Unit 2 - Notes

ELE205

Unit 2: Network Theorems

1. Introduction to Network Theorems

Network theorems are standard methods used to simplify complex electrical networks into manageable equivalent circuits. They allow for the analysis of voltage, current, and power in linear, bilateral networks without solving complex mesh or nodal equations repeatedly.

Key Definitions:

Linear Circuit: A circuit where parameters (R, L, C) are constant with respect to current or voltage (Ohm's law applies).
Bilateral Circuit: A circuit where current flows in both directions with the same magnitude for a given voltage (e.g., a resistor is bilateral; a diode is unilateral).
Active Network: Contains at least one independent source.
Passive Network: Contains no independent sources.

2. Superposition Theorem

Statement

In any linear, bilateral network containing two or more independent sources, the response (current through or voltage across an element) is the algebraic sum of the responses produced by each independent source acting alone, while all other independent sources are replaced by their internal impedances.

Operational conditions for "Turning Off" Sources:

Independent Voltage Sources: Short Circuit ( $V = 0$ ).
Independent Current Sources: Open Circuit ( $I = 0$ ).
Dependent Sources: Must NOT be touched; they remain active in the circuit.

Steps for Analysis:

Select one independent source and turn off all other independent sources.
Calculate the desired current/voltage due to the single active source.
Repeat steps 1 and 2 for each independent source in the network.
Add the individual responses algebraically (paying attention to direction/polarity) to find the total response.

Limitations:

Applicable only to linear networks (V-I relationship is linear).
Not applicable to Power calculations directly. Power is a non-linear function ( $P = I^2R$ ). To find power, you must first find the total current/voltage using Superposition, then calculate power.

3. Thevenin’s Theorem

Statement

Any linear, active, bilateral two-terminal network can be replaced by a simple equivalent circuit consisting of a single independent voltage source ( $V_{th}$ ) in series with a single equivalent impedance ( $Z_{th}$ or $R_{th}$ ).

Thevenin Equivalent Circuit:

$V_{th}$ (Thevenin Voltage): The open-circuit voltage across the load terminals.
$R_{th}$ (Thevenin Resistance): The equivalent resistance looking back into the terminals with all independent sources turned off (Voltage sources shorted, Current sources opened).

Steps for Analysis:

Remove the Load Resistor ( $R_L$ ): Identify the two terminals ( $A$ and $B$ ).
Find $V_{th}$ : Calculate the open-circuit voltage across terminals A-B using standard analysis (KVL, KCL, Nodal).
Find $R_{th}$ :
- Case 1 (Independent Sources only): Turn off all sources. Calculate equivalent resistance across A-B.
- Case 2 (Dependent Sources present): Keep dependent sources active. Turn off independent sources. Connect a test voltage source ( $V_{test}$ ) or test current source ( $I_{test}$ ) at terminals A-B. Calculate $R_{th} = V_{test} / I_{test}$ .
Draw Equivalent Circuit: Connect $V_{th}$ in series with $R_{th}$ , then reconnect $R_L$ .
Calculate Load Current:
$I_L = \frac{V_{th}}{R_{th} + R_L}$

4. Norton’s Theorem

Statement

Norton’s theorem is the dual of Thevenin’s theorem. It states that any linear, active, two-terminal network can be replaced by an equivalent circuit consisting of a single current source ( $I_N$ ) in parallel with a single equivalent impedance ( $Z_N$ or $R_N$ ).

Norton Equivalent Circuit:

$I_N$ (Norton Current): The short-circuit current flowing through the load terminals when shorted.
$R_N$ (Norton Resistance): The equivalent resistance looking into the terminals (calculated exactly the same way as $R_{th}$ ).

Steps for Analysis:

Remove the Load Resistor ( $R_L$ ).
Find $I_N$ ( $I_{sc}$ ): Short the terminals A-B with a wire. Calculate the current flowing through this short circuit.
Find $R_N$ : Procedure is identical to finding $R_{th}$ in Thevenin’s Theorem.
Draw Equivalent Circuit: Connect $I_N$ in parallel with $R_N$ , then reconnect $R_L$ in parallel.
Calculate Load Current: Using current divider rule:
$I_L = I_N \times \frac{R_N}{R_N + R_L}$

Source Transformation Relationship:

R_N = R_{th}

V_{th} = I_N \times R_N

5. Maximum Power Transfer Theorem (MPTT)

Statement

A resistive load ( $R_L$ ) connected to a DC network receives maximum power when the load resistance is equal to the internal resistance (Thevenin resistance) of the source network as seen from the load terminals.

Conditions for Maximum Power:

1. DC Circuits (Resistive Load):

Condition:

R_L = R_{th}

2. AC Circuits (Complex Impedance):

If the source impedance is $Z_{th} = R_{th} + jX_{th}$ and load is $Z_L = R_L + jX_L$ :

Case A (Load is fully variable): Max power transfer occurs when $Z_L$ is the complex conjugate of $Z_{th}$ .
$Z_L = Z_{th}^*$
$(R_L = R_{th} \text{ and } X_L = -X_{th})$
Case B (Only $R_L$ is variable, $X_L$ is fixed):
$R_L = \sqrt{R_{th}^2 + (X_{th} + X_L)^2}$

Calculation of Maximum Power (DC):

When $R_L = R_{th}$ :

P_{max} = I_L^2 R_L = \left(\frac{V_{th}}{2R_{th}}\right)^2 R_{th} = \frac{V_{th}^2}{4R_{th}}

Efficiency at MPTT:

At maximum power transfer condition ( $R_L = R_{th}$ ), the voltage drops equally across the source resistance and the load. Therefore, the efficiency is 50%.

6. Millman’s Theorem

Statement

Millman's theorem (also known as the Parallel Generator Theorem) allows the simplification of a circuit containing several parallel voltage sources with internal resistances into a single equivalent voltage source in series with a single equivalent resistance.

Formulae:

Consider $n$ voltage sources $V_1, V_2, \dots, V_n$ with internal resistances $R_1, R_2, \dots, R_n$ connected in parallel.

Equivalent Voltage ( $V_{eq}$ or $V_{M}$ ):

V_{eq} = \frac{\frac{V_1}{R_1} + \frac{V_2}{R_2} + \dots + \frac{V_n}{R_n}}{\frac{1}{R_1} + \frac{1}{R_2} + \dots + \frac{1}{R_n}}

Equivalent Resistance ( $R_{eq}$ or $R_{M}$ ):

\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \dots + \frac{1}{R_n}

R_{eq} = \frac{1}{G_{total}}

For Current Sources:
If converting parallel current sources ( $I_1, R_1$ || $I_2, R_2$ ...):

I_{eq} = I_1 + I_2 + \dots + I_n

R_{eq} = R_1 || R_2 || \dots || R_n

Application:

Very useful for solving networks with only two nodes (common rail systems) or calculating the voltage across parallel branches quickly without writing multiple nodal equations.

7. Reciprocity Theorem

Statement

In any linear, bilateral, passive network, if a single voltage source $V$ in branch $A$ produces a current $I$ in branch $B$ , then moving the voltage source $V$ to branch $B$ will produce the exact same current $I$ in branch $A$ .

Ratio Representation:

The ratio of Response to Excitation is constant.

\frac{\text{Response}}{\text{Excitation}} = \text{Constant}

\frac{I_2}{V_1} = \frac{I_1}{V_2}

(Where transfer resistance $Z_{12} = Z_{21}$ ).

Conditions for Applicability:

Network must be Linear and Bilateral.
Network must be Passive (No dependent sources).
Applicable only to networks with a single independent source.
Initial conditions must be zero.

Verification Steps:

Calculate current $I$ in Branch 2 due to voltage $V$ in Branch 1.
Swap positions: Place voltage $V$ in Branch 2.
Calculate current in Branch 1 (short circuit the original voltage location).
If the current magnitude is identical, the theorem holds.

8. Tellegen’s Theorem

Statement

Tellegen's Theorem is based on the law of conservation of energy. It states that for any lumped network (linear, non-linear, passive, active, time-variant, or time-invariant), the algebraic sum of the power delivered by the sources is equal to the algebraic sum of the power absorbed by the elements at any instant.

Mathematically:

\sum_{k=1}^{b} v_k(t) \cdot i_k(t) = 0

Where:

$b$ = number of branches.
$v_k$ = voltage across the $k^{th}$ branch.
$i_k$ = current through the $k^{th}$ branch.

Key Characteristics:

Topological Independence: It depends only on KCL and KVL (network topology), not on the type of elements.
Universality: Valid for any type of network provided Kirchhoff's laws are obeyed.

Application Steps:

Find the voltage and current for every branch in the network.
Calculate Power $P = V \times I$ for each branch.
Assign signs based on passive sign convention:
- Current entering positive terminal = Power Absorbed (+).
- Current leaving positive terminal = Power Delivered (-).
Sum all powers. The result must be zero.

Summary Comparison Table

Theorem	Primary Use	Applicability (Strict)
Superposition	Analyzing multi-source circuits	Linear only. Not for power.
Thevenin	Simplifying circuit to V-source + Series Z	Linear, Active, Bilateral.
Norton	Simplifying circuit to I-source + Parallel Z	Linear, Active, Bilateral.
Max Power	Impedance Matching / Efficiency	Linear, Active.
Millman	Solving parallel voltage sources	Linear.
Reciprocity	Analysis of transfer impedance	Linear, Bilateral, Passive (No dependent sources).
Tellegen	Power balance verification	Universal (Linear/Non-linear, Active/Passive).

Unit 2 - Notes

Table of Contents

Unit 2: Network Theorems

1. Introduction to Network Theorems

2. Superposition Theorem

Statement

Operational conditions for "Turning Off" Sources:

Steps for Analysis:

Limitations:

3. Thevenin’s Theorem

Statement

Thevenin Equivalent Circuit:

Steps for Analysis:

4. Norton’s Theorem

Statement

Norton Equivalent Circuit:

Steps for Analysis:

Source Transformation Relationship:

5. Maximum Power Transfer Theorem (MPTT)

Statement

Conditions for Maximum Power:

1. DC Circuits (Resistive Load):

2. AC Circuits (Complex Impedance):

Calculation of Maximum Power (DC):

Efficiency at MPTT:

6. Millman’s Theorem

Statement

Formulae:

Application:

7. Reciprocity Theorem

Statement

Ratio Representation:

Conditions for Applicability:

Verification Steps:

8. Tellegen’s Theorem

Statement

Key Characteristics:

Application Steps:

Summary Comparison Table