Network Analysis (EC-305) - Unit 5 Short Notes

UNIT 5: Network Analysis - Short Notes

1. Basic Circuit Laws, Theorems, and Analysis Techniques

Kirchhoff's Laws

• KCL (Current Law): At any node, the algebraic sum of currents leaving the node is zero.

$$\sum_{k=1}^{n} i_k = 0$$

> [!TIP] Assign consistent current directions (e.g., all leaving).

• KVL (Voltage Law): Around any closed loop, the algebraic sum of voltages is zero.

$$\sum_{k=1}^{n} v_k = 0$$

> [!TIP] Traverse loop in one direction, assign + to voltage if entering + terminal of an element.

Mesh Analysis (Planar Circuits)

1. Identify independent meshes (windows).

2. Apply KVL to each mesh, expressing voltages in terms of mesh currents.

3. Solve simultaneous equations.

• Supermesh: Formed when a current source lies on a common branch of two meshes. The supermesh excludes the current source and its branch. Use KVL on supermesh and the current source constraint equation.

Nodal Analysis

1. Select a reference node (ground).

2. Assign node voltages to remaining (n-1) nodes.

3. Apply KCL at each non-reference node, expressing currents in terms of node voltages.

4. Solve equations.

• Supernode: Formed when a voltage source connects two non-reference nodes. Enclose the source and connected nodes. Apply KCL to the supernode and use the voltage source constraint.

Superposition Theorem

For a linear circuit with multiple independent sources:

• The response (voltage/current) in any element = Algebraic sum of responses caused by each independent source acting alone.

• Procedure: Kill all other independent sources (voltage sources → short, current sources → open). Repeat for each source and sum results.

Pitfall: Does NOT apply to power calculations directly.

Thevenin's Theorem

Any linear two-terminal network can be replaced by an equivalent circuit of:

• V_TH: Open-circuit voltage across terminals.

• R_TH (DC) / Z_TH (AC): Equivalent impedance seen from terminals with all independent sources killed.

  • For AC: Z_TH = R_TH + jX_TH

  • With dependent sources, apply test source.

Norton's Theorem

Equivalent circuit: I_N (short-circuit current) in parallel with R_N / Y_N.

• R_N = R_TH, Y_N = 1/Z_TH

• I_N = V_TH / Z_TH

Maximum Power Transfer Theorem

• Condition: Load impedance Z_L = Z_TH** (complex conjugate for AC).

$$Z_L = R_{TH} + jX_{TH} \quad \text{(For max power)}$$

• Maximum Power:

$$P_{max} = \frac{|V_{TH}|^2}{4R_{TH}}$$

• Efficiency at max power:

$$\eta = \frac{P_{load}}{P_{total}} = \frac{P_{max}}{P_{max} + P_{loss\,in\,R_{TH}}} = 50\%$$

Millman's Theorem

For parallel voltage sources (V₁...V_n) each with series resistance (R₁...R_n):

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

The equivalent resistance:

$$R_{eq} = \frac{1}{\sum_{k=1}^{n} \frac{1}{R_k}}$$

Tellegen's Theorem

For any network (linear/nonlinear, passive/active, time-variant/invariant):

\boxed{\sum_{m=1}^{b} v_m i_m = 0}

Where the sum is over all b branches, v_m and i_m are branch voltage and current satisfying KVL & KCL.

Interpretation: Total instantaneous power in any network is zero (power absorbed = power delivered).

Compensation Theorem

If a branch impedance Z in a network is changed by ΔZ, the change in any branch current/voltage is the same as that produced by injecting a compensating source (V = -IΔZ or I = -VΔZ) in series/parallel with the changed branch, with all other sources killed.

Use: Sensitivity analysis.

Substitution Theorem

If the voltage across and current through any branch of a network are known, that branch can be replaced by any combination of elements that maintains the same v-i relationship (e.g., ideal voltage source of that voltage, or ideal current source of that current), without altering the currents/voltages elsewhere.

2. Network Topology

Graph & Oriented Graph

• Graph: Set of nodes (junctions) and branches (elements) showing connectivity, ignoring element sizes.

• Oriented Graph: Graph with arrows on branches indicating reference direction for current.

Tree & Co-Tree

• Tree (T): A connected subgraph containing all nodes and no loops. Has (n-1) branches for n nodes.

• Twigs: Branches of the tree.

• Links (or Chords): Branches not in the tree. Number of links = b - (n-1) = l.

• Co-Tree: Set of all links. Each link, when added to the tree, forms a fundamental loop.

Tie Set & Basic Tie Set Matrix

• Tie Set (Fundamental Loop): A loop containing exactly one link and the tree branches connecting its end nodes.

• Basic Tie Set Matrix (B_f): (l x b) matrix. Rows = fundamental loops (links). Columns = all branches.

  • Entry: +1 if branch reference direction aligns with loop direction, -1 if opposite, 0 if not in loop.

  • Property: B_f · A = 0 (A = incidence matrix).

Cut Set & Basic Cut Set Matrix

• Cut Set: A set of branches whose removal disconnects the graph into two parts, with exactly one branch from the tree connecting each part.

• Fundamental Cut Set: A cut containing exactly one twig.

• Basic Cut Set Matrix (Q_f): ((n-1) x b) matrix. Rows = fundamental cuts (twigs). Columns = all branches.

  • Property: Q_f · A = I (identity matrix).

Incidence Matrix

• Complete Incidence Matrix (A): (n x b) matrix.

  • a_ij = +1 if branch j leaves node i.

  • a_ij = -1 if branch j enters node i.

  • a_ij = 0 otherwise.

  • Property: Sum of any column = 0. Sum of any row = number of branches incident.

• Reduced Incidence Matrix (A_red): Formed by deleting the row corresponding to the reference node. Size: ((n-1) x b). Used in nodal analysis: A_red · V = E (source vector).

Matrix Relationships

\boxed{A_{red}^T \cdot Q_f^T = I} \quad \text{and} \quad \boxed{B_f \cdot A = 0}

Where I is identity matrix.

3. Mutual Inductance and Coupled Circuits

Self & Mutual Inductance

• Self Inductance (L): Flux linking a coil due to its own current:

$$v = L \frac{di}{dt}$$

• Mutual Inductance (M): Flux in one coil due to current in another:

$$v_2 = M \frac{di_1}{dt}$$

• Dot Convention: Dots indicate polarity of induced voltage. If currents enter dotted terminals simultaneously, mutual voltage is positive.

  • Equation:

$$v_1 = L_1 \frac{di_1}{dt} \pm M \frac{di_2}{dt}$$

*   Sign: **+** if i<sub>2</sub> enters dotted terminal, **-** if enters undotted.

Coefficient of Coupling (k)

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

• k = 1: Perfect coupling (all flux links).

• k = 0: No coupling (M = 0).

Equivalent Inductance

• Series Aiding: L_eq = L₁ + L₂ + 2M

• Series Opposing: L_eq = L₁ + L₂ - 2M

• Parallel Aiding:

$$L_{eq} = \frac{L_1 L_2 - M^2}{L_1 + L_2 + 2M}$$

• Parallel Opposing:

$$L_{eq} = \frac{L_1 L_2 - M^2}{L_1 + L_2 - 2M}$$

Analysis of Coupled Circuits

1. Write KVL for each mesh, including mutual terms using dot convention.

2. Solve simultaneous differential equations.

   • Example (two coils, dots on left):

$$v_1 = L_1 \frac{di_1}{dt} + M \frac{di_2}{dt}$$

$$v_2 = M \frac{di_1}{dt} + L_2 \frac{di_2}{dt}$$

4. Resonance

Series Resonance (RLC Series)

• Resonant Frequency (ω₀): When X_L = X_C.

$$\omega_0 = \frac{1}{\sqrt{LC}} \quad \text{or} \quad f_0 = \frac{1}{2\pi\sqrt{LC}}$$

• Impedance: Z = R (minimum, purely resistive).

• Current: I = V/R (maximum).

• Voltage Magnification: Voltage across L or C = Q × V, where Q-factor =

$$\frac{\omega_0 L}{R} = \frac{1}{\omega_0 C R}$$

• Bandwidth (BW):

$$\text{BW} = \frac{\omega_0}{Q} = \frac{R}{L} \text{ (rad/s)}$$

Parallel Resonance

• Ideal Case (R=0): ω₀ = 1/√(LC), admittance Y = 0 (infinite impedance).

• Practical (with R in series with L):

  • Admittance minimum → Impedance maximum.

  • Resonant frequency:

$$\omega_0 = \sqrt{\frac{1}{LC} - \frac{R^2}{L^2}} \approx \frac{1}{\sqrt{LC}} \text{ (if R small)}$$

*   At ω<sub>0</sub>, current from source is minimum, and currents in L and C branches are large (Q × I<sub>source</sub>).

5. Fourier Series

Trigonometric Fourier Series

For a periodic function f(t) with period T=2π/ω₀:

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

• Coefficients:

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

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

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

• Symmetry Properties:

  • Even function: b_n = 0 (only cosine terms).

  • Odd function: a₀ = a_n = 0 (only sine terms).

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

Exponential Fourier Series

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

• Complex Coefficients:

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

• Relationship:

$$c_0 = a_0, \quad c_n = \frac{a_n - jb_n}{2}, \quad c_{-n} = \frac{a_n + jb_n}{2}$$

• |c_n| gives amplitude of nth harmonic, ∠c_n gives phase.

Waveform Analysis (Common Examples)

• Square Wave (odd, ±A, 50% duty): Only odd harmonics.

$$a_0=0, \quad a_n=0, \quad b_n = \frac{4A}{n\pi} \text{ for n odd}$$

• Triangular Wave (odd, even harmonics only):

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

• Sawtooth Wave (odd):

$$a_0 = 0, \quad a_n = 0, \quad b_n = -\frac{2A}{n\pi}$$

6. Laplace Transform in Network Analysis

Laplace Transform (LT) Properties

• Linearity: L{af(t) + bg(t)} = aF(s) + bG(s)

• Time-shift: L{f(t-a)u(t-a)} = e^-asF(s)

• Differentiation: L{f'(t)} = sF(s) - f(0⁻)

• Integration: L{∫₀ᵗ f(τ)dτ} = F(s)/s

• Convolution: L{f(t)*g(t)} = F(s)G(s)

s-Domain Circuit Elements

• Initial conditions (IC) appear as additional sources in s-domain.

Initial & Final Value Theorems

• Initial Value Theorem (if F(s) has no poles in Re(s)>0):

  \boxed{f(0^+) = \lim_{s \to \infty} sF(s)}

• Final Value Theorem (if sF(s) has no poles in Re(s)≥0):

  \boxed{f(\infty) = \lim_{s \to 0} sF(s)}

Pitfall: Do not apply final value theorem if poles on imaginary axis (e.g., undamped oscillation).

Transient Analysis (Steps)

1. Draw s-domain circuit. Replace L, C with their impedances including IC sources.

2. Apply circuit laws (KVL/KCL) to find desired variable (I(s), V(s)).

3. Simplify F(s) using partial fractions.

4. Apply inverse LT (using tables or pole-zero) to get f(t).

5. For t>0, use final value theorem to check steady-state.

Pole-Zero Analysis

• Poles: Roots of denominator of F(s). → System stability: All poles must have negative real parts for causal, stable system.

• Zeros: Roots of numerator.

• Pole-Zero Plot: s-plane with poles (X) and zeros (O). Shape of |F(jω)| determined by pole-zero locations.

Transfer Functions

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

• Transfer Function (Gain): Ratio of output phasor to input phasor in s-domain, e.g., Voltage Gain = V₂(s)/V₁(s).

7. Two-Port Networks

Z-Parameters (Open-Circuit Impedance)

\boxed{

\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}

}

• Z₁₁ = V₁/I₁ | I₂=0 (input impedance with output open)

• Z₁₂ = V₁/I₂ | I₁=0 (reverse transfer impedance)

• Z₂₁ = V₂/I₁ | I₂=0 (forward transfer impedance)

• Z₂₂ = V₂/I₂ | I₁=0 (output impedance with input open)

For reciprocal networks: Z₁₂ = Z₂₁

Y-Parameters (Short-Circuit Admittance)

\boxed{

\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}

}

• Y_ij = I_i/V_j with other port shorted.

• Y = Z^-1

h-Parameters (Hybrid)

\boxed{

\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}

}

• h₁₁ = Input impedance with output shorted (Ω).

• h₁₂ = Reverse voltage gain with input shorted (dimensionless).

• h₂₁ = Forward current gain with output shorted (dimensionless).

• h₂₂ = Output admittance with input open (S).

ABCD Parameters (Transmission)

\boxed{

\begin{bmatrix}

V_1 \ I_1

\end{bmatrix}

= \begin{bmatrix}

A & B \

C & D

\end{bmatrix}

\begin{bmatrix}

V_2 \ -I_2

\end{bmatrix}

}

• A = V₁/V₂ | I₂=0 (Open-circuit voltage ratio)

• B = -V₁/I₂ | V₂=0 (Negative transfer impedance)

• C = I₁/V₂ | I₂=0 (Transfer admittance)

• D = I₁/(-I₂) | V₂=0 (Short-circuit current ratio)

Cascade: Overall [A] = [A]₁[A]₂ (matrix multiplication).

Parameter Conversions (Key Relationships)

• ΔZ = Z₁₁Z₂₂ - Z₁₂Z₂₁, similarly for ΔY, Δh.

Terminated Two-Port Network

• Input Impedance with Load Z_L:

$$Z_{in} = \frac{AV_2 + B(-I_2)}{CV_2 + D(-I_2)} = \frac{AZ_L + B}{CZ_L + D} \quad (\text{using } V_2 = -I_2 Z_L)$$

• Voltage Gain:

$$A_v = \frac{V_2}{V_1} = \frac{1}{A + B/Z_L} \quad (\text{if } I_1=0)$$

• Current Gain:

$$A_i = \frac{I_2}{I_1} = \frac{1}{C Z_L + D} \quad (\text{if } V_2=0)$$

8. Special Topics for Short Notes

Controlled Sources (Dependent Sources)

Used to model transistors and amplifiers.

Dual Networks

• Principle: Every theorem/equation in a network has a dual obtained by replacing:

  • Voltage ↔ Current

  • Series ↔ Parallel

  • Resistance ↔ Conductance

  • Inductance ↔ Capacitance

  • Open circuit ↔ Short circuit

  • KVL ↔ KCL

• Dual Circuit: Topologically identical graph, but with dual elements.

Network Topology Summary

• Graph: Connectivity only.

• Tree: Loopless, connects all nodes. (n-1) twigs.

• Link: Branch not in tree. (b-n+1) links.

• Fundamental Loop: One link + tree branches.

• Fundamental Cut: One twig + links.

• Matrices: Incidence (A), Tie Set (B_f), Cut Set (Q_f). Relationships: B_fA=0, Q_fA=I.

S-Domain Theorems

• Initial/Final Value Theorems: As above.

• Convolution Theorem: Time-domain convolution ↔ s-domain multiplication.

• Superposition: Holds in s-domain for linear circuits.

• Thevenin/Norton: Valid with s-domain equivalents (Z_TH(s), V_TH(s)).