Circuits IIEE221

Unit 8Instructor: Kevin D. Donohue

2 Port Networks –Impedance/Admittance, Transmission, and Hybird Parameters

2-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: Thévenin and Norton Equivalent Circuits:

10 V 50 100

100 10 i1

i1a

b

Show that Voc=8 V, Isc = 0.08 A, and Rth = 100

2-Port Circuits:

Now take away the source from the previous example:

Why wouldn't it make sense to talk about a Thévenin or Norton equivalent circuit in this case?

The Thévenin 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.

50

100

100 10 ia

ia

Inverse Transmission -Parameter Model:

If the circuit is linear, then a general linear relationship between the terminal voltages and currents can be expressed as:

Geometrically the equations form a linear surface over the v1-i1 plane,

therefore, only three points on the surface are necessary to uniquely determine a, b, c, d, V2, and I2. 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 isolate unknown parameters and simplify the resulting equations. In general, if there are no independent sources in the circuit then V2, and I2 will be 0. This will be the case for problems considered in this unit.

2112

2112

Idicvi

Vbiavv

Example

Determine the transmission parameter model for the given circuit.

50 100

100 10 ia

ia

i1

+v1

-

i2

+v2

-

Show that a =18/5, b= 100, c = 7/250 Siemens, d= 1.

Summary Formula for Inverse Transmission Parameters: If all independent sources deactivated, then set i1 = 0 to find:

If all independent sources deactivated, then set v1 = 0 to find:

01

2

01

2

11

ii

v

ic

v

va

01

2

01

2

11

vv

i

id

i

vb

Equivalent Circuit for Inverse Transmission Parameter Model:

If inverse transmission parameters are known, then the following circuit can be used as an equivalent circuit:

2

1i

c

c

d

1av

1bi

i1

+

v1

-

i2

+

v2

-

+ -

+ -

+ -

This circuit is helpful if implementing on SPICE without knowledgeor details of circuit from which parameters were derived.

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.

50 100

100 10 ia

ia

i1

+v1

-

i2

+v2

-

SPICE Solutions for Two-Port Parameters:

1) Consider setting v1=0, then

2) Excite the circuit with i2=1 then

3) Use SPICE to compute v2

and i1 to solve for b and d.

1

2

i

id

1

2

i

vb

1

100

1

2 i

vb 1

1

1

1

2

i

id

50

R1

100

R2

100

R3

H1

16.67n

VAma1

I2100.00

IVm2

0

V1

-1000.00m

VAm1

SPICE Solutions for Two-Port Parameters:

4) Consider setting i1=0, then

5) Excite the circuit with v2=1, then

6) Use SPICE compute v1 and i2 to solve for a and c.

1

2

v

va

1

2

v

ic

6.32778.0

11

1

v

a m282778.0

00778.

1

2 v

ic

50

R1

100

R2

100

R3

H1

5.56m

VAma

0

I1 277.78m

IVm1

7.78m

VAm2

1

V2

Transmission -Parameter Model:

Transmission parameters are related to the inverse transmission parameters by reversing the independent and dependent variables:

1

1

2

2

112

112

i

v

dc

ba

i

v

dicvi

biavv

221

221

1

1

2

2

1

1

2

2

1

DiCvi

BiAvv

i

v

i

v

DC

BA

i

v

i

v

dc

ba

Impedance/Admittance-Parameter Model:

2

1

2221

1211

2

1

2221212

2121111

i

i

zz

zz

v

v

izizv

izizvImpedance Parameters Admittance Parameters

2221212

2121111

2

1

2

1

2221

1211

2

1

2

1

1

2221

1211

vyvyi

vyvyi

i

i

v

v

yy

yy

i

i

v

v

zz

zz

Hybrid (h)/Inverse Hybrid (g)-Parameter Model:

2

1

2221

1211

2

1

2221212

2121111

v

i

hh

hh

i

v

vhihi

vhihv

Hybrid Parameters Inverse Hybrid Parameters

2221212

2121111

2

1

2

1

2221

1211

2

1

2

1

1

2221

1211

igvgv

igvgi

v

i

i

v

gg

gg

v

i

i

v

hh

hh

Relationship Between 2 Sets of Port Parameters:

Since a single set of network parameters characterizes the linear circuits completely at the input and output terminals, it is possible to derive other network parameters from this set.

Example: Consider the z and y parameter characterization of a given circuit with no independent sources:

Show that:

v

v

z z

z z

i

i

1

2

11 12

21 22

1

2

i

i

y y

y y

v

v

1

2

11 12

21 22

1

2

z z

z z

y y

y y

y

y y y y

y

y y y yy

y y y y

y

y y y y

11 12

21 22

11 12

21 22

1 22

11 22 21 12

12

11 22 21 12

21

11 22 21 12

11

11 22 21 12

y y

y y

z z

z z

z

z z z z

z

z z z zz

z z z z

z

z z z z

11 12

21 22

11 12

21 22

1 22

11 22 21 12

12

11 22 21 12

21

11 22 21 12

11

11 22 21 12

Relationship Between 2 Sets of Port Parameters:

Example: Consider the abcd and h parameter characterization of a given circuit with no independent sources:

Show that:

1

1

2

2

i

v

dc

ba

i

v v

i

h h

h h

i

v

1

2

11 12

21 22

1

2

a

c

a

adbcaa

b

hh

hh1

2221

1211

12

112221

12

22

12

11

12

1

h

hhh

h

h

h

h

h

dc

ba

Solving for Terminal Currents and Voltages from Port Parameters:

Once the port parameters are known, no other information from the circuit is required to determine the behavior of the currents and voltages at the terminals.Example: Given the z-parameter representation of a circuit, determine the resulting terminal voltages and currents when a practical source with internal resistance Rs and voltage Vs is connected to the input (terminal 1) and a load RL is connected to the output (terminal 2):

Show that:

v

v

z z

z z

i

i

1

2

11 12

21 22

1

2

Vs

Rs

RL

+v2

-

+v1

-

i1

-

i2

-

v

z R V z R

z R V R R z R

R R R z R

z R R R z R

L s s

L s L s s

L s L s

L L s s

1

11 12

21 22

12

22 22

v

R R R z R V

z R z R V

R R R z R

z R R R z R

L s L L s

L L s

L s L s

L L s s

2

11

22 21

12

22 22

Combinations of Two-Port Networks: Consider circuits A and B described by their abcd-parameters (assume independent sources zero).

If A and B are connected in series, show that the abcd parameters for the new two-port (from v1a to v2b) is given by:

+v2a

-

+v1a

-

i1a

-

i2a

-A+

v2b

-

+v1b

-

i1b

-

i2b

-B

a

a

abababab

abababab

b

b

i

v

ddbccdac

dbbacbaa

i

v

1

1

2

2

Combinations of Two-Port Networks: Consider circuits A and B described by their y-parameters (assume independent sources zero).

If A and B are connected in parallel, show that the y-parameters for the new two-port (from v1a to v2b) is given by:

i

i

y y y y

y y y y

v

v

a b a b

a b a b

1

2

11 11 12 12

21 21 22 22

1

2

A

B

+v2a

-

+v1a

-

i1a

-

i2a

-

+v2b

-

+v1b

-

i1b

-

i2b

-

+v2

-

+v1

-

i1

-

i2

-