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Network Theorems

Superposition, Thevenin, Norton, Tellegen, Millman, Reciprocity & Maximum Power Transfer theorems.

1. Superposition Theorem

Step-by-Step Response
+ - I_total
Linear Superposition
$ x_{total} = \sum x_k \text{ (Individual responses)} $

Applicable ONLY for linear, bilateral networks.

  • β—† Voltage Source $\rightarrow$ Short Circuit (0V)
  • β—† Current Source $\rightarrow$ Open Circuit (0A)
  • ⚠ Dependent sources are NEVER deactivated.
Limit: Not for Power ($P_{total} \neq P_1 + P_2$).

2. Thevenin's Theorem

Step-by-Step: Find Thevenin Equivalent
Vs R₁ Rβ‚‚ Z_L a b Voc Source killed Rth = R₁ β€– Rβ‚‚ + βˆ’ Vth Rth Z_L a b Thevenin Equivalent Circuit
Thevenin's Theorem
Statement: Any linear bilateral network with voltage/current sources and resistances can be replaced by an equivalent circuit having a voltage source $V_{th}$ in series with a resistance $R_{th}$.

Step 1: Remove load $Z_L$ from terminals a-b

Step 2: Find open-circuit voltage across a-b

$V_{th} = V_{OC} = V_{ab}$

Step 3: Deactivate all independent sources

Voltage source β†’ short circuit, Current source β†’ open circuit

Step 4: Find equivalent resistance from a-b

$R_{th} = R_{ab}$

Key Relations:

$V_{th} = V_{OC}$

$R_{th} = \frac{V_{OC}}{I_{SC}}$

$I_L = \frac{V_{th}}{R_{th} + Z_L}$

Quick Calculator

2b. Norton's Theorem

Step-by-Step: Find Norton Equivalent
Vs R₁ Rβ‚‚ Z_L a b Short Isc Source killed Rn = R₁ β€– Rβ‚‚ = Rth I_N R_N Z_L a b Norton Equivalent Circuit
Norton's Theorem
Statement: Any linear bilateral network can be replaced by an equivalent circuit having a current source $I_N$ in parallel with a resistance $R_N$.

Step 1: Remove load $Z_L$ & short-circuit a-b

Step 2: Find short-circuit current through a-b

$I_N = I_{SC}$

Step 3: Deactivate all independent sources

Same as Thevenin: V→short, I→open

Step 4: Find equivalent resistance from a-b

$R_N = R_{th} = R_{ab}$

Key Relations:

$I_N = I_{SC} = \frac{V_{th}}{R_{th}}$

$R_N = R_{th} = \frac{V_{OC}}{I_{SC}}$

$I_L = I_N \cdot \frac{R_N}{R_N + Z_L}$

Thevenin ↔ Norton Conversion:

$V_{th} = I_N \cdot R_N \quad;\quad I_N = \frac{V_{th}}{R_{th}}$

$R_N = R_{th}$

Quick Calculator

3. Tellegen's Theorem (Conservation)

Instantaneous Power Sum
$\sum_{k=1}^n V_k I_k = 0$
Energy Conservation
$ \sum_{k=1}^{b} V_k I_k = 0 $

Key Insights:

  • β—† Independent of nature of elements.
  • β—† Valid for Linear, Non-Linear, and Time-Varying networks.
  • β—† Simply states that Total Power = 0 at any instant.

4. Millman's Theorem

Parallel Source Simplification
E₁ Eβ‚‚ E₃
Parallel Voltage Sources
$ E_{eq} = \frac{\sum E_k Y_k}{\sum Y_k} \quad ; \quad Y_{eq} = \sum Y_k $

Used for calculating terminal voltage across parallel branches containing different EMFs and internal resistances ($Y_k = 1/R_k$).

Millman's is a specialized form of Nodal Analysis.

5. Reciprocity Theorem

Excitation-Response Swap
PASSIVE NETWORK A
Reading: 5.2 A
Symmetry Property
$ \frac{V_{input}}{I_{output}} = \text{constant} $

Key Insights:

  • β—† Valid for Passive, Linear, Bilateral networks.
  • β—† Transfer impedance $Z_{12} = Z_{21}$.
  • ⚠ NOT valid for networks with dependent sources or active elements.

6. Maximum Power Transfer Theorem

Power vs Load Curve
Power: 120W Eff (Ξ·): 50%
Theorem Condition
$ Z_L = Z_{th}^* \quad \text{(Conjugate Match)} $ $ \text{Efficiency } (\eta) = 50\% \text{ at } P_{max} $

Efficiency Concept:

  • β—† Power systems aim for $\eta \approx 100\%$ ($R_L \gg R_{th}$).
  • β—† Signal systems aim for MPT ($R_L = R_{th}$).

Max power is delivered to the load when its impedance is the complex conjugate of the source impedance.

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