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Circuits II EE221 Unit 8 Instructor: Kevin D. Donohue. 2 Port Networks –Impedance/Admittance, Transmission, and Hybird Parameters. 2-Port Circuits. - PowerPoint PPT Presentation
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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 each equation forms a planar 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-i1 pairs, six equations with six unknowns can be generated and solved.
This problem can be simplified by strategically setting v1 and i1 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 the problems considered in this unit.
2112
2112
IdicviVbiavv
Example
Determine the inverse 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 are deactivated, set i1 = 0 to
find:
If all independent sources are deactivated, set v1 = 0 to find:
01
2
01
2
11
ii
vic
vva
01
2
01
2
11
vv
iid
ivb
Equivalent Circuit for Inverse Transmission Parameter Model:
If inverse transmission parameters are known, then the following circuit can be used as an equivalent circuit:
21 ic
cd
1av
1bi
i1
+
v1
-
i2
+
v2
-
+ -
+ -
+ -
This circuit is helpful when implementing in SPICE withoutknowledge or details of circuit from which parameterswere derived.
SPICE Solutions for Two-Port Parameters:As shown on previous slides, 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 other port can be excited by a unity-valued source (or some other convenient value).
Place meters at the remaining values need to evaluate the ratios.
The two-port parameter can be found computing the ratio values from those found directly by SPICE.
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
iid
1
2
ivb
1100
1
2
iv
b 11
1
1
2
iid
50
R1
100
R2
100
R3
H1
16.67n
VAma1
I2100.00
IVm2
0V1
-1000.00mVAm1
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
vva
1
2
vic
6.32778.011
1
v
a m282778.0
00778.
1
2 vic
50
R1
100
R2
100
R3
H1
5.56m
VAma
0
I1 277.78m
IVm1
7.78mVAm2
1V2
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
iv
dcba
iv
dicvibiavv
221
221
1
1
2
2
1
1
2
21
DiCviBiAvv
iv
iv
DCBA
iv
iv
dcba
TransmissionParameters
Inverse TransmissionParameters
Impedance/Admittance-Parameter Model:
2
1
2221
1211
2
1
2221212
2121111
ii
zzzz
vv
izizvizizv
Impedance Parameters Admittance Parameters
2221212
2121111
2
1
2
1
2221
1211
2
1
2
11
2221
1211
vyvyivyvyi
ii
vv
yyyy
ii
vv
zzzz
Hybrid (h)/Inverse Hybrid (g)-Parameter Model:
2
1
2221
1211
2
1
2221212
2121111
vi
hhhh
iv
vhihivhihv
Hybrid Parameters Inverse Hybrid Parameters
2221212
2121111
2
1
2
1
2221
1211
2
1
2
11
2221
1211
igvgvigvgi
vi
iv
gggg
vi
iv
hhhh
Relationship Between 2 Sets of Port Parameters:
Since a single set of network parameters characterize 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
yy y y y
yy y y y
yy y y y
yy 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
zz z z z
zz z z z
zz z z z
zz 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
iv
dcba
iv v
i
h h
h h
i
v
1
2
11 12
21 22
1
2
ac
aadbc
aab
hhhh
1
2221
1211
12
112221
12
22
12
11
12
1
hhhh
hh
hh
hdcba
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 RR 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 VR 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
a
a
aa
aa
bb
bb
b
b
iv
ddbccdacdbbacbaa
iv
dcba
dcba
iv
1
1
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:
2
1
22222121
12121111
2
1
2221
1211
2221
1211
2
1
vv
yyyyyyyy
vv
yyyy
yyyy
ii
baba
baba
aa
aa
bb
bb
A
B
+v2a
-
+v1a
-i1a
-
i2a
-
+v2b
-
+v1b
-i1b
-
i2b
-
+v2
-
+v1
-
i1
-
i2
-