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Qwestrum Engineering360 · Electrical & Electronics · Network Analysis

Network Theorems

Network theorems replace a complicated linear network seen from two terminals by a single equivalent — Thevenin (V_th in series with R_th) or Norton (I_N in parallel with R_th) — so load calculations become one-line arithmetic.

Exam tip: keep SI units consistent end-to-end, write the governing relation symbolically before substituting, and sanity-check magnitude and sign.

Key formulas & points

Skim these first — then read the full notes below.

Superposition: one independent source at a time
Millman for parallel voltage sources with series resistances
Reciprocity: response interchange for linear bilateral networks

Topic details

Introduction

Thevenin and Norton theorems are exam favourites because they turn a full mesh solution into finding one open-circuit voltage and one equivalent resistance. They apply only to linear bilateral networks, and the equivalent is valid only for the two terminals from which it was derived.

Scope in B.Tech and GATE syllabus

Maximum power transfer (R_L = R_th) is a direct corollary and is repeatedly asked as a 5-mark rider. Note the efficiency at maximum power transfer is only 50%, so it matters for signal circuits, not power systems.

Key relations & formulas

Formulas (Indian textbook notation)

$Thevenin: V_{th} = V_{oc}; R_{th} = \frac{V_{oc}}{I_{sc}}$

Formulas (Indian textbook notation)

$Norton: I_{N} = I_{sc}; R_{N} = R_{th}$

Formulas (Indian textbook notation)

$Maximum power: R_{L} = R_{th}; P_{max} = \frac{V_{th}^{2}}{(4 R_{th})}$

Notation and sign conventions

Relation 1 —

Thevenin: V_{th} = V_{oc}; R_{th} = \frac{V_{oc}}{I_{sc}}

Formulas (Indian textbook notation)

$Thevenin: V_{th} = V_{oc}; R_{th} = \frac{V_{oc}}{I_{sc}}$

Write this relation with symbols exactly as in Network Analysis — Nagrath & Kothari before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.

Relation 2 —

Norton: I_{N} = I_{sc}; R_{N} = R_{th}

Formulas (Indian textbook notation)

$Norton: I_{N} = I_{sc}; R_{N} = R_{th}$

Write this relation with symbols exactly as in Network Analysis — Nagrath & Kothari before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.

Relation 3 —

Maximum power: R_{L} = R_{th}; P_{max} = V_{th}^{2}/

Formulas (Indian textbook notation)

$Maximum power: R_{L} = R_{th}; P_{max} = \frac{V_{th}^{2}}{(4 R_{th})}$

Write this relation with symbols exactly as in Network Analysis — Nagrath & Kothari before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.

Fundamentals and definitions

To find R_th, deactivate independent sources — short voltage sources, open current sources — and look back into the terminals. With dependent sources present you cannot deactivate them; instead apply a 1 V test source and compute R_th = V_test/I_test.

Governing relations in practice

V_th is the open-circuit terminal voltage; I_N is the short-circuit current; the three are linked by R_th = V_th/I_N. Superposition lets you get V_th when several sources act: sum the contribution of each source acting alone.

Design and analysis considerations

After reducing to the equivalent, reconnect the load and solve the trivial series/parallel circuit. Always confirm R_th is positive for a purely passive network.

Assumptions and validity limits

State assumptions explicitly before using any relation for network theorems — steady state, uniform properties, linear elastic material, ideal gas, incompressible flow, etc., as applicable.

Wrong assumptions invalidate the entire solution even when the formula is correct. In Network Analysis viva and GATE descriptive questions, listing valid assumptions often earns separate marks.

Step-by-step problem approach

1. Read the question and list given data with SI units (common in Network Analysis papers).
2. Draw a neat labelled diagram where applicable — examiners in Indian universities award diagram marks even when arithmetic slips.
3. Identify which relation from this topic applies to network theorems.
4. Use equation 1:

Thevenin: V_{th} = V_{oc}; R_{th} = \frac{V_{oc}}{I_{sc}}

.
5. Use equation 2:

Norton: I_{N} = I_{sc}; R_{N} = R_{th}

.
6. Substitute values, compute, and verify units and sign (direction).
7. State conclusion in one line — e.g. safe/unsafe, stable/unstable, feasible/infeasible.

Applications & exam relevance

Network Theorems appears in all electrical engineering circuits. In Indian electrical curricula this topic is tested because it connects theory to DC/AC circuit analysis.

GATE and semester exams often combine network theorems with earlier units — revise prerequisites before attempting mixed problems.

Industry interview panels sometimes ask: "Where did you use network theorems?" — answer with a lab, mini-project, or plant visit example if possible.

Common mistakes in exams

• Deactivating dependent sources (they must stay active — use a test source)
• Shorting current sources or opening voltage sources by mistake
• Using the maximum-power result R_L = R_th when the source resistance is fixed but load already specified
• Forgetting that R_th is defined only for the chosen terminal pair

Quick revision checklist

Before attempting network theorems problems, confirm you can:

1. Superposition: one independent source at a time
2. Millman for parallel voltage sources with series resistances
3. Reciprocity: response interchange for linear bilateral networks

Revise the solved examples in Network Analysis — Nagrath & Kothari and one previous-year GATE or university paper for this unit.

Worked examples

Try the problem first — open the solution when you are ready to check.

Thevenin equivalent and maximum power

Problem

A 24 V source in series with 4 Ω feeds a node that also has an 8 Ω resistor to ground; the load is taken across the 8 Ω. Find V_th, R_th at the load terminals, and the maximum power deliverable to a matched load.

Solution

Remove the load. V_th = open-circuit voltage across 8 Ω = 24 × 8/(4+8) = 16 V.
Deactivate 24 V (short it): R_th = 4 ∥ 8 = (4×8)/12 = 2.67 Ω.
For maximum power, R_L = R_th = 2.67 Ω.
P_max = V_th²/(4 R_th) = 16²/(4 × 2.67) = 256/10.67 = 24 W.

Conceptual check — Network Theorems

Problem

In a Network Analysis semester or GATE paper you are asked: "State the main assumption, the governing relation, and one practical consequence of network theorems." What should a complete answer include?

Exams & GATE

Nagrath & Kothari — Thevenin/Norton equivalence is exam favourite.

📖 Standard books (India)

Network Analysis — Nagrath & Kothari
Read: Syllabus unit
KCL, KVL, theorems, and three-phase circuits

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