Two-Port Networks ntroduction n general, a network ma have n ports. The current entering one terminal leaves through the other terminal so that the net current entering all ports equals zero. Thus, a two-port network has two terminal pairs acting as access points. The current entering one terminal of a pair leaves the other terminal in the pair. Three-terminal devices such as transistors can be configured into two-port networks. Knowing the parameters of a two-port network enables us to treat it as a black box when embedded within a larger network.
ntroduction To characterize a two-port network requires that we relate the terminal quantities V, V,, and, out of which two are independent. The various terms that relate these voltages and currents are called parameters. Our goal is to stud six sets of these parameters. As with op amps, we are onl interested in the terminal behavior of the circuits. Reminder: Onl two of the four variables (V, V,, and ) are independent.
mpedance Parameters V= z+ z V = z + z The z terms are called the impedance parameters, or simpl z parameters, and have units of ohms. The values of the parameters can be evaluated b setting = 0 (input port open-circuited) or = 0 (output port open-circuited). Therefore, the z parameters are also called the open-circuit impedance parameters. 3
mpedance Parameters Note: The two equations associated with a set of parameters determine how the individual parameters are obtained. V = z + z V = z + z z z V [ ] z [ ] = = = 0 = 0 V [ ] z [ ] = = = 0 = 0 V V Sometimes z and z are called driving-point impedances, while z and z are called transfer impedances. A driving-point impedance is the input impedance of a two-terminal (one-port) device. Thus, z is the input driving-point impedance with the output port open, while z is the output driving-point impedance with the input port open. 4
mpedance Parameters Determination of the z parameters When z = z, the two-port network is said to be smmetric. This implies that the network has mirror-like smmetr about some center line; that is, a line can be found that divides the network into two similar halves. When the two-port network is linear and has no dependent sources, the transfer impedances are equal (z = z ), and the two-port is said to be reciprocal. This means that if the points of excitation and response are interchanged, the transfer impedances remain the same. 5
mpedance Parameters An two-port that is made entirel of resistors, capacitors, and inductors must be reciprocal. A reciprocal network can be replaced b the T-equivalent circuit. f the network is not reciprocal, a more general equivalent network is obtained. Reciprocal network (z = z ) Non-reciprocal network Note that both equivalent circuits satisf the z-parameter equations. V = z + z, V = z + z 6
mpedance Parameters Some two-port networks parameters might not exist. As an example, consider the ideal transformer. t is impossible to express the voltages in terms of the currents. An ideal transformer has no z parameters. However, it does have hbrid parameters (later). 7
Example z- Method Determine the z parameters for the following circuit. z z V = = + = 3 = 0 V = = = 0 z z V = = = 0 V = = 4 + = 6 = 0 [ z] é3 ù = ê 6 ú ë û Non-smmetric, reciprocal 8
Example z- Method z = = z z - z = z = + z = 3 z - z = 4 z = 4 + z = 6 [ z] é3 ù = ê 6 ú ë û 9
Example z- Find and in the following circuit. V = 40 + j0 = 00V 0 V = j30 + 50 =-0 = j 00V 0 00V 0 = j80 + j0 = = A -90 j00 = j = A 0 0
Example z-3 Calculate and in the following two-port. z + z = V = V 30 - z + z = V = 0 V 30 = 8 - j4 0 =- j4 + 8 = j0.5 V 30 = 8 + Answer: = 00 30 ma = 00 0 ma
Example z-4 Determine the s domain expressions for the z parameters. z z V 0s + = = + 0 = s s = 0 V = = 0 = 0 z z V = = 0 = 0 V = = + = 0 ( 0.s 0) [ z] é0s + ù 0 = s êë 0 0.s + 0úû Non-smmetric, reciprocal
z Example z-5 Find the z parameters for the following circuit. V = = 5 ( 5+ 0) = 9.375 = 0 0 When = 0, then V = V = 0.8V 0 + 5 z V 0.8V = = = 7.5 V 9.375 = 0 V z = = 0 ( 5+ 5) = 0 = 0 5 When = 0, then V = V = 0.75V 5+ 5 z V 0.75V = = = 7.5 V 0 = 0 [ z] é0 7.5 ù = ê 7.5 9.375 ú ë û Non-smmetric, reciprocal 3
Admittance Parameters The terms are known as the admittance parameters (or, simpl, parameters) and have units of Siemens. = V + V = V+ V Since the parameters are obtained b short-circuiting the input or output port, the are also called the short-circuit admittance parameters. [ S] [ S] = = V V V = 0 V= 0 [ S] [ S] = = V V V = 0 V= 0 4
Admittance Parameters For a two-port network that is linear and has no dependent sources, the transfer admittances are equal ( = ). A reciprocal network ( = ) can be modeled b a - equivalent circuit. f the network is not reciprocal, a more general equivalent network is obtained. Reciprocal network ( = ) Non-reciprocal network Note: mpedance and admittance parameters are collectivel referred to as immittance parameters. 5
Example - Method Obtain the parameters for the network. = = + = 0.5S V 0 5 V = 0 = =- =-0.S V 5 V = 0 = =- =-0.S V 5 V = 0 = = + = 0.67S V 5 5 V = 0 [ ] é 0.5S -0.Sù = ê- 0.S 0.67S ú ë û Non-smmetric, reciprocal 6
Example - Method =- =- 0.S = 5 + = = - = 0.5S 0 0 + = = - = 0.67S 5 5 [ ] é 0.5S -0.Sù = ê- 0.S 0.67S ú ë û 7
Example - Obtain the parameters for the following T network. = = = 0.73S V (+ 4 6) V = 0 4 6 = =- =-0.0909S V 4 6+ 6 V = 0 V V 4 = =- =-0.0909S V 4+ 6 V = 0 = = = 0.364S V (6+ 4) V = 0 [ ] é 0.73S -0.0909Sù = ê- 0.0909S 0.364S ú ë û 8
Example -3 Obtain the s domain expressions of the parameters. + slc = = sc+ = V sl sl V = 0 = =- V sl V = 0 = =- V sl V = 0 + slc = = sc+ = V sl sl V = 0 [ ] = é êë + slc - sl sl + - sl sl slc Smmetric, reciprocal ù úû 9
Example -4 Determine the parameters for the following two-port. Short circuit Port V -V V V -0V = = + + 8 4 o o o V -V V o 0= + 3 8 4 0= V - V + 6V V =-5V o o o o 0
-5Vo-Vo = =-0.75S V 8-0.75S V o = = = V -5Vo Example -4 cont d o 0.5S Vo -0V + + = 0 4 - = 0.5 S V -.5S V =-.5S V.5S V o = = =- V -5Vo o o o 0.5S
Short circuit Port Example -4 cont d 0V-V V V -V = = + + 8 4 o o o Vo Vo Vo- V 0 =- + + 8 4 0=- V + 4V + V -V V =.5V o o o o
Example -4 cont d V o = = =- -V.5V 8 o 0.05S V o - V + + = 4 o = = = 0 Vo.5Vo Vo - = - - =-0.65S V 4 4 8 V 0.65S V.5V o 0.5S [ ] o é 0.5S -0.05Sù = ê- 0.5S 0.5S ú ë û 3
Hbrid Parameters The z and parameters of a two-port network do not alwas exist. So there is a need for developing other sets of parameters. This third set of parameters is based on making V and the dependent variables. V= h+ hv = h + h V The h terms are known as the hbrid parameters (or, simpl, h parameters) because the are a hbrid combination of ratios. The values are: h h V [ ] = h = V V = 0 = 0 = h = V V = 0 = 0 V [ S] 4
Hbrid Parameters To be specific, h = short-circuit input impedance h = open-circuit reverse voltage gain h = short-circuit forward current gain h = open-circuit output admittance For reciprocal networks, h = h. V = h + h V = h + h V The h-parameter equivalent network of a two-port network 5
Example h-: deal Transformer V = V, =-n n V = h + h V = h + h V [ h] é ù 0 n = - 0 êë n úû 6
Example h- Find the hbrid parameters for the following two-port network. V V h h [ ] = h = V V = 0 = 0 = h = V V = 0 = 0 [ S] h h V = = + 3 6 = 4 V = 0 6 = =- =- (3+ 6) 3 V = 0 7
Example h- cont d h h V 6 = = = V (3+ 6) 3 = 0 = = = S V (3+ 6) 9 = 0 [ h] = é ù 4 3 - S êë 3 9 úû Note that h = h since the network is reciprocal 8
Example h-3 Determine the h parameters for the following circuit. h h V = = 0 5 = 4 V = 0 0 = =- =-0.8 (0+ 5) V = 0 [ h] é 4 0.8 ù = ê- 0.8.067S ú ë û h h V 0 = = = 0.8 V (5+ 0) = 0 = = =.067S V 5 5 0 = 0 ( + ) h = h 9
Example h-4 Find the impedance at the input port of the following circuit. = V Z V 50k L () V = h h V h Z () = h h V V Z V hhzl () V= h h Z L L hzl in L h h Z L.667k hzl Z V h 30
nverse Hbrid Parameters A set of parameters closel related to the h parameters are the g parameters or inverse hbrid parameters. These are used to describe the terminal currents and voltages as = g V + g V = g V + g The g-parameter model of a two-port network For reciprocal networks, g = g. 3
nverse Hbrid Parameters The values of the g parameters are determined as = g V + g V = g V + g g g [ S] = g = V = 0 V= 0 V = g = V = 0 V= 0 V [ ] Thus, the inverse hbrid parameters are specificall called g = open-circuit input admittance g = short-circuit reverse current gain g = open-circuit forward voltage gain g = short-circuit output impedance 3
Example g- Find the g parameters as functions of s for the following circuit. V V = g V + g V = g V + g g g = = [S] V s + = 0 V = = V s + = 0 g g = =- s + V = 0 V s s + s+ = = + = [ ] s + s s( s+ ) V = 0 Note that g = g since the network is reciprocal 33
Example g- Find the g parameters from measured data: measured (port open): V 50mV, 5μA 00mV measured (port short): V 0mV, μa V 0.5μA g g 5μA = = = V 50mV = 0 V 00 = = = 4 V 50 = 0 0.mS g g = =- =-4 0.5 V = 0 V 0mV = = = 0 k 0.5μA V = 0 g = g 34