Reciprocity of Parameters

Reciprocity Condition in two port Parameters

Reciprocity Theorem states that the two networks are said to be reciprocal of each other if the ratio of the source voltage t one port to response (current) in other branch remain unchanged upon interchanging the position of the excitation and the response

Figure-1: Network Reciprocity

From Network-1, the Response to Excitation ratio=  I2’/ Vs

From Network-2, the Response to Excitation ratio=  I1’/ Vs

therefore for the two networks to be reciprocal  :       I2’/ Vs  =  I1’/ Vs

Reciprocity in Z Parameter

The generalized Z parameters equations are:

V1  = Z11I1  + Z12I2

V2  = Z21I1 +  Z22I2

Case-1: Port-2 shorted, V2=0, I2=-I2’; and V1=Vs Case-2:  Port-1 shorted, V1=0, I1=-I1’; and V2=Vs
So the equations become:

Vs  = Z11I1  – Z12 I2’————-(1)

0   =  Z21I1 – Z22 I2’  ————–(2)

From (2)

I1 = (Z22/ Z21) I2

Putting it in eqn-1 for Vs

Vs= Z11(Z22/ Z21) I2’   – Z12 I2

Rearranging

   I2’/ Vs= Z21/{ Z11Z22   – Z12 Z21} ———-(5)

So the equations become:

0= -Z11 I1’  +  Z12 I2 —————–(3)

Vs = -Z21 I1’  +  Z22 I2—————-(4)

From (3)

I2 = (Z11/ Z12) I1

Putting it in eqn-4 for Vs

Vs= -Z21I1’  + Z22 (Z11/ Z12) I1

Rearranging

     I1’/ Vs= Z12/{ Z11Z22   – Z12 Z21} ——–(6)

Comparing equation (5) and (6) we find the for the LHS to be equal their RHS must also be equal.

Therefore the condition for reciprocity of Z parameters is:           Z12     =     Z21

 

Reciprocity of Y Parameters

The generalized Y parameter equations are:

I1= Y11V1 + Y12V2

I2= Y21V1 + Y22V2

Case-1: Port-2 shorted, V2=0, I2=-I2’; and V1=Vs Case-2:  Port-1 shorted, V1=0, I1=-I1’; and V2=Vs
Substituting these values, the equations become:

-I2’= Y21Vs

Rearranging

     -I2’/ Vs  = Y21———-(7)

Substituting these values, the equations become:

-I1’= + Y12Vs

Rearranging

      -I1’/ Vs  =  Y12——–(8)

Comparing equation (7) and (8) we get the condition for reciprocity of Y parameters as    Y12  = Y21

Condition of Reciprocity of Transmission Parameters:

The generalized equation of the transmission parameters are:

V1= AV2 + BI2

I1= CV2 + DI2

Case-1: Port-2 shorted, V2=0, I2=-I2’; and V1=Vs Case-2:  Port-1 shorted, V1=0, I1=-I1’; and V2=Vs
Substituting these conditions, the equations for  Vbecome:

Vs= 0 + B(-I2)

So the equations become:

Vs= -BI2

Rearranging

-I2’/ Vs= 1/B  ———-(9)

 

Substituting these, the equations become:

0   = AVs + BI2      —————-(10)

-I1’= CVs + DI    —————- (11)

From eqn(10)

I2= -(A/B)Vs

Substituting in eqn(11) we get

-I1’= CVs – D(A/B)Vs

Rearranging:

-I1’= {(AD   –  BC)/B}Vs ——–(12)

Comparing eqn(9) and eqn(12) for the two networks to be reciprocal their ratio of response to excitation must be equal, therefore eqn (9) must be equal to eqn (12)

Therefore the reciprocal condition for transmission parameters is:

(AD   –  BC)/B      = 1/B

AD – BC                =  1

Writing the above equation in matrix form; the reciprocal condition of transmission parameters are:

Condition of Reciprocity in hybrid parameters

The generalized equation of hybrid (h) parameters are:

V1= h11I1 + h12V2

I2= h21I1 + h22V2

Case-1: Port-2 shorted, V2=0, I2=-I2’; and V1=Vs Case-2:  Port-1 shorted, V1=0, I1=-I1’; and V2=Vs
So the equations become:

Vs= h11I1 ………………… (13)

-I2’= h21I1               or  I1 =-I2’/ h21

Substituting value of I1 in eqn(13) we get:

Vs= -(h11 / h21)I2

I2’  / Vs  = – h21 / h11      ———————- (14)

 

So the equations become:

0= -h11I1’ + h12Vs

Rearranging:

h11I1’ =  h12Vs

I1’/ Vs =  h12/ h11   ————–(15)

 

For reciprocity, eqn 14 must be equal to eqn(15), so,

– h21 / h11      =   h12/ h11

Therefore, the condition of reciprocity in h parameter is:     h21 = – h12

Ex-1: Show that network is reciprocal or not

Ex-2: Reciprocity in Y Parameter

 

 

Updated: April 5, 2020 — 2:05 pm

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