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Two Port NetworksTwo Port Networks5.1 Is used to describe the performance of a circuit in terms of

the voltage and current at its input and output ports.5.2 Impedance parameters z(terminal voltages can be related

to the terminal current)5.3 Admittance parameters y (terminal currents can be

expressed in terms of the terminal voltages)5.4 Hybrid parameters h 5.5 Transmission parameters T (expressing the variables at the

input port in terms of the variables at the output port)

Two Port NetworksTwo Port NetworksIt is a pair of terminals through which a current may enter or leave a network is known as a portTwo terminal devices (R,L &C) result in one-port NetworksMost of the circuits we have dealt with so far are two terminal or one port circuits represented in (a)

Two Port NetworksTwo Port NetworksFour terminal or two port (one input and one output) circuits involving op amps, transistors, and transformer as shown in (b).Use of this building block is subject to 3 restrictions.1. No energy stored within the circuit2. No independent sources3. Current in = Current out4. No connection between port

Two Port NetworksTwo Port NetworksOnly the terminal variables (I1, V1, I2 and V2) are of interest.The various terms that relate these voltages and currents are called parametersOur goal is to derive six sets of these parametersWe will show the relationship between these parameters and how two port networks can be connected in series, parallel or cascade

One port or two terminal circuit

Two port or four terminal circuit

• It is an electrical network with two separate ports for input and output.

• No independent sources.

5.1 Introduction (2)5.1 Introduction (2)

Impedance parametersImpedance parametersImpedance and admittance parameters are commonly used in the synthesis of filters Also useful in the design and analysis of impedance matching networks and power distribution networks

(a) Driven by voltage sources, (b) driven by current sources

Assume no independent source in the network

5.2 Impedance parameters (1)5.2 Impedance parameters (1)

2221212

2121111

IzIzVIzIzV

where the z terms are called the impedance parameters, or simply z parameters, and have units of ohms.

The terminal voltages can be related to the terminal currents as

5.2 Impedance parameters5.2 Impedance parameters (2)(2)

z11 = impedance seen looking into port 1 when port 2 is openz21 = Transfer impedance. It is the ratio of the port 2 voltage to the port 1 current when port 2 is open

z12 = Transfer impedance. It is the ratio of the port 1 voltage to the port 2 current when port 1 is openz22 = impedance seen looking into port 2 when port 1 is open

22IVzand

11IVzand

5.2 Impedance parameters5.2 Impedance parameters (2)(2)Therefore the impedance parameters may be either calculated or measured by first opening port 2 and determining the ratios V1/I1 and V2/I1, and then opening port 1 and determining the ratios V1/I2 and V2/I2

5.2 Impedance parameters (2)5.2 Impedance parameters (2)

andIVz

11IVzand

•When z11 = z22, the two-port network is said to be symmetrical. •When the two-port network is linear and has no dependent sources, the transfer impedances are equal (z12 = z21), and the two-port is said to be reciprocal.

VV1 VV2

II2II1

Example 1

Determine the Z-parameters of the following circuit.

5.2 Impedance parameters 5.2 Impedance parameters

IV zand

IVzand

AnswerAnswer::

70404060

5.3 Admittance parameters (1)5.3 Admittance parameters (1)

2221212

2121111

VyVyIVyVyI

y yy y

where the y terms are called the admittance parameters, or simply y parameters, and they have units of Siemens.

VIyand

y11 = Short-circuit output admittancey21 = Short-circuit transfer admittance from port 2 to port 1

y12 = Short-circuit transfer admittance from port 1 to port 2y22 = Short-circuit input admittance

VIyand

VV1 VV2

II1 II2

Example 2

Determine the y-parameters of the following circuit.

VIyand

S 0.6250.5

0.50.75y

AnswerAnswer::

VIyand

S yyyy

VV1 VV2

II2II1=i

Example 3

Determine the y-parameters of the following circuit.

5.3 Admittance parameters (4)

2221212

2121111

VyVyIVyVyI

S 0.250.250.050.15

AnswerAnswer::

0.25V0.25VI0.05V0.15VI

)I2(I)I4(2iV)I2(I8IV

Apply KVL

5.4 Hybrid parameters (1)5.4 Hybrid parameters (1)

The h-parameter equivalent network of a two port network

2221212

2121111

VhIhIVhIhV

where the h terms are called the hybrid parameters, or simply h parameters, and each parameter has different units, refer above.

h11= short-circuit input impedance ()

h21 = short-circuit forward current gain

h12 = open-circuit reverse voltage-gain

h22 = open-circuit output admittance (S)

VV1 VV2

II1 II2

Example 4

Determine the h-parameters of the following circuit.

IIhand

AnswerAnswer::

VIhand

ShhhΩh

Assume no independent source

in the network

2221212

2121111

IgVgVIgVgI

where the g terms are called the inverse hybrid parameters, and each parameter has different units.

5.5 Transmission parameters (1)5.5 Transmission parameters (1)

Assume no independent source

in the network

DICVIBIAVV

where the T terms are called the transmission parameters, or simply T or ABCD parameters, and each parameter has different units.

A=open-circuit voltage ratio

C= open-circuit transfer admittance

B= negative short-circuit transfer impedance ()

D=negative short-circuit current ratio

VV11 VV22

Example 5

Determine the T-parameters of the following circuit.

1.1760.059S15.294Ω1.765

TAnswerAnswer::

DICVIBIAVV

I1720V

I17260V

)I20(I3IV)I20(I10IV

Apply KVL

VV11 VV22

II11 II22

The output port is connected to a variable load for maximum power transfer. Find RL and the maximum power transferred.

Example 6

The ABCD parameters of the two-port network below are

AnswerAnswer: V: VTH = 10V V; R = 10V V; RL = 8 = 8; Pm = 3.125W.; Pm = 3.125W.

dIcVIbIaVV

where the t terms are called the inverse transmission parameters, and each parameter has different units.

t parameter may be defined by expressing the variables at the output port in terms of the variables at the input port.

5.5 Relationship between 5.5 Relationship between parameters (1)parameters (1)

• Since the six sets of parameters relate the same input and output terminal variables of the same two port network, they should be interrelated.

• If we know one set of parameters, we can derive all the other sets from the known set

• Given the z parameters, let us obtain the y parameters

zII 1-

• Also from eq.

• Comparing with eq.

shows that

The adjoint of the [z] matrix is

and its determinant is zy -1

y yy y

zII 1-

21122211 zzzzz

• Substituting these into eq. we get

Equating terms yields

1211 zzzz

y yy y

5.5 Analysis of the terminated two 5.5 Analysis of the terminated two port circuitport circuit

• In the typical application of a two port model, the circuit is driven at port 1 and loaded at port 2.• This Fig. shows a typically terminated two port model.

• Zg= internal impedance of the source• Vg= internal voltage of the source• ZL= load impedance• Analysis of this circuit involves expressing the terminal currents and voltages as functions of two

port parameters, Vg, Zg, and ZL

-V1 V2

• Six characteristics of the terminated two port circuit define its terminal behavior

• The input impedance Zin=V1/I1• The output current I2

• The Thevenin voltage and impedance with respect to port 2• The current gain I2 /I1• The voltage gain V2/V1

• The voltage gain V2/Vg

-V1 V2

• To shows how these six characteristics are derived, we develop the expressions using z parameters to model the two port portion of the circuit

• The derivation of any one of the desired expressions involves the algebraic manipulation of the two port eqs. along with the two constraint eqs. imposed by the terminations

-V1 V2

2221212

ZIVVIzIzVIzIzV

• The impedance seen looking into port 1, that is Zin=V1/I1• Replace V2 with –I2ZL and solve the resulting expression for I2

• Then substitute this eq. into and solve for Zin

2221212

ZIVVIzIzVIzIzV

1212 zZ

21111 IzIzV 12

Lin Zz

211211

• To find I2, first solve for I1 after replacing

V1 with the RHS of , the result is

Now replace eq. into and solve

the resulting eq. for I2

2221212

ZIVVIzIzVIzIzV

21111 IzIzV 12

212gg ZIVV 11

1212 zZ

21122211

zzZzZzVz

• The Thevenin voltage with respect to port 2 equals V2 when I2=0• With I2=0. eqs. and

combine to yield

But and

Therefore

2221212

ZIVVIzIzVIzIzV

21111 IzIzV 12 2221212 IzIzV

12112102 z

VzIzVI2

gg ZIVV 11 g

ThI2 VzZ

• The Thevenin, or output, impedance is the ratio V2/I2 when Vg is replaced by a short circuit. When Vg is zero eq.

• reduces to

• Substituting eq. into eq.

• gives

• Use eq. to replace

2221212

ZIVVIzIzVIzIzV

gg ZIVV 11 gZIV 11

gZIV 11 21111 IzIzV 12

2221212 IzIzV

• With the result that

• The current gain I2/I1 comes directly from eq.

For the voltage gain V2/V1, replace I2 ineq. with its value from

eq. ; thus

2221212

ZIVVIzIzVIzIzV

ZzzzzZ

211222

1212 zZ

2221212 IzIzV

LZIV 22

• Next solve eq. for I1 as a function of V1

• and V2

• or

2221212

ZIVVIzIzVIzIzV

2221212 Z

V-zIzV

21111 IzIzV 12

2111 Z

-VzVIz 121

• Now replace I1 in eq.

• with eq. and solve the resulting

• expression for V2/V1

2221212

ZIVVIzIzVIzIzV

zzzzZzZz

2112221111

2221212 Z

V-zIzV

• To derive the voltage ratio V2/Vg, combine eqs.

• and to find I1 as a function • of V2 and Vg

• Now use eq.

• and eq. in conjunction with

21111 IzIzV 12

gL1 Zz

2221212

ZIVVIzIzVIzIzV

gg ZIVV 11 LZIV 22

gL1 Zz

LZIV 22

• Eq. To derive an expression involving

• only V2 and Vg; that is

• Which can manipulate to get the desired voltage ratio:

2221212

ZIVVIzIzVIzIzV

21122211

)( zzZzZzZz

2221212 IzIzV

212212 )()(

ZzZVzzV

5.6 Interconnected two port circuit5.6 Interconnected two port circuit• Two port circuits may be interconnected five ways:In

cascade, in series, in parallel, in series parallel and parallel series

5.6 Interconnected two port circuit5.6 Interconnected two port circuit• Consider the series connection of two port networks shown below

• For network 1 For network 2

2a22a1a21a2a

2aa1a11a1a

IzIzVIzIzV

I1a I2I2a

V2I1b I2b

2b22b1b21b2b

2bb1b11b1b

IzIzVIzIzV

ba1 III 11 ba2 III 22

5.6 Interconnected two port circuit5.6 Interconnected two port circuit• and that

• Thus, the z parameters for all the overall network are

• Or showing that the z • parameters for the overall network are the sum of the z parameters for the individual networks

222ba121b21a2ba2

212ba111b11a1ba1

)Izz)Iz(zVVV)Izz)Iz(zVVV

zzzzzzzz

22222121

12121111

ba zzz

5.6 Interconnected two port circuit5.6 Interconnected two port circuit• Consider the parallel connection of two port networks shown below

2a22a1a21a2a

2aa1a11a1a

VyVyIVyVyI

2b22b1b21b2b

2bb1b11b1b

VyVyIVyVyI

ba1 III 11 ba2 III 22

I1 I1a I2a

V1a V2a

V2bV1b

I1b I2bV1 V2

ba1 VVV 11 ba2 VVV 22

• Thus, the y parameters for all the overall network are

• Or showing that the y • parameters for the overall network are the sum of the y parameters for the individual networks

222ba121b21a2

212ba111b11a1

)Vyy)Vy(yI)Vyy)Vy(yI

yyyyyyyy

22222121

12121111

ba yyy

5.6 Interconnected two port circuit5.6 Interconnected two port circuit• Consider the cascaded connection of two port networks shown below

1 2+-V1 V1a

I1a I2I2a

V1bV2a V2b V2

I1b I2b

• Thus, the transmission parameters for all the overall network are the product of the transmission parameters for the individual transmission parameters

• or

ba TTT

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