Two-Port Networks Definitions Impedance Parameters Admittance - PowerPoint PPT Presentation

Two-Port Networks • Definitions • Impedance Parameters • Admittance Parameters • Hybrid Parameters • Transmission Parameters • Cascaded Two-Port Networks • Examples • Applications J. McNames Portland State University ECE 222 Two-Port Networks Ver. 1.11 1

One-Port Networks i 1 + One-Port v Network - i' 1 • A pair of terminals at which a signal (voltage or current) may enter or leave is called a port • A network having only one such pair of terminals is called a one-port network • 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 2

Two-Port Networks: Definitions & Requirements i 1 i 2 + + Two-Port v 1 v 2 Network - - i' 1 i' 2 • Two-port networks are used to describe the relationship between a pair of terminals • The analysis methods we will discuss require the following conditions be met 1. Linearity 2. No independent sources inside the network 3. No stored energy inside the network (zero initial conditions) 4. i 1 = i ′ 1 and i 2 = i ′ 2 J. McNames Portland State University ECE 222 Two-Port Networks Ver. 1.11 3

Two-Port Networks: Defining Equations I 1 ( s ) I 2 ( s ) + + Two-Port V 1 ( s ) V 2 ( s ) Network - - • If the network contains dependent sources, one or more of the equivalent resistors may be negative • Generally, the network is analyzed in the s domain • Each two-port has exactly two governing equations that can be written in terms of any pair of network variables • Like Th´ evenin and Norton equivalents of one-ports, once we know a set of governing equations we no longer need to know what is inside the box J. McNames Portland State University ECE 222 Two-Port Networks Ver. 1.11 4

Impedance Parameters + + Two-Port I 1 ( s ) V 1 ( s ) V 2 ( s ) I 2 ( s ) Network - - V 1 = z 11 I 1 + z 12 I 2 � � � � � � V 1 z 11 z 12 I 1 = 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 • Alternatively, if the circuit in the box is known, V 1 and V 2 can be calculated based on circuit analysis • Relationship can be written in terms of the impedance parameters • We can also calculate the impedance parameters after making two sets of measurements J. McNames Portland State University ECE 222 Two-Port Networks Ver. 1.11 5

Impedance Parameter Measurements + + Two-Port I 1 ( s ) V 1 ( s ) V 2 ( s ) Network - - = z 11 I 1 + z 12 I 2 V 1 = z 21 I 1 + z 22 I 2 V 2 If the right port is an open circuit ( I 2 = 0 ), then we can easily solve for two of the impedance parameters: � � z 11 = V 1 z 21 = V 2 � � � � I 1 I 1 � � I 2 =0 I 2 =0 J. McNames Portland State University ECE 222 Two-Port Networks Ver. 1.11 6

Impedance Parameter Measurements Continued + + Two-Port V 1 ( s ) V 2 ( s ) I 2 ( s ) Network - - = z 11 I 1 + z 12 I 2 V 1 = z 21 I 1 + z 22 I 2 V 2 If the left port is an open circuit ( I 1 = 0 ), then we can easily solve for 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

Impedance Parameter Measurements Summarized + + Two-Port I 1 ( s ) V 1 ( s ) V 2 ( s ) I 2 ( s ) Network - - � � z 11 = V 1 z 12 = V 1 � � � � I 1 I 2 � � I 2 =0 I 1 =0 � � z 21 = V 2 z 22 = V 2 � � � � I 1 I 2 � � I 2 =0 I 1 =0 J. McNames Portland State University ECE 222 Two-Port Networks Ver. 1.11 8

Impedance Parameter Equivalent I 1 ( s ) I 2 ( s ) z 11 z 22 + + V 1 ( s ) z 12 I 2 z 21 I 1 V 2 ( s ) - - = z 11 I 1 + z 12 I 2 V 1 = 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

Example 1: Impedance Parameters 200 Ω 40 Ω I 1 + I 2 + 500 Ω 800 Ω V 1 V 2 1 k Ω - - Find the z parameters of the circuit. J. McNames Portland State University ECE 222 Two-Port Networks Ver. 1.11 10

Example 1: Workspace J. McNames Portland State University ECE 222 Two-Port Networks Ver. 1.11 11

Example 2: Parameter Conversion I 1 ( s ) I 2 ( s ) + + Two-Port V 1 ( s ) V 2 ( s ) Network - - = z 11 I 1 + z 12 I 2 V 1 = z 21 I 1 + z 22 I 2 V 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 12

Example 2: Workspace J. McNames Portland State University ECE 222 Two-Port Networks Ver. 1.11 13

Example 2: Workspace Continued J. McNames Portland State University ECE 222 Two-Port Networks Ver. 1.11 14

Impedance & Admittance Parameters I 1 ( s ) I 2 ( s ) + + Two-Port V 1 ( s ) V 2 ( s ) Network - - Impedance Parameters V 1 = z 11 I 1 + z 12 I 2 � � � � � � V 1 z 11 z 12 I 1 = V 2 = z 21 I 1 + z 22 I 2 V 2 z 21 z 22 I 2 Admittance Parameters I 1 = y 11 V 1 + y 12 V 2 � � � � � � I 1 y 11 y 12 V 1 = I 2 = y 21 V 1 + y 22 V 2 I 2 y 21 y 22 V 2 J. McNames Portland State University ECE 222 Two-Port Networks Ver. 1.11 15

Hybrid Parameters I 1 ( s ) I 2 ( s ) + + Two-Port V 1 ( s ) V 2 ( s ) Network - - Hybrid Parameters � h 11 � � I 1 V 1 = h 11 I 1 + h 12 V 2 � V 1 � � h 12 = I 2 h 21 h 22 V 2 I 2 = h 21 I 1 + h 22 V 2 Inverse Hybrid Parameters � g 11 � � V 1 I 1 = g 11 V 1 + g 12 I 2 � I 1 � � g 12 = 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 16

Transmission Parameters I 1 ( s ) I 2 ( s ) + + Two-Port V 1 ( s ) V 2 ( s ) Network - - Transmission Parameters � a 11 V 1 = a 11 V 2 − a 12 I 2 � V 1 � � � � � � b 12 V 2 V 2 = = A I 1 a 21 a 22 − I 2 − I 2 I 1 = a 21 V 2 − a 22 I 2 Inverse Transmission Parameters � b 11 V 2 = b 11 V 1 − b 12 I 1 � V 2 � � � � � � b 12 V 1 V 2 = = B I 2 = I 2 b 21 b 22 − I 1 − I 2 b 21 V 1 − b 22 I 1 J. McNames Portland State University ECE 222 Two-Port Networks Ver. 1.11 17

Transmission Parameter Conversion I 1 ( s ) I 2 ( s ) + + Two-Port V 1 ( s ) V 2 ( s ) Network - - • Altogether there are 6 sets of parameters • Each set completely describes the two-port network • Any set of parameters can be converted to any other set • We have seen one example of a conversion • A complete table of conversions is listed in the text (Pg. 933) • 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 18

Example 3: Two-Port Measurements The following measurements were taken from a two-port network. Find the transmission parameters. Port 2 Open = 150 cos(4000 t ) V applied V 1 25 cos(4000 t − 45 ◦ ) A measured = I 1 1000 cos(4000 t + 15 ◦ ) V measured = V 2 Port 2 Shorted V 1 = 30 cos(4000 t ) V applied 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

Example 3: Workspace J. McNames Portland State University ECE 222 Two-Port Networks Ver. 1.11 20

Example 4: Two-Port Analysis 800 Ω i 1 i 2 40 Ω 160 Ω + + + v 1 v 3 16.2 v 3 v 2 200 Ω - - - Find the hybrid parameters for the circuit shown above. J. McNames Portland State University ECE 222 Two-Port Networks Ver. 1.11 21

Example 4: Workspace 800 Ω i 1 i 2 40 Ω 160 Ω + + + v 1 v 3 16.2 v 3 v 2 200 Ω - - - J. McNames Portland State University ECE 222 Two-Port Networks Ver. 1.11 22

Example 4: Workspace Continued J. McNames Portland State University ECE 222 Two-Port Networks Ver. 1.11 23

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 = − 0 . 5 µ A I 1 V 2 = 10 V = 80 µ A I 2 I 2 = 200 µ A = 5 V V 2 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 24

Example 5: Workspace J. McNames Portland State University ECE 222 Two-Port Networks Ver. 1.11 25

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 26

Example 6: Workspace 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 J. McNames Portland State University ECE 222 Two-Port Networks Ver. 1.11 27

Example 6: Workspace Continued (1) 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 J. McNames Portland State University ECE 222 Two-Port Networks Ver. 1.11 28