7.1
Two 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:
Show that Voc=8 V, Isc = 0.08 A, and Rth = 100
Now take away the source from the previous example:
Why 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:
If the circuit is linear, then a most general linear relationship between the terminal voltages and currents can be expressed as:
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.
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.
Show that a =4, b= -30, c = 1/2 Siemens, d= -4, V2 =-10, and I2 = -1.
Summary Formula for the ABCD-Parameter Model:
For contribution from independent sources, set v1= i1 = 0:
If all independent sources deactivated, then set i1 = 0 to find:
If all independent sources deactivated, then set v1 = 0 to find:
Equivalent Circuit using the ABCD-Parameter Model:
If abcd parameters are known, then the following circuit can be used as an equivalent circuit:
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:
- Port variables can be constrained by attaching a zero-valued voltage or current source
- The variable in the denominator for that constraint set to a unity-valued voltage or current source
- The two-port parameter can be found directly by commanding SPICE to print out the numerator values in the ratio.
Example:
Determine the SPICE commands to find the abcd parameters for the circuit below.
1) Since circuit contains no independent sources, V2=I2=0.
2) Consider setting v1=0, then and
3) Excite the circuit with i2=1, then and
4) So have SPICE compute v2 and i1 to solve for b and d.
* Circuit example to compute abcd* parameters
V1 1 0 DC 0
R1 1 2 50
V4 2 0 DC 0
H1 1 3 V4 10
R2 3 0 100
R3 3 4 100
I2 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+00
Therefore b = 100, d = 1
5) Consider setting i1=0, then and
6) Excite the circuit with v2=1, then and
7) So have SPICE compute v1 and i2 to solve for a and c.
* Circuit example to compute abcd* parameters
I1 1 0 DC 0
R1 1 2 50
V4 2 0 DC 0
H1 1 3 V4 10
R2 3 0 100
R3 3 4 100
V2 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-03
Therefore 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:
Note the units of the z-parameters are in ohms (impedance).
Example:
Find the z-parameters of circuit below:
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:
If all independent sources deactivated, then set i2 = 0 to find:
If all independent sources deactivated, then set i1 = 0 to find:
Equivalent Circuit using the z-Parameter Model:
If z parameters are known, then the following circuit can be used as an equivalent circuit:
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:
Note the units of the y-parameters are in Siemens (admittance).
Example:
Find the y-parameters of circuit below:
Show that I1=0, I2=0, y11 = 1/22 S, y21 = -3/110 S, y12 = -3/110, y22 = 2/55 S
H (hybrid) -Parameter Model:
If the circuit is linear, then a most general linear relationship between the terminal voltages and currents can be expressed as:
The above expressions can also be represented in matrix and vector form:
See examples in text.