Network Theorems MCQ Quiz - Objective Question with Answer for Network Theorems - Download Free PDF

Kirchhoff's Voltage Law (KVL)

According to KVL, the algebraic sum of the voltages in a closed loop is always zero.

ΣV = 0

\(-V_1+V_2+V_3+V_4=0\)

Calculation

Applying KVL as per the direction ABCD:

+ 50 - 30 - V_S - 10 + 20 = 0

V_S = 30 V

Apparent Power:

• Apparent power is the total power that appears to be transferred between the source and load in an AC circuit.
• It is given by S = VI
• The unit of apparent power is Volt-Amperes (VA).

Calculation

Given, V = 230 volts

I = 10 A

S = 230 × 10

S = 2300 VA = 2.3 kVA

Additional Information 
Active Power:

• Active power is the real power consumed by the circuit.
• It is given by: P = VI cosϕ
• The unit of active power is watts (W).

Reactive Power:

• Reactive power is the electrical power that oscillates between the source and the load without performing useful work.
• It is given by P = VI sinϕ
• The unit of reactive power is Volt-Ampere Reactive (VAR).

Concept:

Norton’s Theorem states that any linear electrical network with voltage and current sources and resistors can be replaced at terminals A-B by an equivalent circuit consisting of a current source in parallel with a resistor.

Calculation

The Norton equivalent circuit includes: - Norton Current (I_N): The short-circuit current through the output terminals. - Norton Resistance (R_N): The equivalent resistance seen from the terminals when all independent sources are turned off. These two elements are connected in parallel.

Hence, the equivalent circuit consists of a current source in parallel with a resistor.

Thevenin Theorem

Thevenin’s theorem states that it is possible to simplify any linear circuit, into an equivalent circuit with a single voltage source and a series resistance.

Steps to calculate:

1. Thevenin Resistance (Rth): Open the circuit to the terminal from where the Thevenin resistance has to be found. While finding the Rth, short circuit the independent voltage source and open circuit the independent current source.
2. Thevenin Voltage (Vth): This is the open-circuit voltage across the terminals where the load was connected.
    

Calculation

Open circuit the terminal XY to calculate R_th

3 kΩ, 1 kΩ and 2 kΩ are connected in series.

R = 3 + 2 + 1 = 6 kΩ 

This 6 kΩ and 3 kΩ are connected in parallel.

\(R_{XY}={6\times 3\over 6+3}\)

R_XY = 2 kΩ 

Concept

The maximum power transfer theorem states that the maximum power is delivered to a load when the load resistance is equal to the source resistance.

The maximum power across the load is given by:

 \(P_{max}={V_{th}^2\over 4R_{th}}\) at condition \(R_L=R_{th}\)

Now, the power transferred to the load at other loading condition is given by:

\(P=I^2R_L\)

\(P=({V_{th}\over R_{th}+R_L})^2 \times R_L\)

Calculation

Given, P_max = 50 W at R_L = R_th =  2Ω 

\(50={V_{th}^2\over 4\times 2}\)

V_th = 20V

\(P=({20\over 2+8})^2 \times 8\)

P = 32 W

Concept

The power transfer efficiency is:

​\(η={I^2R_L\over VI}\times 100\)

\(η={IR_L\over V}\times 100\)

The current across any resistor is given by:

\(I={V\over R}\)

where, I = Current

V = Voltage

R = Resistance

Calculation

Let the voltage and internal resistance of the voltage source be V and R respectively.

Case 1: When the current of 2 A flows through 5 Ω resistance.

\(2={V\over 5+R}\) .... (i)

Case 2: When the current of 1.6 A flows through 10 Ω resistance.

\(1.6={V\over 10+R}\) .....(ii)

Solving equations (i) and (ii), we get:

2(5+R)=1.6(10+R)

10 + 2R = 16 + 1.6R

0.4R = 6

R = 15Ω

Putting the value of R = 15Ω in equation (i):

V = 40 volts

Case 3: Current when the load is 15Ω

\(I={V\over R+R_L}\)

\(I={40\over 15+15}={4\over 3}A\)

\(η={{4\over 3}\times 15\over 40}\times 100\)

η = 50%

Additional Information Condition for Maximum Power Transfer Theorem:

When the value of internal resistance is equal to load resistance, then the power transferred is maximum.

Under such conditions, the efficiency is equal to 50%.

Concept:

Thevenin's Theorem:

Any two terminal bilateral linear DC circuits can be replaced by an equivalent circuit consisting of a voltage source and a series resistor.

To find V_oc: Calculate the open-circuit voltage across load terminals. This open-circuit voltage is called Thevenin’s voltage (V_th).

To find I_sc: Short the load terminals and then calculate the current flowing through it. This current is called Norton current (or) short circuit current (i_sc).

To find R_th: Since there are Independent sources in the circuit, we can’t find Rth directly. We will calculate Rth using Voc and Isc and it is given by

\({{\rm{R}}_{{\rm{th}}}} = \frac{{{{\rm{V}}_{{\rm{oc}}}}}}{{{{\rm{i}}_{{\rm{sc}}}}}}\)  

Application:

Given: Load line equation = V + i = 100

To obtain open-circuit voltage (Vth) put i = 0 in load line equation 

⇒ Vth = 100 V

To obtain short-circuit current (isc) put V = 0 in load line equation

⇒ isc = 100 A

So, \({R_{th}} = \frac{{{V_{th}}}}{{{i_{sc}}}} = \frac{{100}}{{100}} = 1{\rm{\Omega }}\)

Equivalent circuit is

Current (i) = 100/2 = 50 A

Applying loop-law in the given circuit.

- V + i × R = 0

- V + I × 1 = 0

⇒ V = i

Given Load line equation is V + i = 100

Putting V = i 

then i + i = 100 

⇒ i = 50 A

Distributed Network:

• If the network element such as resistance, capacitance, and inductance are not physically separated, then it is called a Distributed network.
• Distributed systems assume that the electrical properties R, L, C, etc. are distributed across the entire circuit.
• These systems are applicable for high (microwave) frequency applications.

Lumped Network:

• If the network element can be separated physically from each other, then they are called a lumped network.
• Lumped means a case similar to combining all the parameters and considering it as a single unit.
• Lumped systems are those systems in which electrical properties like R, L, C, etc. are assumed to be located on a small space of the circuit.
• These systems are applicable to low-frequency applications.

Kirchoff's Laws:

• Kirchhoff’s laws are used for voltage and current calculations in electrical circuits.
• These laws can be understood from the results of the Maxwell equations in the low-frequency limit.
• They are applicable for DC and AC circuits at low frequencies where the electromagnetic radiation wavelengths are very large when we compare with other circuits. So they are only applicable for lumped parameter networks.

Kirchhoff's current law (KCL) is applicable to networks that are:

• Unilateral or bilateral 
• Active or passive 
• Linear or non-linear
• Lumped network

KCL (Kirchoff Current Law): According to Kirchhoff’s current law (KCL), the algebraic sum of the electric currents meeting at a common point is zero.

Mathematically we can express this as:

\(\mathop \sum \limits_{n = 1}^M {i_n} = 0\)

Where in represents the nth current

M is the total number of currents meeting at a common node.

KCL is based on the law of conservation of charge.

Kirchhoff’s Voltage Law (KVL):

It states that the sum of the voltages or electrical potential differences in a closed network is zero. 

• According to Tellegen’s theorem, the summation of instantaneous powers for the n number of branches in an electrical network is zero.
• Let n number of branches in an electrical network have I1, I2, I3, ….. In respective instantaneous currents through them.
• These branches have instantaneous voltages across them are V1, V2, V3, ….. Vn respectively.
• According to Tellegen’s theorem, \(\mathop \sum \limits_{k = 1}^n {V_k}.{I_k} = 0\)
• It is based on the conservation of energy.
• It is applicable to both linear and non-linear circuits.

Concept:

Maximum power transfer theorem:

Maximum power transfer theorem states that " In a linear bilateral network if the entire network is represented by its Thevenin's equivalent circuit then the maximum power transferred from source to the load when the load impedance is equal to the complex conjugate of  Thevenin's impedance.

Let's consider variable resistive load and Thevenin's equivalent network as shown below,

\({P_m} = \frac{{V_{th}^2}}{{4{R_{th}}}}\)

Where, 

Pm is the maximum power 

Vth is the source voltage or Thevenin's voltage

Rth is the Thevenin's resistance (Rth = RL = RS)

The efficiency of the maximum power transfer theorem will be 50 %

The voltage across the load resistance/impedance is VL = VS / 2

Calculation:

Given the circuit diagram

Source voltage VS = 200 V

Va = 60 V

As V is the voltage across the load.

V = VS / 2 = 200 / 2 = 100 V

Load current i = V / RL (When maximum power is transferred RL = RS = Rth = 10 Ω) 

i = 100 / 10 = 10 A

By applying nodal analysis at node V

\( - i + \frac{V}{{20}} + \frac{{V - {V_a}}}{R} = 0\)

\( - 10 + \frac{{100}}{{20}} + \frac{{100 - 60}}{R} = 0\)

R = 8 Ω

Therefore, the value of R is 8 Ω when V_a is 60 V and maximum power is transferred from N₁ to N₂

Concept:

Superposition theorem is used to solve a circuit that contains multiple current and/or voltage sources acting together.

Theorem:

• The superposition theorem states that "in a linear circuit with several sources, the current and voltage for any element in the circuit is the sum of the currents and voltages produced by each source acting independently."
• The superposition theorem applies only when all the components of the circuit are linear, which is the case for resistors, capacitors, and inductors it is not applicable to networks containing nonlinear elements.

​Calculation:

case 1:

When 8 V source are there

I = 8 / 16 = 0.5 A

case 2:

When only 2 A current source present

Apply KVL in loop

6 I + 2 ( I - 2 )+ 8 I = 0

I = 0.25 A

case 3:

When 6 V source are there

I = - 6 / 16 = - 0.375 A

Now using superposition theorem

Total current I = 0.5 + 0.25 + ( - 0.375 ) A

= 0.375 A

= 375 mA

Important Points

Various Theorem and the circuits where they are applicable is shown below in the table:

Concept:

Maximum power transfer theorem:

• Maximum power transfer theorem states that " In a linear bilateral network if the entire network is represented by its Thevenin's equivalent circuit then the maximum power transferred from source to the load when the load resistance is equal to the Thevenin's resistance."
• P=VS2.RL(RS+RL)2" role="presentation" style="display: inline; position: relative;" tabindex="0">P=VS2.RL(RS+RL)2For maximum power transfer, RL = Rth 
• Then the maximum power transferred is given by \({{\rm{P}}_{max}} = {\rm{\;}}\frac{{V_S^2}}{{4{R_L}}}\)

Explanation:

Circuit Diagram

Given,

R_s = 5 to 25 Ω (variable)

R_L = 10 Ω (fixed)

Here Maximum Power Transfer theorem is not applicable as the load resistor is not variable.

Current, \(I=\frac{V}{R_s+R_L}\)

Power transferred to load R_L,

\(P=I^2R_L=[\frac{V}{R_S+R_L}]^2\times R_L\)

It is clear that for P to be maximum, R_S should be minimum.

∴ R_S = 5 Ω 

Additional Information 

Properties of maximum power transfer theorem: 

• This theorem is applicable only for linear networks i.e networks with R, L, C, transformer, and linear controlled sources as elements.
• The presence of dependent sources makes the network active and hence, MPPT is used for both active as well as passive networks.
• This theorem is applicable when the load is variable.

Maximum power transfers at RL = Rs

The current at this condition is,

\(I_L=\frac{V_S}{2R_L}=\frac{V_S}{2R_S}\)

The maximum value of current occurs at RL­ = 0 and is given by
\(I_L=\frac{V}{R_S}\)

Therefore, the current at maximum power is equal to 50% of the maximum current

Key Points

•  If source impedance is complex then load impedance has to be a complex conjugate of source impedance for maximum power transfer to occur.
•  Maximum efficiency is not related to maximum power transfer.

Reciprocity theorem:

Reciprocity theorem states that in any branch of a network, the current (I) due to a single source of voltage (V) elsewhere in the network is equal to the current through the branch in which the source was originally placed when the source is placed in the branch in which the current (I) was originally obtained.

In the circuit (a), the value I_a is obtained for a voltage source V. According to reciprocity theorem, this current is equivalent to I_b in the circuit B.

Limitations of reciprocity theorem:

• The network should be bilateral linear and time-invariant.
• It can apply only to the single-source network and not for multi-source.
• It is also applicable for passive networks consisting L,C.
• Not applicable for circuits containing dependent sources even if it is linear.

Concept:

Maximum power transfer for DC circuit:

According to the MPT the maximum power transfer to the load when the load resistance is equal to the source resistance or Thevenin resistance.

RL = Rth 

RL = load resistance

Rth = Thevenin or source resistance

The power at maximum power transfer (P_max) = V_th² / 4R_th

The maximum power transfer theorem is used in electrical circuits.

Calculation:

R_th = R_L

= ( 30 × 150 )  / 180

= 25 Ω 

V_th = V_ab 

= ( 150 × 180 ) / (150 + 30 )

= 150 V

From above concept,

\(P_{max}=\frac{V_{th}^2}{4R_{th}}=\frac{150^2}{4\times25}=225\ W\)

Pmax = 225 W

Reciprocity theorem: It states that the current I in any branch of a network, due to single voltage source (E) anywhere in the network is equal to the current of the branch in which source was placed originally and when the source is again put in the branch in which current is obtained originally.

Limitations of reciprocity theorem:

• The network should be linear and time-invariant
• It can apply only to the single-source network