Concepts of Network functions (driving point and transfer function)

General Terminologies of Two-Port networks:

Network: A network is an interconnection of passive elements and dependent sources. There are no active or independent sources inside the box. The network is treated to be part of the square box shown in the figure.

Terminal: A terminal is the endpoint of the conductor which is connected to the node of the network which is brought out of the designated square box representing the network

Port:  port is a pair of terminal that is used to connect to the sources of energy i.e. independent sources and or the excitation.

A network may be one-port in which there is only one driving force or multi-port network to which multiple sources of energy are connected. We consider a 2-port network for our study.

In a multi-port network of specifically in our study a 2-port network, one port may be connected to driving force and other port may be used for measurement, for connecting to load or for inter-connecting to the input of some other networks.

Driving Point: The ports which are used for connecting sources or load and for interconnecting networks is called driving points of a network.

Network functions:

Transform impedance:  It is written as Z(s) = V(s) /I(s) is defined as the ratio of the voltage transform and the current transform at the same port. They are also called as driving point impedance.

Transform admittance: It is written as Y(s) and equals the inverse of transform impedance i.e. Y(s) = 1/Z(s) is defined as the ratio of current transform to the voltage transform at the same port.

 

Transform /Driving Point  Port-1 voltage and current Port-2 voltage and current
Impedance Z11   =   V1(s) / I1(s) Z22      =      V2(s) / I2(s)
Admittance Y11    =   I1/V1 Y22      =       I2 /V2

Transfer Function: Transfer function is defined as the ratio of the voltage or current transform at the output to the transform of voltage or current at input port:

 

The transfer function ( Z, Y, G, α)

 

 

Transfer impedance (Z21) Ratio of Voltage transform at output port (V2(s) ) to the transform of current (I1(s))at the input port Z21      =    V2(s) / I1(s)
Transfer admittance (Y21 ) Ratio of current transform (I2(s)) at output port to the transform of Voltage (V1(s) ) at the input port Y21   =     I2(s) / V1(s)
Voltage Transfer Ratio  (G21) Ratio of Voltage transform at output port (V2(s) ) to the voltage transform (V1(s)) at the input port G21     =     V2(s)  /  V1(s)
Current Transfer Ratio (α21) Ratio of Current transform at output port (I2(s) ) to the current transform (I1(s)) at the input port α21      =    I2(s)  /  I1(s)

 

When solving for the transfer function, the degree of the polynomial may be found to be greater than the degree of the denominator polynomial, in which case we can reduce it by simple division to make the degree of the numerator less than or equal to the degree of denominator polynomial.

For determining the above quantities, we should be well conversant with the series and parallel combination of the components. Using the transform of the R, L, and C components it is convenient and easy to determine as the transform allow us to find these quantities as if we are calculating the values for series and parallel combination of resistance values. It should be clear from the following examples.

 

Example-1:  For the following one-port network find the transform impedance and admittance

Solution:

In figure(a) applying transform to the components

R1      –>     R1

L1      –>      sL1

The network is a series connection of R1 and L1.

The driving point impedance of the network is:

Z(s) = R1  +  sL1

The driving point admittance is:

Y(s) = 1/Z(s)   =  1/( R1  +  sL1)

Solution:

From figure(b), applying the transform to the network components.

R2   –>     R2

L2    –>     sL2

Impedance of parallel network =  (ZAZB)/(ZA + ZB); therefore we get the driving point impedance of the parallel network  :

=   (R2.sL2) / (R2 + sL2)

The driving point admittance = 1 / Z

= (R2 + sL2) / (R2.sL2)

Practice Problem:

 

Examples of determining the driving point impedance, driving point admittance, and the transfer function of a 2-port network.

 Ex-2  Determine the transfer impedance of the network

 

Solution: The transfer impedance of the network is V2(s) / V1(s)

Writing the network components in the transform form R1 –>  R1;  R2 –>  R2; and L  –>  sL; V1 –>  V1(s) and V2 –>  V2(s) and I1 –> I1(s)

We can write down the KVL equation with the port-2 as open-circuited. With port-2 open the current I2=0. Therefore the two-loop equations are:

V1(s)  = I1(s)(R1   + R2)

V2(s)  = I1(s)R2

The transfer impedance

G21    = V2(s) / V1(s)

                             = I1(s)R2 / I1(s)(R1   + R2)

=   R2 / (R1   + R2)

The transform/driving point impedance

Z11      = V1(s) / I1(s)

                  V1(s)    = I1(s)(R1   + R2)

Z11     = V1(s) / I1(s)  =  (R1   + R2)

 

Ex-3: determine the transfer functions and the driving point impedance for the following network

Solution:

Writing the network components in transform form

L–>  sL1;            L2   –> sL2;        C1   — >  1/sC1;         V1 –>   V1(s);

V2   –>   V2(s);       I1  –>  I1(s)

With output port open (I2=0) the loop equations gives:

V1(s)       = ( sL1 + R1 +1/sC1)I1(s)

= {(s2L1C1 + sR1C1  +1)/sC1} I1(s)

V2(s)      = (R1+1/sC1)I1

                        ={(1+sR1C1)/sC1}I1

The transfer function G21

= V2(s) /V1(s)

=(1+sR1C1)/sC1} / {(s2L1C1 + sR1C1  +1)/sC1}

=(1+sR1C1) / (s2L1C1 + sR1C1  +1)

Z11           = V1(s) /I1(s)

= (s2L1C1 + sR1C1  +1)/sC1

Example-4: Determine the driving point impedance  Z11 and the transfer ratio G21 for the following network.

Updated: June 4, 2020 — 2:06 pm

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