UNIT-3
Network Theorems – I
Q.1) State Millman’s theorem, using the same calculate current through the load in the circuit shown in
Fig.
JAN.2015, JAN.2014
Q.2) Using superposition theorem, obtain the response I for the network shown in Fig.
JAN.2014, JUNE 2013, JUNE 2015
Sol : Step 1 : Consider 8Ω 1350 volt voltage source alone in the network. Replace remaining current
sources by open circuit as shown in the Fig.(a). As two current source are open circuited, 2 resistor is
connected to open terminals. It can be neglected as current can not flow through it. Hence the equivalent
network can be drawn as shown in the Fig.(b).
Q.3) In the circuit shown in Fig. Find Vx and prove reciprocity theorem
JUNE 2015, JUNE 2013
Ans.: Now Vx = I2 (-j2)I2 can be obtained by current division in parallel circuits.
Now interchange the positions of current I and Vx as shown in Fig. 11 (a).
The voltage Vx is same as before. Thus Reciprocity theorem is verified, as the ratio of Vx to I remains
same in both cases.
Q. 4) State and explain Reciprocity theorem.
JAN.2015
Sol. : Reciprocity Theorem: In any linear network consisting of linear and bilateral impedances and
active sources, the ratio of voltage V introduced in one loop to the current I in other loop is same as the
ratio obtained if the positions of V and I are interchanged in the network. While calculating the ratio, the
sources other than one which is considered to obtain the ratio must be replaced by their internal
impedances.
Q. 5) State and explain Super position theorem.
JUNE 2014
Ans. : Superposition Theorem This Theorem is applicable for linear and bilateral networks. Let us see
the statement of the theorem. Statement : In any multisource complex network consisting of linear
bilateral elements, the voltage across or current through, any given element of the network is equal to
the sum of the individual voltages or currents, produced independently across or in that element by each
source acting independently, when all the remaining sources are replaced by their respective internal
impedances. If the internal impedance of the sources are unknown then the independent voltage
sources must be replaced by short circuit while the independent current sources must be replaced by
an open circuit. The theorem is also known as superposition principle. In other words, it can be stated
as, response in any element of linear, bilateral network containing more than one sources is the sum of
the responses produced by the sources, each acting independently. The response means the voltage
across the element or the current in the element. The superposition theorem does not apply to the power
as power is proportional to square of the current, which is not a linear function.
Explanation of superposition theorem Consider a network, shown in the Fig having two voltage
sources E1 and E2..let us calculate, the current in branch A-B of the network, using Superposition
theorem.
Case i) :-According to Superposition theorem, consider a network each source independently. Let
source V is acting independently. At this time, other sources must be replaced by internal impedances.
But as internal impedances of E2 is not given, the source E2 must be replaced by short circuit. Hence
circuit becomes, as shown in the Fig. Using any of the network reduction techniques discussed in last
chapter, obtain the current through branch A-B i.e. I’AB due to source E1 alone. Case ii) Now consider
source E2 alone, with E1 replaced by a short circuit, to obtain the current, through branch A-B. The
corresponding circuit is shown in the Fig. Obtain I’’AB due to E2 alone, by using any of the network
reduction techniques discussed in the last chapter.
Case iii) According to superposition theorem, the total current through branch A-B is the gebraic sum of
the currents through branch A-B, produced by each source acting independently. ∴ Total IAB = I’AB
due to E1 + I’’AB due to E2
Q. 6) State and explain Reciprocity theorem.
JUNE 2013
Millmann‟s Theorem : If n voltage sources V1, V2,,,,,,Vn having internal impedances, Z1, Z2, …...Zn
respectively, are in parallel, then these sources may be replaced by a single voltage source of voltage
VM having a series impedance ZM where VM and ZM are given by,
Where Y1,Y2, ....Yn are the admittances corresponding to the impedances Z1, Z2, …...Zn. Consider the
n voltage sources V1, V2,,,,,,Vn having series impedances Z1, Z2, …...Zn. connected in parallel as
shown in Fig.
Then according to Millamann’s theorem, all the voltage sources can be combined to get a single voltage
source VM with a series impedance ZM as shown in Fig. 8(a).