Network Analysis (EC-305) - Unit 4 Short Notes

UNIT 4: Network Analysis - Short Notes

I. Fundamental Laws and Basic Concepts

Kirchhoff's Current Law (KCL)

Statement: The algebraic sum of currents meeting at any node (junction) in a network is zero.

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

• Explanation: Based on conservation of charge. Currents entering a node are positive; currents leaving are negative (or vice-versa, but consistency is key).

• Example: For a node with three branches, if $$\displaystyle I_1 $$ enters and $$\displaystyle I_2 $$, $$\displaystyle I_3 $$ leave, then $$\displaystyle I_1 - I_2 - I_3 = 0 $$ or $$\displaystyle I_1 = I_2 + I_3 $$.

Kirchhoff's Voltage Law (KVL)

Statement: The algebraic sum of voltages around any closed loop in a network is zero.

$$ \sum_{k=1}^{m} V_k = 0 $$

• Explanation: Based on conservation of energy. Assign polarity signs to voltage drops; traverse the loop in one direction.

• Example: For a loop with a voltage source $$\displaystyle V_s $$ and two resistors with drops $$\displaystyle V_1 $$, $$\displaystyle V_2 $$, traversing from - to + across $$\displaystyle V_s $$ gives $$\displaystyle +V_s - V_1 - V_2 = 0 $$.

II. Graph Theory and Network Topology

Graph and Oriented Graph

• Graph: A representation of a circuit where branches (circuit elements) are shown as lines and nodes (connection points) as dots. It ignores physical layout.

• Oriented Graph: A graph with directions (arrows) assigned to all branches. Used for analysis (e.g., KCL/KVL equations).

Tree and Co-Tree

• Tree: A connected subgraph of a graph that includes all nodes but no closed loops. Number of twigs (tree branches) = $b - n + 1$, where $b$ = total branches, $n$ = total nodes.

• Co-Tree: The set of branches not in the tree. Each branch in the co-tree is a link or chord. Links = $n - 1$.

Twigs and Links

• Twigs: Branches that form the tree.

• Links (Chords): Branches that form the co-tree.

Cut Set and Basic Cut Set Matrix

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

• Basic Cut Set Matrix ($Q$): A matrix where rows represent fundamental cut sets (one twig removed, rest are links). Entries: $+1$ if branch leaves the isolated part, $-1$ if it enters, $0$ otherwise. For a graph with $n$ nodes, there are $(n-1)$ fundamental cut sets.

Tie Set and Basic Tie Set Matrix

• Tie Set (Loop): A minimal set of branches forming a closed loop with exactly one link and the remaining twigs.

• Basic Tie Set Matrix ($B$): Rows represent fundamental tie sets. Entries: $+1$ if branch direction matches loop direction, $-1$ if opposite, $0$ if not in loop. Number of fundamental tie sets = $b - n + 1$ (links).

Incidence Matrix (Complete Incidence Matrix $A$)

• Definition: A matrix representing the connection between branches and nodes.

• Construction: Rows = nodes ($n$), Columns = branches ($b$). For branch $k$ connecting node $i$ to $j$:

  • $$\displaystyle A_{ik} = +1 $$ (branch leaves node $i$)

  • $$\displaystyle A_{jk} = -1 $$ (branch enters node $j$)

  • All other entries = 0.

• Property: Sum of elements in any column = 0 (KCL). Rank = $n-1$ for a connected graph.

Network Topology

• Significance: Provides a systematic, algebraic method (via matrices $A$, $B$, $Q$) to write KCL/KVL equations without drawing loops/nodes repeatedly. Essential for computer-aided circuit analysis.

III. Circuit Theorems

Thevenin's Theorem (DC & AC)

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

• Procedure:

  1. Remove load. Find $$\displaystyle V_{th} $$ = open-circuit voltage across terminals.

  2. Find $$\displaystyle Z_{th} $$:

     • DC: $$\displaystyle Z_{th} = R_{th} $$ = (V_oc / I_sc) or deactivate independent sources.

     • AC: $$\displaystyle Z_{th} $$ = (V_oc / I_sc) with phasors; deactivate independent sources.

  3. Connect load $$\displaystyle Z_L $$ to $$\displaystyle V_{th} $$ and $$\displaystyle Z_{th} $$.

• Key: $$\displaystyle V_{th} $$ is the open-circuit voltage; $$\displaystyle Z_{th} $$ is the impedance seen into the network with sources killed.

Norton's Theorem (DC & AC)

Statement: Any linear bilateral network can be replaced by an equivalent circuit consisting of a current source $$\displaystyle I_N $$ in parallel with an impedance $$\displaystyle Z_N $$.

• Procedure:

  1. Remove load. Find $$\displaystyle I_N $$ = short-circuit current across terminals.

  2. Find $$\displaystyle Z_N $$ = $$\displaystyle Z_{th} $$ (same as Thevenin).

  3. Connect load $$\displaystyle Z_L $$ to $$\displaystyle I_N $$ and $$\displaystyle Z_N $$.

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

Superposition Theorem

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

• Application: Useful for circuits with one source at a time. Not applicable for power calculations (non-linear).

Maximum Power Transfer Theorem (DC)

Condition: Maximum power is delivered to the load when load resistance $$\displaystyle R_L $$ equals Thevenin resistance $$\displaystyle R_{th} $$ seen from the load terminals.

$$ R_L = R_{th} $$

• Maximum Power:

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

• Efficiency Proof: Efficiency $$\displaystyle \eta = \frac{P_L}{P_{total}} $$. At max power, $$\displaystyle P_L = V_{th}^2/(4R_{th}) $$, $$\displaystyle P_{total} = V_{th}^2/R_{th} $$. Thus $$\displaystyle \eta = 0.5 $$ or 50%.

[!TIP] Efficiency is 50% only at maximum power condition. For higher efficiency, $$\displaystyle R_L > R_{th} $$.

Tellegen's Theorem

Statement: For any two networks (not necessarily the same) that have the same topology (same incidence matrix), the following holds:

$$ \sum_{k=1}^{b} v_k i_k = 0 $$

where $$\displaystyle v_k $$ and $$\displaystyle i_k $$ are branch voltages and currents of the two networks respectively.

• Significance: A topological theorem independent of element characteristics. Used to verify network solutions and derive other theorems.

Millman's Theorem

Statement: For a circuit with several parallel branches, each containing a voltage source $$\displaystyle V_k $$ in series with impedance $$\displaystyle Z_k $$, the equivalent voltage $$\displaystyle V_{eq} $$ and impedance $$\displaystyle Z_{eq} $$ across the parallel combination are:

$$ V_{eq} = \frac{\sum_{k=1}^{n} \frac{V_k}{Z_k}}{\sum_{k=1}^{n} \frac{1}{Z_k}}, \quad Z_{eq} = \frac{1}{\sum_{k=1}^{n} \frac{1}{Z_k}} $$

• Application: Directly finds the common voltage in parallel voltage sources. For current sources, use dual form.

Substitution Theorem

Statement: 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 voltage and current, without affecting the rest of the network.

• Conditions: The replacing network must have identical $v$ and $i$ for the given branch. Useful for simplifying networks during analysis.

Compensation Theorem

Statement: If a branch with impedance $Z$ has a current $I$, and a voltage $$\displaystyle V = IZ $$ is injected in series opposition to $V$, the effect on the rest of the network is the same as replacing that branch with a voltage source of value $V$.

• Application: Network modification. Useful in sensitivity analysis and fault studies.

IV. Resonance

Series Resonant Circuit

• Circuit: $R$, $L$, $C$ in series with voltage source $V$.

• Resonance Condition: Impedance is purely resistive and minimum.

$$ X_L = X_C \implies \omega_0 L = \frac{1}{\omega_0 C} \implies \omega_0 = \frac{1}{\sqrt{LC}} $$

$$ f_0 = \frac{1}{2\pi\sqrt{LC}} $$

• Quality Factor (Q): Measure of sharpness.

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

• Voltage Magnification: At resonance, voltage across $L$ or $C$ is $Q$ times the input voltage.

$$ |V_L| = |V_C| = Q \cdot V $$

Parallel Resonant Circuit

• Circuit: $R$, $L$, $C$ in parallel (often $R$ represents inductor resistance).

• Resonance Condition: Admittance is purely conductive and minimum (impedance maximum).

$$ B_L = B_C \implies \frac{1}{\omega_0 L} = \omega_0 C \quad \text{(if } R \text{ large)} \implies \omega_0 = \frac{1}{\sqrt{LC}} $$

• Admittance at Resonance: $$\displaystyle Y_{res} = 1/R_{eq} $$ (purely real). For ideal parallel RLC, $$\displaystyle Y_{res} = 1/R $$.

V. Laplace Transform Analysis

Laplace Transform of Standard Waveforms

Application to RLC Circuits (Transient Analysis)

1. Model circuit in s-domain:

   • $$\displaystyle R \rightarrow R $$

   • $$\displaystyle L \rightarrow sL + L i(0^-) $$ (voltage source in series)

   • $$\displaystyle C \rightarrow \frac{1}{sC} + \frac{v_C(0^-)}{s} $$ (current source in parallel)

2. Write KVL/KCL equations in s-domain.

3. Solve for desired variable (e.g., $I(s)$).

4. Apply Inverse Laplace to get $i(t)$.

Initial Value Theorem (IVT)

Statement: If $F(s)$ is the Laplace transform of $f(t)$, and $sF(s)$ has no poles in the right-half plane (RHP) and possibly a simple pole at $$\displaystyle s=0 $$, then:

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

• Application: Find initial value of $i(t)$ or $v(t)$ directly from $I(s)$ or $V(s)$.

Final Value Theorem (FVT)

Statement: If $sF(s)$ has no poles in the RHP and possibly a simple pole at $$\displaystyle s=0 $$, then:

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

• Verification: Check poles of $sF(s)$. Fails if poles on imaginary axis (except origin) or in RHP.

• Example: For $$\displaystyle I(s) = \frac{2s+3}{(s+1)(s+3)} $$, poles at $$\displaystyle s=-1,-3 $$ (LHP). FVT applicable.

Transfer Function, Pole-Zero Plot, Inverse Laplace

• Driving Point Impedance: $$\displaystyle Z(s) = \frac{V(s)}{I(s)} $$ (ratio at same port).

• Transfer Function: $$\displaystyle H(s) = \frac{Output(s)}{Input(s)} $$ (e.g., $$\displaystyle V_2(s)/V_1(s) $$).

• Pole-Zero Plot: Plot poles ($\times$) and zeros ($\circ$) on s-plane. Determines stability and response shape.

• Inverse Laplace: Use partial fraction expansion and transform tables to obtain $i(t)$ or $v(t)$.

VI. Fourier Series Analysis

Trigonometric Fourier Series

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

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

Coefficients:

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

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

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

Exponential Fourier Series

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

Coefficient:

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

Relation to Trigonometric:

$$ c_n = \frac{1}{2} (a_n - j b_n), \quad c_{-n} = c_n^* $$

$$ a_0 = c_0, \quad a_n = c_n + c_{-n}, \quad b_n = j(c_n - c_{-n}) $$

VII. Two-Port Network Parameters

Z-Parameters (Impedance Parameters)

Definition: Open-circuit impedance parameters.

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

$$ V_2 = Z_{21} I_1 + Z_{22} I_2 $$

• Calculation: $$\displaystyle Z_{11} = \left. \frac{V_1}{I_1} \right|_{I_2=0} $$, $$\displaystyle Z_{12} = \left. \frac{V_1}{I_2} \right|_{I_1=0} $$, etc.

Y-Parameters (Admittance Parameters)

Definition: Short-circuit admittance parameters.

$$ I_1 = Y_{11} V_1 + Y_{12} V_2 $$

$$ I_2 = Y_{21} V_1 + Y_{22} V_2 $$

• Relation to Z: $$\displaystyle [Y] = [Z]^{-1} $$.

h-Parameters (Hybrid Parameters)

Definition: Mix of voltage and current.

$$ V_1 = h_{11} I_1 + h_{12} V_2 $$

$$ I_2 = h_{21} I_1 + h_{22} V_2 $$

• Common in transistor circuits. $$\displaystyle h_{11} $$: input impedance (V/I), $$\displaystyle h_{12} $$: reverse voltage gain, $$\displaystyle h_{21} $$: forward current gain, $$\displaystyle h_{22} $$: output admittance.

ABCD Parameters (Transmission Parameters)

Definition: Cascadable parameters.

$$ V_1 = A V_2 + B I_2 $$

$$ I_1 = C V_2 + D I_2 $$

• Cascade Property: For two networks in cascade, $$\displaystyle [ABCD]_{total} = [ABCD]_1 \cdot [ABCD]_2 $$.

• Relation to Z: $$\displaystyle A = \frac{Z_{11}}{Z_{21}} $$, $$\displaystyle B = \frac{Z_{11}Z_{22} - Z_{12}Z_{21}}{Z_{21}} $$, etc.

Conversions between Parameters (Key Relations)

• $$\displaystyle \Delta_h = h_{11}h_{22} - h_{12}h_{21} $$, $$\displaystyle \Delta = AD - BC $$.

Terminated Two-Port Network

• Concept: A two-port network with a load $$\displaystyle Z_L $$ connected at output port 2.

• Analysis: Use two-port equations (e.g., Z-parameters) with $$\displaystyle V_2 = -Z_L I_2 $$ (assuming port 2 reference direction into network).

• Gain Calculations:

  • Voltage Gain: $$\displaystyle A_v = \frac{V_2}{V_1} = \frac{-Z_L}{Z_{22} + Z_L} $$ (for Z-params).

  • Current Gain: $$\displaystyle A_i = \frac{I_2}{I_1} $$.

  • Power Gain: $$\displaystyle A_p = \frac{P_L}{P_{in}} $$.

VIII. Additional Topics

Controlled Sources

• Types:

  1. VCVS: Voltage-Controlled Voltage Source ($$\displaystyle v_o = \mu v_c $$)

  2. VCCS: Voltage-Controlled Current Source ($$\displaystyle i_o = g v_c $$)

  3. CCVS: Current-Controlled Voltage Source ($$\displaystyle v_o = r i_c $$)

  4. CCCS: Current-Controlled Current Source ($$\displaystyle i_o = \beta i_c $$)

• Symbol: Diamond shape for controlled source. Controlling variable ($$\displaystyle v_c $$ or $$\displaystyle i_c $$) shown separately.

Mutual Inductance and Coefficient of Coupling

• Mutual Inductance ($M$): Voltage induced in one coil due to current change in another: $$\displaystyle v_1 = M \frac{di_2}{dt} $$.

• Coefficient of Coupling ($k$): Measures magnetic coupling between two coils.

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

• Calculation from Inductance Measurements:

  • Series aiding: $$\displaystyle L_{eq} = L_1 + L_2 + 2M $$

  • Series opposing: $$\displaystyle L_{eq} = L_1 + L_2 - 2M $$

  • Solve for $M$, then $k$.

Dual Networks

• Concept: Two networks are duals if their equations are identical when dual quantities are exchanged.

• Duality Principle: Replace:

  • Voltage $$\displaystyle \leftrightarrow $$ Current

  • Resistance $$\displaystyle R \leftrightarrow $$ Conductance $G$

  • Inductance $$\displaystyle L \leftrightarrow $$ Capacitance $C$

  • Series $$\displaystyle \leftrightarrow $$ Parallel

  • Open circuit $$\displaystyle \leftrightarrow $$ Short circuit

  • Node $$\displaystyle \leftrightarrow $$ Mesh (Loop)

• Example: Series RLC circuit $$\displaystyle \leftrightarrow $$ Parallel GLC circuit.

s-Domain Superposition Theorem

• Clarification: In s-domain, Superposition Theorem still applies to linear circuits with independent sources (voltage/current sources in s-domain). Deactivate all but one independent source at a time (set to zero). Dependent sources remain active.

• Application: Simplifies solving circuits with multiple independent sources in transient analysis.

Exam Tips from Past Papers:

• KCL/KVL (Dec 24, Jun 23): Always draw a clear circuit, label currents/voltages with assumed directions/polarities. Write equations systematically.

• Graph Theory (Dec 24, Jun 23): Practice drawing trees, identifying twigs/links, and constructing $A$, $B$, $Q$ matrices for small graphs (3-4 nodes).

• Thevenin/Norton (Dec 24, Jun 24, Dec 23): Very frequent. Master finding $$\displaystyle V_{th}/I_N $$ and $$\displaystyle Z_{th} $$. For AC, use phasors.

• Superposition (Dec 24, Jun 23): Kill sources correctly (voltage source → short, current source → open). Remember: Do not kill dependent sources.

• Max Power (Jun 24, Dec 23): State condition $$\displaystyle R_L = R_{th} $$ clearly. Derive/state efficiency = 50%.

• Laplace (Jun 24, Dec 23, Nov 22): Be flawless in s-domain modeling (initial conditions as sources). Practice IVT/FVT with pole analysis.

• Fourier Series (Dec 24, Jun 23, Nov 22): Know both forms. For given waveform, write expression first, then compute $$\displaystyle a_0, a_n, b_n $$ or $$\displaystyle c_n $$.

• Two-Port (Dec 24, Jun 24, Dec 23): Be able to calculate Z/Y/h/ABCD from a given circuit. Conversions are crucial. For terminated networks, use correct two-port equation with load condition.

• Short Notes (Dec 24, Jun 24): Topics like Cut Set, Compensation, Incidence Matrix, Dual Networks, Terminated Two-Port are often asked for 7m. Prepare 3-4 bullet points with a key formula or example.