GATE EE

Network Theorems Notes for GATE Electrical Engineering (EE)

Lecture Notes

SEC 01

Kirchhoff’s Laws

SEC 02

Kirchhoff’s Current Law (KCL)

1Kirchhoff’s Current Law (KCL)
1Statement

The algebraic sum of currents entering a node is zero.

Mathematical Form:

\[\sum_{k=1}^{n} I_k = 0\]

Alternative Form:

\[\sum I_{in} = \sum I_{out}\]

Physical Basis: Conservation of charge

Currents entering or leaving the node
Currents entering or leaving the node

Example: \(I_1 - I_2 - I_3 - I_4 = 0\)

1GATE Tip

Sign convention: Current entering = +ve, Current leaving = -ve

SEC 03

Kirchhoff’s Voltage Law (KVL)

1Kirchhoff’s Voltage Law (KVL)
1Statement

The algebraic sum of voltages around any closed loop is zero.

Mathematical Form:

\[\sum_{k=1}^{n} V_k = 0\]

Physical Basis: Conservation of energy

Steps:

  1. Choose loop direction

  2. Apply sign convention

  3. Write KVL equation

Voltage in a closed loop as per KVL
Voltage in a closed loop as per KVL

Example: \(V_s - I R_1 - I R_2 = 0\)

1GATE Tip

Voltage rise = +ve, Voltage drop = -ve (or vice versa, be consistent)

SEC 04

Circuit Analysis Methods

SEC 05

Nodal Analysis

1Nodal Analysis
1Procedure

  1. Select reference node (ground)

  2. Assign node voltages

  3. Apply KCL at each non-reference node

  4. Solve simultaneous equations

For node with voltage \(V_n\):

\[\sum \dfrac{V_n - V_{adjacent}}{R_{connecting}} = I_{sources}\]

Supernode: When voltage source connects two non-reference nodes

\[V_2 - V_1 = V_s \text{ (constraint equation)}\]

1When to Use

  • More current sources than voltage sources

  • Number of nodes < Number of meshes

  • Circuits with supernodes

GATE Formula: For n nodes, write (n-1) KCL equations

SEC 06

Mesh Analysis

1Mesh Analysis
1Procedure

  1. Identify independent meshes

  2. Assign mesh currents (clockwise)

  3. Apply KVL to each mesh

  4. Solve simultaneous equations

For mesh with current \(I_m\):

\[\sum R_{self} \cdot I_m - \sum R_{mutual} \cdot I_{adjacent} = \sum V_{sources}\]

Supermesh: When current source is common to two meshes

\[I_1 - I_2 = I_s \text{ (constraint equation)}\]

1When to Use

  • More voltage sources than current sources

  • Number of meshes < Number of nodes

  • Planar circuits only

GATE Formula: For planar circuit: Meshes = Branches - Nodes + 1

SEC 07

Network Theorems

SEC 08

Superposition Theorem

1Superposition Theorem
1Statement

In a linear circuit with multiple sources, the response is the algebraic sum of responses due to individual sources acting alone.

1Procedure

  1. Consider one source at a time

  2. Replace other voltage sources with short circuits

  3. Replace other current sources with open circuits

  4. Calculate response due to active source

  5. Repeat for all sources

  6. Add all responses algebraically

1GATE Important

  • Applicable only to LINEAR circuits

  • Power cannot be calculated using superposition

  • Dependent sources remain active

  • \(P_{total} \neq P_1 + P_2 + ... + P_n\)

SEC 09

Thevenin’s Theorem

1Thevenin’s Theorem
1Statement

Any linear two-terminal network can be replaced by a voltage source \(V_{th}\) in series with resistance \(R_{th}\).

Steps:

  1. Remove load resistance

  2. Find \(V_{th}\) = Open circuit voltage

  3. Find \(R_{th}\):

    • Kill independent sources

    • Look back into terminals

    • For dependent sources: \(R_{th} = \dfrac{V_{test}}{I_{test}}\)

  4. Draw Thevenin equivalent

Thevenin equivalent circuit
Thevenin equivalent circuit
1GATE Tip

Alternative method: \(R_{th} = \dfrac{V_{oc}}{I_{sc}}\) (open circuit voltage / short circuit current)

SEC 10

Norton’s Theorem

1Norton’s Theorem
1Statement

Any linear two-terminal network can be replaced by a current source \(I_N\) in parallel with resistance \(R_N\).

Steps:

  1. Remove load resistance

  2. Find \(I_N\) = Short circuit current

  3. Find \(R_N = R_{th}\)

  4. Draw Norton equivalent

Source Transformation:

\[I_N = \dfrac{V_{th}}{R_{th}}\]
\[R_N = R_{th}\]
\[V_{th} = I_N \cdot R_N\]

Norton equivalent circuit
Norton equivalent circuit
1GATE Important

Norton and Thevenin are dual theorems. Use Norton when current calculation is easier.

SEC 11

Maximum Power Transfer Theorem

1Maximum Power Transfer Theorem
1Statement

Maximum power is transferred to load when load resistance equals source resistance.

Condition: \(R_L = R_{th}\)

Maximum Power:

\[P_{max} = \dfrac{V_{th}^2}{4R_{th}}\]

Load Current:

\[I_L = \dfrac{V_{th}}{2R_{th}}\]

Efficiency at Maximum Power:

\[\eta = \dfrac{P_L}{P_{total}} = 50\%\]

Thevenin’s circuit with load resistor
Thevenin’s circuit with load resistor
Power vs Load Resistance
Power vs Load Resistance
1GATE Note

For AC circuits: \(Z_L = Z_{th}^*\) (complex conjugate matching)

SEC 12

Additional Important Theorems

SEC 13

Reciprocity Theorem

1Reciprocity Theorem
1Statement

In a linear, bilateral network, if a voltage source in branch A produces current in branch B, then the same voltage source in branch B will produce the same current in branch A.

Mathematical Form:

\[\dfrac{I_2}{V_1} = \dfrac{I_1}{V_2}\]

Transfer Impedance:

\[Z_{12} = \dfrac{V_1}{I_2} = \dfrac{V_2}{I_1} = Z_{21}\]

1Conditions

  • Linear network

  • Bilateral elements (R, L, C)

  • No dependent sources

  • Same frequency for AC analysis

GATE Application: Useful in analyzing two-port networks and transmission lines.

SEC 14

Millman’s Theorem

1Millman’s Theorem
1Statement

Used to find voltage across parallel branches with different voltage sources.

Formula:

\[V = \dfrac{\sum \dfrac{V_k}{R_k}}{\sum \dfrac{1}{R_k}} = \dfrac{\sum G_k V_k}{\sum G_k}\]

where \(V_k\) are source voltages, \(R_k\) are branch resistances, and \(G_k = \dfrac{1}{R_k}\) (conductance).

An example circuit
An example circuit

GATE Tip: Very useful for parallel voltage divider circuits and finding common voltage.

SEC 15

Compensation Theorem

1Compensation Theorem
1Statement

When impedance in any branch changes from \(Z_1\) to \(Z_2\), the change in current can be calculated by considering a compensating voltage source.

Compensating Voltage:

\[V_c = I_1(Z_2 - Z_1)\]

where \(I_1\) is the original current through the branch.

Change in Current:

\[\Delta I = \dfrac{V_c}{Z_{th} + Z_2}\]

1Applications

  • Fault analysis in power systems

  • Incremental analysis of circuits

  • Sensitivity analysis

  • Finding change in circuit parameters

GATE Usage: Useful for analyzing the effect of component variations on circuit behavior.

SEC 16

Substitution Theorem

1Substitution Theorem
1Statement

Any branch in a network can be replaced by an equivalent branch that has the same voltage across it and current through it.

Equivalent Replacements:

  • Voltage source = Branch voltage (\(V_s = V_{branch}\))

  • Current source = Branch current (\(I_s = I_{branch}\))

  • Impedance = \(Z_{eq} = \dfrac{V_{branch}}{I_{branch}}\)

1GATE Application

Very useful for:

  • Simplifying complex circuits

  • Finding equivalent circuits

  • Network reduction techniques

  • Analyzing one part of circuit independently

Note: The rest of the circuit behavior remains unchanged.

SEC 17

Tellegen’s Theorem

1Tellegen’s Theorem
1Statement

In any lumped network, the sum of instantaneous powers delivered to all branches is zero.

Mathematical Form:

\[\sum_{k=1}^{n} v_k(t) \cdot i_k(t) = 0\]

Physical Interpretation: Total power generated = Total power consumed

1Applications

  • Power balance verification

  • Network analysis validation

  • Fundamental theorem for all network theorems

  • Circuit simulation algorithms

1GATE Importance

Tellegen’s theorem is the most fundamental theorem - all other network theorems can be derived from it.

SEC 18

Star-Delta Transformation

1Star-Delta Transformation
1Statement

Any three-terminal star (Y) network can be transformed to equivalent delta (\(\Delta\)) network and vice versa.

Star to Delta:

\[R_a = \dfrac{R_1 R_2 + R_2 R_3 + R_3 R_1}{R_3}\]
\[R_b = \dfrac{R_1 R_2 + R_2 R_3 + R_3 R_1}{R_1}\]
\[R_c = \dfrac{R_1 R_2 + R_2 R_3 + R_3 R_1}{R_2}\]

General Form:

\[R_{ab} = \dfrac{\text{Sum of products}}{\text{Opposite Star resistance}}\]

Delta to Star:

\[R_1 = \dfrac{R_b R_c}{R_a + R_b + R_c}\]
\[R_2 = \dfrac{R_a R_c}{R_a + R_b + R_c}\]
\[R_3 = \dfrac{R_a R_b}{R_a + R_b + R_c}\]

General Form:

\[R_n = \dfrac{\text{Product of adjacent }~ \Delta \text{resistances}}{\text{Sum of all }~\Delta ~\text{resistances}}\]

1GATE Tip

For equal resistances: \(R_Y = \dfrac{R_\Delta}{3}\) and \(R_\Delta = 3R_Y\)

SEC 19

Advanced Concepts

SEC 20

Dependent Sources and Network Theorems

1Dependent Sources and Network Theorems
1Key Points

  • Dependent sources are never killed/deactivated

  • They remain active during Thevenin/Norton analysis

  • Superposition: Only independent sources are considered one at a time

  • For \(R_{th}\) with dependent sources: Apply test voltage/current method

Test Source Method:

  1. Kill all independent sources

  2. Apply test voltage \(V_t\) (or current \(I_t\))

  3. Calculate resulting current \(I_t\) (or voltage \(V_t\))

  4. \(R_{th} = \dfrac{V_t}{I_t}\)

1GATE Common Error

Students often try to kill dependent sources - this is incorrect!

SEC 21

AC Circuit Analysis

1AC Circuit Analysis
1Impedance Form

All theorems apply to AC circuits using complex impedances:

  • Resistance \(R ~\to\) Impedance \(Z\)

  • DC voltage/current \(\to\) Phasor voltage/current

  • \(R_{th}\) becomes \(Z_{th}\)

Maximum Power Transfer (AC):

\[Z_L = Z_{th}^*\]
\[P_{max} = \dfrac{|V_{th}|^2}{4 \cdot \text{Re}(Z_{th})}\]

Phasor Analysis:

  • Use \(j\omega L\) for inductors

  • Use \(\dfrac{1}{j\omega C}\) for capacitors

  • All calculations in complex domain

1GATE Tip

For purely resistive loads: \(R_L = |Z_{th}|\) for maximum power transfer

SEC 22

Quick Reference

SEC 23

GATE Quick Reference Card

1GATE Quick Reference Card
1Key Formulas

  • KCL: \(\sum I = 0\) at any node

  • KVL: \(\sum V = 0\) around any loop

  • Thevenin: \(V_{th}\) (open circuit), \(R_{th}\) (kill sources or \(V_{oc}/I_{sc}\))

  • Norton: \(I_N\) (short circuit), \(R_N = R_{th}\)

  • Max Power: \(R_L = R_{th}\), \(P_{max} = \dfrac{V_{th}^2}{4R_{th}}\), \(\eta = 50\%\)

  • Millman: \(V = \dfrac{\sum G_k V_k}{\sum G_k}\)

  • Star-Delta: \(R_Y = \dfrac{R_\Delta}{3}\) (equal resistances)

1Method Selection

  • Nodal: More current sources, fewer nodes

  • Mesh: More voltage sources, planar circuit

  • Thevenin/Norton: Load analysis, source equivalents

  • Superposition: Multiple sources, linear circuits

  • Star-Delta: Non-planar circuits, bridge circuits

SEC 24

Common GATE Problem Types

1Common GATE Problem Types
1Numerical Problems

  1. Direct Application: Find current/voltage using specific theorem

  2. Equivalent Circuits: Thevenin/Norton equivalents

  3. Maximum Power: Load resistance and power calculations

  4. Comparative Analysis: Which method is most efficient?

  5. Mixed Circuits: AC and DC sources, dependent sources

  6. Network Simplification: Using multiple theorems

1Common GATE Mistakes

  • Killing dependent sources in Thevenin analysis

  • Using superposition for power calculations

  • Wrong sign conventions in KCL/KVL

  • Forgetting complex conjugate in AC maximum power transfer

  • Not considering all constraint equations in supernode/supermesh

SEC 25

Time-Saving Techniques

1Time-Saving Techniques
1Quick Methods

  • Voltage Divider: \(V_R = V_s \dfrac{R}{R_{total}}\)

  • Current Divider: \(I_R = I_s \dfrac{R_{other}}{R + R_{other}}\)

  • Source Transformation: Convert between voltage and current sources

  • Series/Parallel Combinations: Simplify before applying theorems

  • Symmetry: Use circuit symmetry to reduce calculations

1GATE Strategy

  • Identify the quickest method first (30 seconds)

  • Check if answer choices give clues about approach

  • Use approximations when exact values aren’t needed

  • Verify answers using alternative methods if time permits

SEC 26

Memory Aids

1Memory Aids
1Mnemonics

  • KCL: "Current In = Current Out"

  • KVL: "Voltage Rises = Voltage Falls"

  • Thevenin: "Open circuit voltage, Kill sources for resistance"

  • Norton: "Short circuit current, Same resistance"

  • Max Power: "Match the load, Get half efficiency"

  • Superposition: "One source at a time, others killed"

1Quick Checks

  • Power balance: \(\sum P_{generated} = \sum P_{consumed}\)

  • Dimension analysis: Check units in final answer

  • Limiting cases: What happens when \(R \to 0\) or \(R \to \infty\)?

  • Symmetry: Equal components should have equal currents/voltages