UNIT 5: NETWORK ANALYSIS - EXAM-FOCUSED SHORT NOTES

I. FUNDAMENTAL CIRCUIT LAWS AND ANALYSIS TECHNIQUES

Kirchhoff's Current Law (KCL)

Definition: The algebraic sum of currents entering a node (or closed boundary) is zero.

Mathematical Form: $$\displaystyle \sum_{k=1}^{n} i_k = 0 $$

Principle: Based on charge conservation. Currents entering are positive, leaving are negative (or vice-versa, but consistent).

Example: For node A, $$\displaystyle i_1 + i_2 - i_3 - i_4 = 0 $$.

Kirchhoff's Voltage Law (KVL)

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

Mathematical Form: $$\displaystyle \sum_{k=1}^{m} v_k = 0 $$

Principle: Based on energy conservation. Traverse loop in a direction; voltage rises are positive, drops are negative.

Example: For loop ABCDA, $$\displaystyle v_{AB} + v_{BC} + v_{CD} + v_{DA} = 0 $$.

[!TIP] Exam Alert:

• KCL applies to any closed surface (not just a single node).

• KVL applies to any closed path, even if it doesn't follow a single mesh.

• Always assign polarities and direction before writing equations.

Nodal Analysis

Objective: Find node voltages using KCL.
Steps:

1. Select reference node (ground).

2. Assign voltages $$\displaystyle V_1, V_2, ... $$ to remaining $(n-1)$ nodes.

3. Apply KCL at each non-reference node (except voltage source terminals).

4. Solve simultaneous equations.

For circuits with voltage sources: Use supernode method when a voltage source is between two non-reference nodes.

Example: For a 3-node circuit (1 reference), write 2 KCL equations.

Mesh Analysis

Objective: Find mesh currents using KVL.
Steps:

1. Identify meshes (independent loops).

2. Assign mesh currents $$\displaystyle I_1, I_2, ... $$ (clockwise convention).

3. Apply KVL around each mesh.

4. Solve equations.

For circuits with current sources:

• If current source is in a mesh branch, that mesh current is known.

• If current source is shared by two meshes, form a supermesh (exclude the source).

Example: For a planar circuit with 2 meshes, write 2 KVL equations in terms of $$\displaystyle I_1, I_2 $$.

II. NETWORK THEOREMS

Superposition Theorem

Statement: In a linear circuit with multiple sources, the response (voltage/current) in any element is the algebraic sum of responses caused by each source acting alone, with all other independent sources deactivated (voltage sources → short, current sources → open).

Note: Dependent sources are never deactivated; they remain with their controlling variables.

Power calculation: Cannot use superposition directly (non-linear).

Thevenin's Theorem (DC & AC)

Statement: Any linear two-terminal network can be replaced by an equivalent circuit consisting of a voltage source $$\displaystyle V_{th} $$ in series with impedance $$\displaystyle Z_{th} $$.

$$\displaystyle V_{th} $$: Open-circuit voltage across terminals.

$$\displaystyle Z_{th} $$: Input impedance with all independent sources deactivated (for AC, replace sources with their internal impedances).

For AC: $$\displaystyle Z_{th} $$ is complex; $$\displaystyle V_{th} $$ is phasor.

Steps:

1. Remove load.

2. Find $$\displaystyle V_{oc} = V_{th} $$.

3. Find $$\displaystyle Z_{th} $$ (or $$\displaystyle R_{th} $$ for DC).

4. Reconnect load.

Norton's Theorem (DC & AC)

Statement: Equivalent to a current source $$\displaystyle I_N $$ in parallel with impedance $$\displaystyle Z_N $$.

$$\displaystyle I_N $$: Short-circuit current across terminals.

$$\displaystyle Z_N $$: Same as $$\displaystyle Z_{th} $$ from Thevenin.

Relation: $$\displaystyle I_N = V_{th} / Z_{th} $$, $$\displaystyle Z_N = Z_{th} $$.

Maximum Power Transfer Theorem

DC Condition: Load resistance $$\displaystyle R_L = R_{th} $$ (Thevenin resistance).

AC Condition: Load impedance $$\displaystyle Z_L = Z_{th}^* $$ (complex conjugate of Thevenin impedance).

Maximum Power: $$\displaystyle P_{max} = \frac{V_{th}^2}{4 R_{th}} $$ (DC).

Efficiency Proof:

Total power delivered $$\displaystyle P_{total} = I^2 (R_{th} + R_L) $$.

Power to load $$\displaystyle P_L = I^2 R_L $$.

At max power, $$\displaystyle R_L = R_{th} \Rightarrow \eta = \frac{P_L}{P_{total}} = \frac{R_L}{2 R_L} = 0.5 $$ (50%).

\boxed{\eta_{\text{max}} = 50%}

Tellegen's Theorem

Statement: For any two networks (not necessarily same topology) with the same graph and branch voltages/currents satisfying KVL and KCL respectively:

$$\sum_{b=1}^{B} v_b i_b = 0$$

where $$\displaystyle v_b $$ are voltages in one network, $$\displaystyle i_b $$ are currents in the other (both obey same reference directions).

Interpretation: Conservation of energy in network form.

Verification: Compute instantaneous power in each branch; sum must be zero.

Millman's Theorem

Application: Parallel voltage sources with series resistances.

Voltage at common node:

$$V = \frac{\sum_{k=1}^{n} \frac{V_k}{R_k}}{\sum_{k=1}^{n} \frac{1}{R_k}}$$

Equivalent resistance: $$\displaystyle R_{eq} = \left( \sum \frac{1}{R_k} \right)^{-1} $$.

Compensation Theorem

Statement: If the impedance of a branch changes by $\Delta Z$, the change in any current/voltage anywhere in the network is the same as that produced by injecting a compensating source of value $-\Delta Z \cdot i$ (where $i$ is original branch current) in series with the changed branch.

Use: Sensitivity analysis.

Substitution Theorem

Statement: If the voltage across a branch and current through it are known (from network solution), that branch can be replaced by any combination of elements that maintains the same $v$ and $i$ (e.g., voltage source, current source, or impedance), without affecting the rest of the network.

Controlled Sources (Dependent Sources)

Note: Controlling variable may be in same or different branch; use subscript notation (e.g., $$\displaystyle v_{be} $$).

III. GRAPH THEORY AND NETWORK TOPOLOGY

Graph & Oriented Graph

• Graph: Set of nodes (vertices) and branches (edges) showing connectivity, ignoring component values/directions.

• Oriented Graph: Graph with assigned directions to all branches.

Tree & Co-Tree

• Tree: Connected subgraph containing all nodes and no loops; has $n-1$ branches ($n$ = number of nodes).

• Co-Tree: Branches not in the tree; each co-tree branch forms a fundamental loop with tree branches.

• Twigs: Branches of the tree.

• Links: Branches of the co-tree.

Tie Set & Basic Tie Set Matrix

• Tie Set (Loop): Set of branches forming a loop when a link is added to the tree.

• Basic Tie Set Matrix ($B$): $(n-1) \times b$ matrix ($b$ = total branches). Rows = fundamental loops (one per link). Entries:

  $$\displaystyle b_{ij} = +1 $$ if branch $j$ is in loop $i$ and same direction as loop current,

  $-1$ if opposite,

  $0$ if not in loop.

KVL in matrix form: $$\displaystyle B \cdot \mathbf{v} = \mathbf{0} $$.

Cut Set & Cut Set Matrix

• Cut Set: Minimal set of branches whose removal disconnects the graph into two parts.

• Basic Cut Set: Formed by one twig and possibly some links; there are $(n-1)$ basic cut sets.

• Cut Set Matrix ($Q$): $(n-1) \times b$ matrix. Rows = basic cut sets. Entries:

  $$\displaystyle q_{ij} = +1 $$ if branch $j$ is in cut set $i$ and directed from + to - side,

  $-1$ if opposite,

  $0$ if not in cut set.

KCL in matrix form: $$\displaystyle Q \cdot \mathbf{i} = \mathbf{0} $$.

Incidence Matrix (Complete)

• Definition: $n \times b$ matrix describing branch connections to nodes.

• Entry $$\displaystyle a_{ij} $$:

  $+1$ if branch $j$ leaves node $i$,

  $-1$ if branch $j$ enters node $i$,

  $0$ otherwise.

• Properties: Sum of any column = 0; rank = $n-1$ for connected graph.

[!TIP] Exam Pattern:

• Often asked: "Draw graph, write cut set matrix" or "Find incidence matrix from figure."

• Remember: Tree has $n-1$ branches, co-tree has $b-n+1$ links.

• $B$ matrix relates to KVL, $Q$ to KCL, Incidence to both.

IV. TRANSIENT ANALYSIS USING LAPLACE TRANSFORM

Laplace Transform of Standard Waveforms

Application to RC, RL, RLC Circuits

General Approach:

1. Draw s-domain equivalent circuit:

   • $R \to R$, $L \to sL$, $$\displaystyle C \to \frac{1}{sC} $$.

   • Initial conditions:

     • Capacitor voltage $$\displaystyle v_C(0^-) \to $$ voltage source $$\displaystyle \frac{v_C(0^-)}{s} $$ in series with $$\displaystyle \frac{1}{sC} $$.

     • Inductor current $$\displaystyle i_L(0^-) \to $$ current source $$\displaystyle \frac{i_L(0^-)}{s} $$ in parallel with $sL$.

2. Apply circuit laws (KVL/KCL) in s-domain.

3. Solve for desired $I(s)$ or $V(s)$.

4. Inverse Laplace to get $i(t)$ or $v(t)$.

Step Response of RC Circuit (source $V u(t)$, $$\displaystyle v_C(0)=0 $$):

$$I(s) = \frac{V}{s(R + \frac{1}{sC})} = \frac{V}{R} \cdot \frac{1}{s + \frac{1}{RC}}$$

Inverse:

$$i(t) = \frac{V}{R} e^{-t/(RC)} u(t)$$

Graph: Exponential decay from $V/R$ to 0.

Initial Value Theorem (IVT) & Final Value Theorem (FVT)

• IVT: If $sF(s)$ has no poles in RHP and no pole at $$\displaystyle s=0 $$ (except simple), then:

$$f(0^+) = \lim_{s \to \infty} s F(s)$$

• FVT: If $sF(s)$ has all poles in LHP (except possibly simple pole at $$\displaystyle s=0 $$), then:

$$f(\infty) = \lim_{s \to 0} s F(s)$$

[!TIP] Common Pitfall:

• Check pole locations before applying FVT. If poles on imaginary axis (e.g., undamped oscillation), FVT fails (limit doesn't exist).

• IVT requires $f(t)$ to be continuous at $$\displaystyle t=0^+ $$ (no impulses).

Pole-Zero Plot & Inverse Laplace

• Poles: Roots of denominator of $F(s)$ (where $F(s) \to \infty$).

• Zeros: Roots of numerator (where $$\displaystyle F(s)=0 $$).

• Plot: s-plane (Re vs Im).

• Time response: Each pole contributes:

  • Real negative pole: decaying exponential.

  • Complex pole pair: damped sinusoid.

  • Repeated pole: $$\displaystyle t e^{at} $$ terms.

• Inverse: Use partial fraction expansion, then lookup table.

Example: $$\displaystyle I(s) = \frac{20s}{(s+5)(s+2)} $$ → poles at $-5, -2$; zero at $0$.
$$\displaystyle i(t) = A e^{-5t} + B e^{-2t} $$.

V. FOURIER SERIES ANALYSIS

Trigonometric Fourier Series (TFS)

For periodic $f(t)$ with period $T$, fundamental $$\displaystyle \omega_0 = 2\pi/T $$:

$$f(t) = a_0 + \sum_{n=1}^{\infty} \left( a_n \cos n\omega_0 t + b_n \sin n\omega_0 t \right)$$

where:

$$a_0 = \frac{1}{T} \int_{0}^{T} f(t) dt$$

$$a_n = \frac{2}{T} \int_{0}^{T} f(t) \cos n\omega_0 t \, dt$$

$$b_n = \frac{2}{T} \int_{0}^{T} f(t) \sin n\omega_0 t \, dt$$

Symmetry shortcuts:

• Even function: $$\displaystyle b_n = 0 $$, only cosine terms.

• Odd function: $$\displaystyle a_0 = a_n = 0 $$, only sine terms.

• Half-wave symmetry: $$\displaystyle f(t+T/2) = -f(t) $$ → only odd harmonics ($$\displaystyle n=1,3,5... $$).

Exponential Fourier Series (EFS)

$$f(t) = \sum_{n=-\infty}^{\infty} c_n e^{j n \omega_0 t}$$

$$c_n = \frac{1}{T} \int_{0}^{T} f(t) e^{-j n \omega_0 t} dt$$

Relation to TFS:
$$\displaystyle c_n = \frac{1}{2}(a_n - j b_n) $$ for $$\displaystyle n>0 $$,
$$\displaystyle c_{-n} = \frac{1}{2}(a_n + j b_n) $$,
$$\displaystyle c_0 = a_0 $$.

Fourier Series of Triangular Wave

Definition: Symmetric triangle wave, period $T$, amplitude $A$.
Properties: Even + half-wave symmetric → only odd cosine harmonics.
Coefficients:

$$a_0 = 0 \quad (\text{symmetric about zero})$$

$$a_n = \frac{8A}{n^2 \pi^2} \quad \text{for odd } n$$

$$a_n = 0 \quad \text{for even } n$$

$$b_n = 0$$

TFS:

$$f(t) = \frac{8A}{\pi^2} \left( \cos \omega_0 t - \frac{1}{9} \cos 3\omega_0 t + \frac{1}{25} \cos 5\omega_0 t - \cdots \right)$$

Note: Amplitudes decay as $$\displaystyle 1/n^2 $$ (faster than square wave's $1/n$).

VI. TWO-PORT NETWORK PARAMETERS

Z-Parameters (Impedance)

$$\begin{bmatrix} V_1 \\ V_2 \end{bmatrix} = \begin{bmatrix} Z_{11} & Z_{12} \\ Z_{21} & Z_{22} \end{bmatrix} \begin{bmatrix} I_1 \\ I_2 \end{bmatrix}$$

• $$\displaystyle Z_{11} = \left. \frac{V_1}{I_1} \right|_{I_2=0} $$ (open-circuit output)

• $$\displaystyle Z_{12} = \left. \frac{V_1}{I_2} \right|_{I_1=0} $$

• $$\displaystyle Z_{21} = \left. \frac{V_2}{I_1} \right|_{I_2=0} $$

• $$\displaystyle Z_{22} = \left. \frac{V_2}{I_2} \right|_{I_1=0} $$

Y-Parameters (Admittance)

$$\begin{bmatrix} I_1 \\ I_2 \end{bmatrix} = \begin{bmatrix} Y_{11} & Y_{12} \\ Y_{21} & Y_{22} \end{bmatrix} \begin{bmatrix} V_1 \\ V_2 \end{bmatrix}$$

• $$\displaystyle Y_{11} = \left. \frac{I_1}{V_1} \right|_{V_2=0} $$ (short-circuit output)

• $$\displaystyle Y_{12} = \left. \frac{I_1}{V_2} \right|_{V_1=0} $$

• $$\displaystyle Y_{21} = \left. \frac{I_2}{V_1} \right|_{V_2=0} $$

• $$\displaystyle Y_{22} = \left. \frac{I_2}{V_2} \right|_{V_1=0} $$

Relation: $$\displaystyle Y = Z^{-1} $$ (if $Z$ nonsingular).

h-Parameters (Hybrid)

$$\begin{bmatrix} V_1 \\ I_2 \end{bmatrix} = \begin{bmatrix} h_{11} & h_{12} \\ h_{21} & h_{22} \end{bmatrix} \begin{bmatrix} I_1 \\ V_2 \end{bmatrix}$$

• $$\displaystyle h_{11} = \left. \frac{V_1}{I_1} \right|_{V_2=0} $$ (input impedance with output shorted)

• $$\displaystyle h_{12} = \left. \frac{V_1}{V_2} \right|_{I_1=0} $$ (reverse voltage gain with input open)

• $$\displaystyle h_{21} = \left. \frac{I_2}{I_1} \right|_{V_2=0} $$ (forward current gain with output shorted)

• $$\displaystyle h_{22} = \left. \frac{I_2}{V_2} \right|_{I_1=0} $$ (output admittance with input open)

Units: $$\displaystyle h_{11} $$: $\Omega$, $$\displaystyle h_{12} $$: dimensionless, $$\displaystyle h_{21} $$: dimensionless, $$\displaystyle h_{22} $$: S.

ABCD Parameters (Transmission)

$$\begin{bmatrix} V_1 \\ I_1 \end{bmatrix} = \begin{bmatrix} A & B \\ C & D \end{bmatrix} \begin{bmatrix} V_2 \\ -I_2 \end{bmatrix}$$

• $$\displaystyle A = \left. \frac{V_1}{V_2} \right|_{I_2=0} $$ (open-circuit voltage ratio)

• $$\displaystyle B = \left. -\frac{V_1}{I_2} \right|_{V_2=0} $$ (short-circuit impedance)

• $$\displaystyle C = \left. \frac{I_1}{V_2} \right|_{I_2=0} $$ (open-circuit admittance)

• $$\displaystyle D = \left. -\frac{I_1}{I_2} \right|_{V_2=0} $$ (short-circuit current ratio)

For reciprocal network: $$\displaystyle AD - BC = 1 $$.
For symmetric network: $$\displaystyle A = D $$.

Parameter Conversions (Key Relations)

1. Y in terms of ABCD:

$$Y_{11} = \frac{D}{B}, \quad Y_{12} = -\frac{1}{B}, \quad Y_{21} = -\frac{1}{B}, \quad Y_{22} = \frac{A}{B}$$

(provided $B \neq 0$)

2. Z in terms of ABCD:

$$Z_{11} = \frac{A}{C}, \quad Z_{12} = \frac{AD - BC}{C}, \quad Z_{21} = \frac{1}{C}, \quad Z_{22} = \frac{D}{C}$$

(provided $C \neq 0$)

3. h in terms of ABCD:

$$h_{11} = \frac{A}{C}, \quad h_{12} = \frac{AD - BC}{C}, \quad h_{21} = \frac{1}{C}, \quad h_{22} = -\frac{D}{B}$$

[!TIP] Cascade Connection:

For two-port networks in cascade, ABCD parameters multiply:

$$\begin{bmatrix} A & B \\ C & D \end{bmatrix}_{\text{total}} = \begin{bmatrix} A_1 & B_1 \\ C_1 & D_1 \end{bmatrix} \begin{bmatrix} A_2 & B_2 \\ C_2 & D_2 \end{bmatrix}$$

This is why ABCD is preferred for cascaded systems (e.g., filters).

Terminated Two-Port Network

Given two-port with parameters (say Z) and load $$\displaystyle Z_L $$ at port 2:

• $$\displaystyle V_2 = -Z_L I_2 $$

• Solve:

  $$\displaystyle V_1 = Z_{11} I_1 + Z_{12} I_2 $$

  $$\displaystyle -Z_L I_2 = Z_{21} I_1 + Z_{22} I_2 $$

• Find voltage gain $$\displaystyle G_V = V_2/V_1 $$, current gain, etc.

VII. RESONANCE IN AC CIRCUITS

Series Resonant Circuit

Circuit: $R$, $L$, $C$ in series with voltage source $V$.
Impedance: $$\displaystyle Z = R + j(\omega L - 1/(\omega C)) $$
Resonant Frequency $$\displaystyle \omega_0 $$:

Imaginary part zero:

$$\omega_0 L = \frac{1}{\omega_0 C} \quad \Rightarrow \quad \boxed{\omega_0 = \frac{1}{\sqrt{LC}}}$$

At resonance:

• $$\displaystyle Z = R $$ (minimum, purely resistive)

• Current $$\displaystyle I = V/R $$ (maximum)

• $$\displaystyle V_L = V_C = Q \cdot V $$, where Quality Factor $$\displaystyle Q = \frac{\omega_0 L}{R} = \frac{1}{R \omega_0 C} $$

• Bandwidth (BW): $$\displaystyle \text{BW} = \frac{\omega_0}{Q} $$ (between half-power frequencies).

Parallel Resonant Circuit

Circuit: $R$, $L$, $C$ in parallel (or practical inductor with $$\displaystyle R_p $$).
Admittance: $$\displaystyle Y = \frac{1}{R} + j\left( \frac{1}{\omega L} - \omega C \right) $$
Resonant Frequency $$\displaystyle \omega_0 $$:

$$\frac{1}{\omega_0 L} = \omega_0 C \quad \Rightarrow \quad \boxed{\omega_0 = \frac{1}{\sqrt{LC}}}$$

(same as series, but for ideal parallel RLC; with resistance in inductor, $$\displaystyle \omega_0 $$ shifts slightly).
At resonance:

• $$\displaystyle Y = 1/R $$ (minimum admittance, maximum impedance)

• Current from source minimum.

VIII. MAGNETIC COUPLING

Mutual Inductance $M$

• Voltage induced in coil 2 due to current in coil 1: $$\displaystyle v_2 = M \frac{di_1}{dt} $$

• Similarly, $$\displaystyle v_1 = M \frac{di_2}{dt} $$

• $M$ depends on geometry, orientation, core material.

• Sign convention: Dot notation. If currents enter dotted terminals, mutual voltage adds to self-induced voltage.

Coefficient of Coupling $k$

$$k = \frac{M}{\sqrt{L_1 L_2}}, \quad 0 \le k \le 1$$

• $$\displaystyle k=1 $$: Perfect coupling (all flux links both coils).

• $$\displaystyle k<1 $$: Partial coupling.

Coupled Coils in Series/Parallel

Series:

• Aiding (dots same side): $$\displaystyle L_{eq} = L_1 + L_2 + 2M $$

• Opposing (dots opposite): $$\displaystyle L_{eq} = L_1 + L_2 - 2M $$

Parallel:

• Aiding: $$\displaystyle L_{eq} = \frac{L_1 L_2 - M^2}{L_1 + L_2 + 2M} $$

• Opposing: $$\displaystyle L_{eq} = \frac{L_1 L_2 - M^2}{L_1 + L_2 - 2M} $$

Example: Two coils $$\displaystyle L_1=L_2=L $$, in series aiding: $$\displaystyle L_{eq}=2L+2M $$; opposing: $$\displaystyle L_{eq}=2L-2M $$.

IX. ADVANCED TOPICS AND SPECIAL THEOREMS

Dual Networks

• Definition: Two networks are dual if their equations are identical when:

  $$\displaystyle R \leftrightarrow G $$, $$\displaystyle L \leftrightarrow C $$, $$\displaystyle V \leftrightarrow I $$, series $$\displaystyle \leftrightarrow $$ parallel, open $$\displaystyle \leftrightarrow $$ short.

• Procedure: Replace all elements and connections with their duals.

• Property: If a theorem holds for a network, it holds for its dual.

Network Functions & Transfer Functions

• Driving Point Impedance: $$\displaystyle Z(s) = V(s)/I(s) $$ at a port with other ports terminated.

• Transfer Function: Ratio of output to input in s-domain, e.g.,

  $$\displaystyle G_{21}(s) = I_2(s)/I_1(s) $$ (current gain),

  $$\displaystyle Z_2(s) = V_2(s)/I_2(s) $$ (driving point at port 2).

s-Domain Analysis (General)

• Replace $j\omega$ with $s$ in phasor impedance:

  $R \to R$, $L \to sL$, $C \to 1/(sC)$.

• Initial conditions handled as independent sources in s-domain.

• System function $H(s)$: poles → natural response, zeros → forced response characteristics.

FINAL EXAM STRATEGY:

1. Prioritize based on past papers: KCL/KVL, Nodal/Mesh, Superposition, Thevenin/Norton, Max Power, Fourier Series (triangular/square), Laplace (step response, IVT/FVT), Two-port Z/Y/ABCD conversions, Graph (Incidence/Cut set).

2. Diagrams are crucial for graph theory and two-port connections.

3. Always state conditions for theorems (e.g., linearity for superposition, conjugate matching for AC max power).

4. Conversions between two-port parameters are high-yield—memorize key formulas.

5. For Laplace, show s-domain circuit clearly with initial condition sources.