7.1Two Port Circuits:Network parameters characterize linear circuits that have both input and output terminals, in terms of linear equations that describe the voltage and current relationships at those terminals. This model provides critical information for understanding the effects of connecting circuits, loads, and sources together at the input and output terminals of a two-port circuit. A similar model was used when dealing with one-port circuits.Review example: Thevenin and Norton Equivalent Circuits:10 V50 100 100 10 i1i1abShow that Voc=8 V, Isc = 0.08 A, and Rth = 100Now take away the source from the previous example:7.250 100 100 10 i1i1Why wouldn't it make sense to talk about a Thevenin or Norton equivalent circuit in this case?The Thevenin and Norton models must be extended to describe circuit behavior at two ports.Label the terminal voltage and currents as v1, i1, v2, and i2 and develop a mathematical relationship to show their dependencies.ABCD (or Chain) -Parameter Model:7.3If the circuit is linear, then a most general linear relationship between the terminal voltages and currents can be expressed as: v av bi Vi cv di I2 1 1 22 1 1 2    Since the above system of equations forms a linear surface over the v1 - i1 plane, only three points on the surface are necessary to determine the a, b, c, d, V2, and I2 values that uniquely determine the surface. So if the circuit response is known for three different values of the v1 and i1 pair, six equations with six unknowns can be generated and solved.This problem can be simplified by strategically setting the v1 and i1 values to zero in order to isolated certain unknown parameters and simplify the resulting equations. Example: Determine the abcd-parameter model for the given circuit.7.450 100 100 10 iaiai1+v1-i2+v2-Show that a =18/5, b= -100, c = 7/250 Siemens, d= -1, V2 =0, and I2 = 0.Example: Determine the abcd-parameter model for the given circuit.+ -10 30 10 10 Vi1+v1-i2+v2-Show that a =4, b= -30, c = 1/2 Siemens, d= -4, V2 =-10, and I2 = -1.Summary Formula for the ABCD-Parameter Model:7.5For contribution from independent sources, set v1= i1 = 0:V v I iv i v i2 21 102 21 10     If all independent sources deactivated, then set i1 = 0 to find:avvcivi i  21102110 If all independent sources deactivated, then set v1 = 0 to find:bvidiiv v  21102110 Equivalent Circuit using the ABCD-Parameter Model:If abcd parameters are known, then the following circuit can be used as an equivalent circuit:7.6+-+-+-12cidc12cIav1bi1V2i1+v1-i2+v2-This circuit is helpful for implementing on SPICE.SPICE Solutions for Two-Port Parameters:As shown on a previous slide, by strategically selecting the constraints on certain port variables, the two-port parameters are equal to ratios of other port variables. Therefore:1. Port variables can be constrained by attaching a zero-valued voltage or current source7.72. The variable in the denominator for that constraint set to a unity-valued voltage orcurrent source3. The two-port parameter can be found directly by commanding SPICE to print outthe numerator values in the ratio. Example:Determine the SPICE commands to find the abcd parameters for the circuit below.50 100 100 10 iaiai1+v1-i2+v2-1) Since circuit contains no independent sources, V2=I2=0.2) Consider setting v1=0, then12ivb  and 12iid 3) Excite the circuit with i2=1, then 12ivb  and 11id V1=0V4=0 R1=50H1 = 10iaia R2=100 R3=100I2=1i1+v2-134207.84) So have SPICE compute v2 and i1 to solve for b and d.* Circuit example to compute abcd* parametersV1 1 0 DC 0R1 1 2 50V4 2 0 DC 0H1 1 3 V4 10R2 3 0 100R3 3 4 100I2 0 4 DC 1.DC I2 1 1 1.PRINT DC V(4) I(V1).END Results: I2 V(4) I(V1) 1.000E+00 1.000E+02 1.000E+00Therefore b = 100, d = 15) Consider setting i1=0, then12vva  and 12vic 6) Excite the circuit with v2=1, then 11va  and 12vic +v1-V4=0 R1=50H1 = 10iaia R2=100 R3=100i2I1=0V2=1134207.97) So have SPICE compute v1 and i2 to solve for a and c.* Circuit example to compute abcd* parametersI1 1 0 DC 0R1 1 2 50V4 2 0 DC 0H1 1 3 V4 10R2 3 0 100R3 3 4 100V2 0 4 DC 1.DC V2 1 1 1.PRINT DC V(1) I(V2).END Results: V2 V(1) I(V2) 1.000E+00 -2.778E-01 -7.778E-03Therefore a = 1/(-2.771E-1),c = (7.778E-3)/(2.778E-1)Z (impedance) -Parameter Model:If the circuit is linear, then a most general linear relationship between the terminal voltages and currents can be expressed as:7.10 v z i z i Vv z i z i V1 11 1 12 2 12 21 1 22 2 2    Note the units of the z-parameters are in ohms (impedance).Example:Find the z-parameters of circuit below:+ -10 30 10 10 Vi1+v1-i2+v2-Show that V1=2, V2=-2, z11 = 8, z21 = 2, z12 = 2, z22 = 8Summary Formula for the z-Parameter Model:For contribution from independent sources, set i1= i2 = 0:V v vi i i i1 102 202 1 2 1     V7.11If all independent sources deactivated, then set i2 = 0 to find:zvizvii i11110212102 2   If all independent sources deactivated, then set i1 = 0 to find:zvizvii i12120222201 1   Equivalent Circuit using the z-Parameter Model:If z parameters are known, then the following circuit can be used as an equivalent circuit:7.12z11zi12 2V1zi21 1z22V2i1+v1-i2+v2-This circuit is helpful for implementing on SPICE.Y (Admittance) -Parameter Model:If the circuit is linear, then a most general linear relationship between the terminal voltages and currents can be expressed as:7.13 i y v y v Ii y v y v I1 11 1 12 2 12 21 1 22 2 2    Note the units of the y-parameters are in Siemens (admittance).Example:Find the y-parameters of circuit below:10 30 20 i1+v1-i2+v2-Show that I1=0, I2=0, y11 = 1/22 S, y21 = -3/110 S, y12 = -3/110, y22 = 2/55 SH (hybrid) -Parameter Model:If the circuit is linear, then a most general linear relationship between the terminal voltages and currents can be expressed as: v h i h v Vi h i h v I1 11 1 12 2 12 21 1 22 2 2    7.14The above expressions