1 y 11 y 12 V 1 g 12 = = I 2 = y 21 V 1 + y 22 V 2 I 2 y 21 y 22 V 2 V 2 = g 21 V 1 + g 22 I 2 V 2 g 21 g 22 I 2 J. McNames Portland State University ECE 222 Two-Port Networks Ver. 1.11 15 J. McNames Portland State University ECE 222 Two-Port Networks Ver. 1.11 16
Transmission Parameters Transmission Parameter Conversion I 1 ( s ) I 2 ( s ) I 1 ( s ) I 2 ( s ) + + + + Two-Port Two-Port V 1 ( s ) V 2 ( s ) V 1 ( s ) V 2 ( s ) Network Network - - - - Transmission Parameters • Altogether there are 6 sets of parameters � a 11 V 1 = a 11 V 2 − a 12 I 2 • Each set completely describes the two-port network � V 1 � � � � � � b 12 V 2 V 2 = = A I 1 = I 1 a 21 a 22 − I 2 − I 2 a 21 V 2 − a 22 I 2 • Any set of parameters can be converted to any other set • We have seen one example of a conversion Inverse Transmission Parameters � b 11 V 2 = b 11 V 1 − b 12 I 1 • A complete table of conversions is listed in the text (Pg. 933) � V 2 � � � � � � b 12 V 1 V 2 = = B I 2 b 21 b 22 − I 1 − I 2 I 2 = b 21 V 1 − b 22 I 1 • You should have a copy of this in your notes for the final J. McNames Portland State University ECE 222 Two-Port Networks Ver. 1.11 17 J. McNames Portland State University ECE 222 Two-Port Networks Ver. 1.11 18 Example 3: Two-Port Measurements Example 3: Workspace The following measurements were taken from a two-port network. Find the transmission parameters. Port 2 Open V 1 = 150 cos(4000 t ) V applied 25 cos(4000 t − 45 ◦ ) A measured = I 1 1000 cos(4000 t + 15 ◦ ) V measured V 2 = Port 2 Shorted = 30 cos(4000 t ) V applied V 1 1 . 5 cos(4000 t + 30 ◦ ) A measured I 1 = 0 . 25 cos(4000 t + 150 ◦ ) A measured = I 2 J. McNames Portland State University ECE 222 Two-Port Networks Ver. 1.11 19 J. McNames Portland State University ECE 222 Two-Port Networks Ver. 1.11 20
Example 4: Two-Port Analysis Example 4: Workspace 800 Ω 800 Ω i 1 i 2 i 1 i 2 40 Ω 160 Ω 40 Ω 160 Ω + + + + + + v 1 v 3 16.2 v 3 v 2 v 1 v 3 16.2 v 3 v 2 200 Ω 200 Ω - - - - - - Find the hybrid parameters for the circuit shown above. J. McNames Portland State University ECE 222 Two-Port Networks Ver. 1.11 21 J. McNames Portland State University ECE 222 Two-Port Networks Ver. 1.11 22 Example 4: Workspace Continued Example 5: Two-Port Measurements The following measurements were taken from a two-port network. Find the transmission parameters. Port 1 Open Port 1 Shorted V 1 = 1 mV I 1 = − 0 . 5 µ A V 2 = 10 V = 80 µ A I 2 I 2 = 200 µ A V 2 = 5 V Hint: △ b = b 11 b 22 − b 12 b 21 , a 11 = b 22 △ b , a 12 = b 12 △ b , a 21 = b 21 △ b , and a 22 = b 11 △ b . J. McNames Portland State University ECE 222 Two-Port Networks Ver. 1.11 23 J. McNames Portland State University ECE 222 Two-Port Networks Ver. 1.11 24
Example 5: Workspace Example 6: Two-Port Analysis i 1 R 1 R 3 v + ( t ) R 4 i 2 v - ( t ) + v 1 ( t ) C 1 R 2 v 2 ( t ) - C 2 Find an expression for the transfer function, h 11 , z 11 , g 12 , g 22 , a 11 , and y 21 . J. McNames Portland State University ECE 222 Two-Port Networks Ver. 1.11 25 J. McNames Portland State University ECE 222 Two-Port Networks Ver. 1.11 26 Example 6: Workspace Example 6: Workspace Continued (1) i 1 R 1 R 3 i 1 R 1 R 3 v + ( t ) v + ( t ) R 4 i 2 R 4 i 2 v - ( t ) v - ( t ) + + v 1 ( t ) v 1 ( t ) C 1 R 2 C 1 R 2 v 2 ( t ) v 2 ( t ) - - C 2 C 2 J. McNames Portland State University ECE 222 Two-Port Networks Ver. 1.11 27 J. McNames Portland State University ECE 222 Two-Port Networks Ver. 1.11 28
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