Network Theory - Two-Port Networks

Network Theory - Two-Port Networks

InIn general, itit isis easy toto analyze any electrical network, ifif itit isis representedrepresented with an equivalent model, which gives thethe relationrelation between inputinput and output variables. For this,this, we can use twotwo portport networknetwork rrepresentations. As thethe name suggests, twotwo port networks contain twotwo ports. Among which, one port isis used as an inputinput port and thethe other port isis used as an output port. The firstfirst and second ports are called as port1 and port2 respectivelyrespectively..

One port network iis a twotwo terminalterminal electrical network inin which, current enters throughthrough one terminalterminal and leavesleaves throughthrough another terminal.terminal. Resistors, inductorsinductors and capacitors are thethe examples of one port network because each one has twotwo terminals.terminals. One port network representationrepresentation isis shownshown inin thethe followingfollowing figure.figure.

Here, the pair of terminals, 1 & 1’ represents a port. In this case, we are having only one port since it is a one port network. Similarly,, twotwo port network isis a pair of twotwo terminalterminal electrical network inin which, current enters throughthrough one terminalterminal and leavesleaves throughthrough another terminal of each port. Two port network representation is shown in the following figure. Here, one pair of terminals,terminals, 1 & 1’ representsrepresents one port, which isis called as port1 and thethe other pair of terminals,terminals, 2 & 2’ representsrepresents another port, which is called as port2..

There are fourfour variables V1,, V2,, II1 and II2 inin a twotwo port network as shown inin tthe figure.figure. Out of which, we can choose twotwo variables as independentindependent and another twotwo variables as dependent. So, we will get six possible pairs of equations. These equations representrepresent thethe dependent variables inin termsterms of indepeindependent variables. The coefficientsficients of independentindependent variables are called as parameters.. So, each pair of equations will give a set of four parameters.

Two Port Network Parameters

The parameters of a twotwo port network are called as twotwo port network parameters or simply,, twotwo port parameters. Following are thethe typestypes of twotwo portport networknetwork parameters.parameters.

Z parameters Y parameters T parameters T’ parameters h-parameters g-parameters

Now,, letlet usus discussdiscuss aboutabout thesethese twotwo portport networknetwork parametersparameters oneone byby one.one.

Z parameters

We willill get thethe followingfollowing set of twotwo equations by considering thethe variables V1 & V2 as dependent and II1 & II2 as independent.independent. The coefficientsficients of independent variables, I1 andand II2 areare calledcalled asas Z parameters.. V1 = Z11I1 + Z12I2

V2 = Z21I1 + Z22I2

The Z parameters areare

V1 Z11 = , when I2 = 0 I1

V1 Z12 = , when I1 = 0 I2

V2 Z21 = , when I2 = 0 I1

V2 Z22 = , when I1 = 0 I2

Z parameters are called as impedanceimpedance parametersparameters because thesethese are simply thethe ratiosratios of voltages and currents. Units of Z parameters are Ohm (Ω).

We can calculate twotwo Z parameters, Z11 and Z21,, by doing open circuit of port2. Similarly,, we can calculate thethe other twotwo Z parameters, Z12 and Z22 byby doingdoing openopen circuitcircuit ofof port1.port1. Hence,Hence, thethe ZZ parametersparameters areare alsoalso calledcalled asas open-circuit impedance parameters..

Y parameters

We willill get thethe followingfollowing set of twotwo equations by considering thethe variables II1 & II2 as dependent and V1 & V2 as independent.independent. The coefficientsficients of independent variables, V1 andand VV2 areare calledcalled asas Y parameters.. I1 = Y11 V1 + Y12 V2

I2 = Y21 V1 + Y22 V2

The Y parameters areare

I1 Y11 = , when V2 = 0 V1

I1 Y12 = , when V1 = 0 V2

I2 Y21 = , when V2 = 0 V1

I2 Y22 = , when V1 = 0 V2

Y parameters are called as admittance parameters because thesethese are simply,, thethe ratiosratios of currents and voltages. Units of Y parameters are mho.

We can calculate twotwo Y parameters, Y11 and Y21 by doing short circuit of port2. Similarly,, we can calculate thethe other twotwo Y parameters, Y12 and Y22 byby doingdoing shortshort circuitcircuit ofof port1.port1. Hence,Hence, thethe YY parametersparameters areare alsoalso calledcalled asas short-circuit admittance parameters..

T parameters

We willill get thethe followingfollowing set of twotwo equations by considering thethe variables V1 & II1 as dependent and V2 & II2 as independent.independent. The coefficientsficients of V2 andand -I-I2 areare calledcalled asas T parameters.. V1 = AV2 − BI2

I1 = CV2 − DI2

The T parameters areare

V1 A = , when I2 = 0 V2

V1 B = − , when V2 = 0 I2

I1 C = , when I2 = 0 V2

I1 D = − , when V2 = 0 I2

T parameters are called as transmissiontransmission parameters or ABCD parameters.. The parameters, A and D do not have any units, since thosethose are dimension less. The units of parameters, B and C are ohm and mho respectively.. We can calculate twtwo parameters, A and C by doing open circuit of port2. Similarly,, we can calculate thethe other twotwo parameters, B and D by doing short circuit of port2.

T ’ parameters

We willill get thethe followingfollowing set of twotwo equations by considering thethe variables V2 & II2 as dependent and V1 & II1 as independent.independent. The coefficientsficients of V1 andand -I-I1 areare calledcalled asas T’ parameters.. ′′ ′′ V2 = A V1 − B I1

′′ ′′ I2 = C V1 − D I1

The T’ parameters areare

′′ V2 A = , when I1 = 0 V1

′′ V2 B = − , when V1 = 0 I1

′′ I2 C = , when I1 = 0 V1

′′ I2 D = − , when V1 = 0 I1

T’ parametersrameters are called as inverseinverse tratransmission parameters or A’B’C’D’’B’C’D’ parametersparameters.. The parameters A’’ and D’ do not have any units, since thosethose areare dimensiondimension less.less. TheThe unitsunits ofof parameters,parameters, B’B’ andand C’,C’, areare OhmOhm andand MhoMho respectivelyrespectively.. We can calculate twtwo parameters, A’’ and C’, by doing an open circuit of port1. Similarly,, we can calculate thethe other twotwo parameters, B’ and D’, by doing a short circuit of port1. h-parameters

We willill get thethe followingfollowing set of twotwo equations by considering thethe variables V1 & II2 as dependent and II1 & V2 as independent.independent. The coefficientsficients of independent variables, I1 andand VV2,, areare calledcalled asas h-parameters.. V1 = h11I1 + h12V2

I2 = h21I1 + h22V2

The h-parameters are

V1 h11 = , when V2 = 0 I1

V1 h12 = , when I1 = 0 V2

I2 h21 = , when V2 = 0 I1

I2 h22 = , when I1 = 0 V2

h-parameters are called as hybrid parameters.. The parameters, h12 and h21,, do not have any units, since thosethose arere dimension-less. The units of parameters, h11 andand hh22,, areare OhmOhm andand MhoMho respectivelyrespectively..

We can calculate twtwo parameters, h11 andand hh21 by doing short circuitit of port2. Similarly,, we can calculate thethe other twotwo parameters, h12 andand hh22 by doing open circuit of port1. The h-parameters or hybrid parameters are useful in transistor modelling circuits (networks). g-parameters

We willill get thethe followingfollowing set of twotwo equations by considering thethe variables II1 & V2 as dependent and V1 & II2 as independent.independent. The coefficientsficients of independent variables, V1 andand II2 areare calledcalled asas g-parameters.. I1 = g11V1 + g12I2

V2 = g21V1 + g22I2

The g-parameters areare

I1 g11 = , when I2 = 0 V1

I1 g12 = , when V1 = 0 I2

V2 g21 = , when I2 = 0 V1

V2 g22 = , when V1 = 0 I2

g-parameters are called as inverseinverse hybrid parameters.. The parameters, g12 and g21 do not have any units, since thosethose are dimension less.less. The units of parameters, g11 andand gg22 areare mhomho andand ohmohm respectivelyrespectively..

We can calculate twtwo parameters, g11 andand gg21 by doing open circuitit of port2. Similarly,, we can calculate thethe other twotwo parameters, g12 andand gg22 by doing short circuit of port1.