Network Analysis (EX-304) - Unit 2 Short Notes

I. Fundamental Circuit Laws and Analysis Techniques

Kirchhoff's Current Law (KCL)

The algebraic sum of currents entering a node is zero.

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

Example: At node A, if $$\displaystyle I_1 $$ enters and $$\displaystyle I_2, I_3 $$ leave, then $$\displaystyle I_1 = I_2 + I_3 $$.

Kirchhoff's Voltage Law (KVL)

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

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

Example: In a loop with sources $$\displaystyle V_s $$ and drops $$\displaystyle V_R, V_L $$, $$\displaystyle V_s - V_R - V_L = 0 $$.

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Common Pitfall: KCL applies to nodes (junctions), KVL to loops (closed paths. Always assign consistent sign conventions (currents entering +, leaving -; voltage rises +, drops -).

Mesh Analysis (Planar Circuits)

1. Identify meshes (independent loops).

2. Assign mesh currents (clockwise assumed).

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

4. Solve simultaneous equations. Key: For shared branches, voltage = impedance × (difference of mesh currents).

Nodal Analysis

1. Choose a reference node (ground).

2. Assign node voltages to remaining nodes.

3. Apply KCL at each non-reference node, expressing currents via Ohm's law.

4. Solve for node voltages. Key: Conductance form simplifies: $$\displaystyle G_{ii} V_i + \sum_{j \neq i} G_{ij} V_j = I_i $$.

II. Network Topology (Graph Theory)

Graph & Oriented Graph

• Graph: Set of nodes (vertices) connected by branches (edges), representing circuit topology without regard to component values.

• Oriented Graph: Graph with assigned directions to all branches (used for analysis).

Tree, Co-tree, Twigs, Links

• Tree: Connected subgraph with all nodes and no loops. Contains $b - n + 1$ branches ($b$ = total branches, $n$ = nodes).

• Twigs: Branches belonging to the tree.

• Co-tree: Branches not in the tree.

• Links: Branches in the co-tree. Each link forms a fundamental loop when added to the tree.

Incidence Matrix (Complete)

• Order: $n \times b$ ($n$ nodes, $b$ branches).

• Element $$\displaystyle a_{ij} $$:

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

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

  • $0$ otherwise.

• Property: Each column has exactly one $+1$ and one $-1$ (for connected graph). Row sum = 0.

Tie Set Matrix (Fundamental)

• Relates branch currents to loop currents.

• For each fundamental loop (one link + tree branches), a row.

• Element $$\displaystyle b_{ij} $$: $+1$ if branch $j$ is in loop $i$ and same direction as loop current, $-1$ if opposite, $0$ if not in loop.

• Equation: $$\displaystyle [I_b] = [B]^T [I_L] $$, where $[B]$ is tie-set matrix.

Cut Set Matrix (Fundamental)

• Relates branch currents to cut-set currents.

• A cut set is a set of branches whose removal disconnects the graph into two parts, with exactly one branch from the tree.

• Equation: $$\displaystyle [I_b] = [Q]^T [I_S] $$, where $[Q]$ is cut-set matrix.

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Exam Focus: Be able to draw a graph from a circuit, identify a tree, and write the complete incidence matrix. Tie-set matrix rows correspond to fundamental loops (one link each). Cut-set matrix rows correspond to fundamental cuts (one twig each).

III. Network Theorems

Thevenin's Theorem

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

• $$\displaystyle V_{th} $$ = Open-circuit voltage at terminals.

• $$\displaystyle Z_{th} $$ = Impedance seen at terminals with all independent sources killed (voltage sources shorted, current sources opened).

• AC Extension: Use phasors; $$\displaystyle Z_{th} $$ is complex.

Norton's Theorem

Equivalent to a current source $$\displaystyle I_N $$ in parallel with admittance $$\displaystyle Y_N $$ (or impedance $$\displaystyle Z_N $$).

• $$\displaystyle I_N $$ = Short-circuit current at terminals.

• $$\displaystyle Y_N = 1/Z_{th} $$ (same as Thevenin impedance).

• AC Case: $$\displaystyle I_N $$ is phasor short-circuit current.

Superposition Theorem

In a linear circuit with multiple sources, the response (voltage/current) is the algebraic sum of responses due to each source acting alone, with all other independent sources killed.

• Important: Power is not linear; cannot use superposition for power calculations directly.

Maximum Power Transfer Theorem (DC)

Maximum power is delivered to load $$\displaystyle R_L $$ when $$\displaystyle R_L = R_{th} $$ (Thevenin resistance).

• Condition: $$\displaystyle R_L = |Z_{th}| $$ for AC (impedance matching).

• Efficiency: $$\displaystyle \eta = \frac{P_{max}}{P_{source}} = \frac{1}{2} = 50\% $$ when $$\displaystyle R_L = R_{th} $$.

  Proof: $$\displaystyle P_L = \frac{V_{th}^2 R_L}{(R_{th}+R_L)^2} $$. Maximize w.r.t $$\displaystyle R_L $$ → $$\displaystyle R_L = R_{th} $$. Then $$\displaystyle P_{max} = V_{th}^2/(4R_{th}) $$, $$\displaystyle P_{source} = V_{th}^2/(2R_{th}) $$, so $$\displaystyle \eta = 1/2 $$.

Millman's Theorem

For multiple parallel branches between two nodes, the common voltage $V$ is:

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

where $$\displaystyle V_k $$ is the voltage source in branch $k$ (if present), and $$\displaystyle Z_k $$ is the branch impedance (if no source, $$\displaystyle V_k=0 $$).

Tellegen's Theorem

For any network (linear/nonlinear, passive/active), the sum of instantaneous power in all branches is zero:

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

• Verification: Compute power in each branch (absorbed positive if current enters positive terminal). Sum must be zero.

Compensation Theorem

If a branch impedance $Z$ in a network is changed by $\Delta Z$, the resulting current/voltage changes are equivalent to injecting a compensating voltage source $-I \Delta Z$ (where $I$ is original branch current) in series with the changed branch, with all other sources killed.

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Common Pitfalls:

• Superposition: Kill independent sources (set voltage=0 → short; set current=0 → open). Dependent sources remain active.

• Max Power: Efficiency is 50% only at max power condition.

• Millman's: Applicable only for parallel branches between two nodes.

IV. Resonance

Series Resonant Circuit ($R$, $L$, $C$ in series)

• Impedance: $$\displaystyle Z = R + j(\omega L - 1/(\omega C)) $$

• Resonant Frequency ($$\displaystyle \omega_0 $$): Imaginary part zero.

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

• At Resonance:

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

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

  • Voltage Magnification: $Q$-factor.

$$Q = \frac{\omega_0 L}{R} = \frac{1}{\omega_0 C R} = \frac{V_L}{V} = \frac{V_C}{V}$$

• Bandwidth (BW): $$\displaystyle \text{BW} = \frac{\omega_0}{Q} $$ (rad/s) or $$\displaystyle f_0/Q $$ (Hz).

Parallel Resonant Circuit ($R$, $L$, $C$ in parallel)

• Admittance: $$\displaystyle Y = \frac{1}{R} + j\left(\omega C - \frac{1}{\omega L}\right) $$

• Resonant Frequency:

$$\omega_0 = \sqrt{\frac{1}{LC} - \frac{R^2}{L^2}} \approx \frac{1}{\sqrt{LC}} \quad \text{if } R \text{ is large (high } Q)$$

• At Resonance:

  • $$\displaystyle Y = 1/R $$ (minimum admittance).

  • Impedance $$\displaystyle Z = R $$ (maximum, purely resistive).

  • Current Magnification: Input current $I$ minimum, branch currents $$\displaystyle I_L = I_C = Q \cdot I $$ (large).

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Key Difference: Series: current max, voltage across L/C magnified. Parallel: voltage max (for given input current), currents through L/C magnified. For parallel, exact $$\displaystyle \omega_0 $$ depends on $R$; approximate formula valid for high $Q$.

V. Laplace Transform Analysis

Laplace Transform (LT)

For $f(t)$ defined for $t \geq 0$:

$$F(s) = \mathcal{L}\{f(t)\} = \int_{0^{-}}^{\infty} f(t) e^{-st} dt$$

where $$\displaystyle s = \sigma + j\omega $$.

Standard Transforms

• $$\displaystyle \mathcal{L}\{1\} = \frac{1}{s} $$

• $$\displaystyle \mathcal{L}\{u(t)\} = \frac{1}{s} $$ (unit step)

• $$\displaystyle \mathcal{L}\{\delta(t)\} = 1 $$ (unit impulse)

• $$\displaystyle \mathcal{L}\{e^{-at}\} = \frac{1}{s+a} $$

• $$\displaystyle \mathcal{L}\{t^n\} = \frac{n!}{s^{n+1}} $$

• $$\displaystyle \mathcal{L}\{e^{-at} \sin \omega t\} = \frac{\omega}{(s+a)^2 + \omega^2} $$

Properties (Key)

• Linearity: $$\displaystyle \mathcal{L}\{a f_1 + b f_2\} = a F_1 + b F_2 $$

• Differentiation: $$\displaystyle \mathcal{L}\{f'(t)\} = sF(s) - f(0^{-}) $$

• Integration: $$\displaystyle \mathcal{L}\{\int_{0}^{t} f(\tau) d\tau\} = \frac{F(s)}{s} $$

Initial Value Theorem (IVT)

If $sF(s)$ has no poles in $$\displaystyle \text{Re}(s) > 0 $$, then:

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

Conditions: $f(t)$ and $f'(t)$ must be Laplace transformable; $F(s)$ proper rational function.

Final Value Theorem (FVT)

If $sF(s)$ has all poles in $$\displaystyle \text{Re}(s) < 0 $$ (except possibly at $$\displaystyle s=0 $$), then:

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

Conditions: System stable (poles in $$\displaystyle \text{Re}(s)<0 $$); $f(\infty)$ exists.

Application to Circuit Analysis

1. Replace circuit elements with s-domain equivalents:

   • $R \to R$

   • $L \to sL$ (with initial current $$\displaystyle I_0 $$: add source $$\displaystyle LI_0 $$)

   • $C \to 1/(sC)$ (with initial voltage $$\displaystyle V_0 $$: add source $$\displaystyle V_0/s $$)

2. Apply KVL/KCL or nodal/mesh in s-domain.

3. Solve for desired variable $F(s)$.

4. Find $$\displaystyle f(t) = \mathcal{L}^{-1}\{F(s)\} $$ (partial fractions).

Transfer Function & Pole-Zero Plot

• Transfer Function: $$\displaystyle H(s) = \frac{\text{Output}}{\text{Input}} $$ (zero initial conditions).

• Poles: Roots of denominator (where $H(s) \to \infty$). Determine natural response & stability.

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

• Plot: s-plane; poles (×), zeros (○). Location dictates time-domain behavior (e.g., real negative poles → exponential decay).

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Critical Checks:

• IVT/FVT: Always verify conditions before applying.

• Initial conditions in s-domain: For inductor current $$\displaystyle i_L(0^-)=I_0 $$, model as $$\displaystyle sL I(s) - L I_0 $$. For capacitor voltage $$\displaystyle v_C(0^-)=V_0 $$, model as $$\displaystyle \frac{1}{sC} I(s) + \frac{V_0}{s} $$.

• Partial fraction inversion: Use standard forms or convolution.

VI. Fourier Series

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:

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

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

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

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

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

Exponential Fourier Series (EFS)

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

with coefficients:

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

• Relation to TFS:

$$C_0 = a_0$$

$$C_n = \frac{1}{2}(a_n - jb_n) \quad (n>0)$$

$$C_{-n} = \frac{1}{2}(a_n + jb_n) = C_n^*$$

• Power: $$\displaystyle \frac{1}{T} \int_{0}^{T} |f(t)|^2 dt = \sum_{n=-\infty}^{\infty} |C_n|^2 $$ (Parseval's theorem).

Common Waveforms

• Square wave (odd, 50% duty): $$\displaystyle b_n = \frac{4A}{n\pi} $$ for odd $n$, $$\displaystyle a_n=0 $$.

• Triangular wave (odd, even harmonics only): $$\displaystyle b_n = \frac{8A}{n^2\pi^2} $$ for odd $n$.

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Computation Strategy:

1. Determine period $T$, fundamental $$\displaystyle \omega_0 $$.

2. Check symmetry (even/odd/half-wave) to simplify integrals.

3. For EFS, integrate over one period with complex exponential.

4. For TFS of even/odd functions, use half-range expansions.

VII. Two-Port Networks

Parameters (All defined with input port 1, output port 2)

Interconversion (Key Relations)

• Y from Z: $$\displaystyle [Y] = [Z]^{-1} $$

• ABCD from Z:

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

• h from Z:

$$h_{11} = \frac{Z_{11}}{Z_{21}}, \quad h_{12} = \frac{Z_{12}Z_{21} - Z_{11}Z_{22}}{Z_{21}}, \quad h_{21} = \frac{1}{Z_{21}}, \quad h_{22} = \frac{Z_{22}}{Z_{21}}$$

• General Approach: Write defining equations, solve for desired variables.

Cascade Connection

For two two-ports in cascade (output of 1 to input of 2), overall transmission matrix is product:

$$[T] = [T_1][T_2] = \begin{bmatrix} A_1 & B_1 \\ C_1 & D_1 \end{bmatrix} \begin{bmatrix} A_2 & B_2 \\ C_2 & D_2 \end{bmatrix}$$

Proof: From cascade, $$\displaystyle V_{1} = A_1 V_{2}' + B_1 I_{2}' $$, $$\displaystyle I_{1} = C_1 V_{2}' + D_1 I_{2}' $$, and $$\displaystyle V_{2}' = A_2 V_2 + B_2 I_2 $$, $$\displaystyle I_{2}' = C_2 V_2 + D_2 I_2 $$. Substitute to get $$\displaystyle V_1, I_1 $$ in terms of $$\displaystyle V_2, I_2 $$.

Terminated Two-Port Network

• Input Impedance with load $$\displaystyle Z_L $$ on port 2:

$$Z_{in} = \frac{V_1}{I_1} = Z_{11} - \frac{Z_{12}Z_{21}}{Z_{22} + Z_L} \quad \text{(using Z-parameters)}$$

• Applications: Matching, amplifier design, filter analysis.

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Parameter Selection:

• Z/Y: Easy for series/parallel connections.

• h-parameters: Useful for transistor circuits (input impedance, current gain).

• ABCD: Ideal for cascade (transmission lines, amplifiers).

Always check conditions: For Z-parameters, port 2 open ($$\displaystyle I_2=0 $$); for h-parameters, $$\displaystyle V_2=0 $$ means short circuit at output.

VIII. Additional Topics

Mutual Inductance & Coefficient of Coupling

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

• Coefficient of Coupling $k$:

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

• Series Connection:

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

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

• From measurements: If aiding gives $$\displaystyle L_a $$, opposing gives $$\displaystyle L_o $$, then:

$$L_1 + L_2 = \frac{L_a + L_o}{2}, \quad M = \frac{L_a - L_o}{4}$$

Controlled Sources (Dependent Sources)

Dual Networks

• Duality Principle: Replace every element/quantity with its dual:

  • Series ↔ Parallel

  • Voltage ↔ Current

  • Resistance ↔ Conductance

  • Open circuit ↔ Short circuit

  • KVL ↔ KCL

• Dual Network: Obtained by dual transformation; has same equation set with dual quantities.

Network Topology (Summary)

• Graph: Nodes/branches.

• Tree: Loopless connected subgraph covering all nodes.

• Fundamental Loops (Tie Sets): One link + tree branches.

• Fundamental Cuts: One twig + co-tree branches.

• Incidence Matrix: Node-branch incidence.

• Tie-set Matrix: Loop-branch incidence.

• Cut-set Matrix: Cut-branch incidence.

s-Domain Network Functions

• Driving-point impedance: $$\displaystyle Z(s) = \frac{V(s)}{I(s)} $$ (single port).

• Transfer function: $$\displaystyle H(s) = \frac{\text{output } V(s) \text{ or } I(s)}{\text{input } V(s) \text{ or } I(s)} $$.

• Properties: Rational function; poles determine stability; zeros determine frequency response.

• Realizability: Must be proper ($\deg N \leq \deg D$) for causal, stable networks.

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Controlled Sources: Always retain when killing independent sources for Thevenin/Norton.

Duality: To find dual, draw graph with meshes as nodes, branches dual to original branches.

s-domain: Initial conditions become sources in s-domain; always include them for complete solution.