Network Analysis (EX-304) - Unit 3 Short Notes

UNIT 3: Network Analysis - High-Impact Short Notes

1. Circuit Laws and Basic Elements

Kirchhoff's Laws

• 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 $$\displaystyle i_1 $$ and $$\displaystyle i_2 $$ enter, $$\displaystyle i_3 $$ leaves: $$\displaystyle i_1 + i_2 - i_3 = 0 $$.

• Kirchhoff's Voltage Law (KVL): The algebraic sum of voltages around any closed loop is zero.

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

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

[!TIP]

Common Pitfall: KCL applies to instantaneous currents; KVL assumes no time-varying magnetic flux linking the loop (valid for lumped circuits).

Basic Circuit Elements

Dependent (Controlled) Sources

Coupled Inductors

• Mutual Inductance (M): Voltage induced in one coil due to current change in the other.

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

• Coefficient of Coupling (k):

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

• Dot Convention: Dots indicate polarity of induced voltage. Aiding (dots same side): $+M$; Opposing (dots opposite): $-M$.

[!TIP]

Exam Focus: Calculate $M$ from given $$\displaystyle L_{series(aiding)} $$ and $$\displaystyle L_{series(opposing)} $$:

$$L_{aiding} = L_1 + L_2 + 2M, \quad L_{opposing} = L_1 + L_2 - 2M$$

2. Circuit Analysis Methods

Mesh Analysis (Planar Circuits)

1. Identify independent meshes (non-overlapping loops).

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

3. Solve simultaneous equations.

• Supermesh: When a current source lies between two meshes, create a supermesh excluding the source and add the constraint equation (current source value equals difference of mesh currents).

Nodal Analysis

1. Choose reference node (ground).

2. Apply KCL at each non-reference node.

3. Express currents in terms of node voltages.

4. Solve equations.

• Supernode: When a voltage source connects two non-reference nodes, enclose them in a supernode. Apply KCL to the supernode and use the voltage source equation as constraint.

[!TIP]

Choosing Method: Use mesh for voltage-source-rich circuits; nodal for current-source-rich circuits. Supermesh/supernode handle dependent sources efficiently.

3. Circuit Theorems

Thevenin's Theorem

• Statement: Any linear two-terminal 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 across terminals.

• $$\displaystyle Z_{th} $$: Impedance seen with all independent sources zero (voltage sources shorted, current sources opened).

• Application: Simplifies analysis for varying load.

Norton's Theorem

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

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

• $$\displaystyle Z_N = Z_{th} $$.

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

Superposition Theorem

• Statement: In a linear circuit with multiple sources, response (voltage/current) is algebraic sum of responses due to each source acting alone, with all other independent sources zeroed (voltage sources → short, current sources → open).

• Limitation: Not applicable for power calculations (non-linear).

Maximum Power Transfer Theorem

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

• Maximum Power:

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

• Efficiency Proof: At max power, power delivered to load = power from source? No.

$$\eta = \frac{P_L}{P_{total}} = \frac{V_{th}^2/(4R_{th})}{V_{th}^2/(2R_{th})} = \boxed{50\%}$$

Millman's Theorem

• For parallel voltage sources $$\displaystyle V_1, V_2, \dots, V_n $$ with series resistances $$\displaystyle R_1, R_2, \dots, R_n $$:

$$V_{eq} = \frac{\sum_{k=1}^{n} \frac{V_k}{R_k}}{\sum_{k=1}^{n} \frac{1}{R_k}}, \quad R_{eq} = \left( \sum_{k=1}^{n} \frac{1}{R_k} \right)^{-1}$$

• Use: Reduces parallel voltage sources to a single equivalent.

Tellegen's Theorem

• Statement: For any network (linear/non-linear, passive/active), sum of power absorbed by all elements is zero.

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

where $$\displaystyle v_k, i_k $$ are voltage and current through branch $k$ with passive sign convention.

• Application: Verifies power balance in a circuit.

Compensation Theorem

• Statement: If impedance $Z$ in a branch carries current $I$, replacing $Z$ by a voltage source $-IZ$ (or current source $-I/Z$) produces no change elsewhere in the circuit.

• Use: Sensitivity analysis, effect of branch parameter change.

Substitution Theorem

• Statement: If voltage across a branch equals $V$ and current through it equals $I$, the branch can be replaced by any combination of elements that maintains $$\displaystyle v = V $$ and $$\displaystyle i = I $$ (e.g., voltage source $V$, current source $I$, or impedance $V/I$).

• Caution: Replacement must not alter the rest of the circuit's topology.

4. Graph Theory and Network Topology

Basic Definitions

• Graph: Set of nodes (vertices) connected by branches (edges).

• Oriented Graph: Graph with assigned direction to each branch.

• Planar Graph: Can be drawn on a plane without crossing branches. Non-planar cannot.

• Tree: Connected subgraph containing all nodes but no loops. Number of twigs = $n-1$ ($n$ = nodes).

• Co-tree: Branches not in the tree; called links. Number of links = $b - (n-1)$ ($b$ = branches).

Incidence Matrix (A)

• Complete Incidence Matrix [A] ($n \times b$):

  • Row: node, Column: branch.

  • $$\displaystyle a_{ij} = 1 $$ if branch $j$ leaves node $i$, $-1$ if enters, $0$ otherwise.

  • Row sum = 0 (KCL).

• Reduced Incidence Matrix [A_r]: Remove reference node row.

Tie Set Matrix (B)

• Fundamental Tie Set: One link + unique tree path forms a loop.

• Tie Set Matrix [B] ($l \times b$): $$\displaystyle l = b - n + 1 $$ (number of links).

  • Rows correspond to fundamental loops.

  • Columns: branches. Entries: $+1$ if branch direction aligns with loop, $-1$ opposite, $0$ if not in loop.

  • Property: $$\displaystyle [A][B]^T = 0 $$ (orthogonal).

Cut Set Matrix (Q)

• Fundamental Cut Set: One twig + links that, when removed, disconnect graph into two parts.

• Cut Set Matrix [Q] ($t \times b$): $$\displaystyle t = n-1 $$ (number of twigs).

  • Rows: fundamental cut sets.

  • Entries: $+1$ if branch leaves cut set, $-1$ if enters, $0$ otherwise.

  • Property: $$\displaystyle [Q][B] = 0 $$.

[!TIP]

Exam Pattern: Often given a graph, draw tree, list twigs/links, then write complete incidence matrix or cut set matrix. Remember: Cut set separates graph into two parts; tie set forms a loop.

5. Resonance and Coupled Circuits

Series Resonance

• RLC Series Circuit:

$$Z = R + j\left(\omega L - \frac{1}{\omega C}\right)$$

• Resonant Frequency ($$\displaystyle \omega_0 $$):

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

• At resonance: $$\displaystyle X_L = X_C $$, $$\displaystyle Z = R $$ (minimum), current $$\displaystyle I = V/R $$ (maximum).

• Quality Factor (Q):

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

• Voltage Magnification:

$$V_L = V_C = Q \cdot V_{source}$$

Parallel Resonance

• RLC Parallel Circuit (ideal, $$\displaystyle G=1/R $$):

$$Y = G + j\left(\omega C - \frac{1}{\omega L}\right)$$

• Resonant Frequency:

$$\omega_0 = \frac{1}{\sqrt{LC}} \quad (\text{same as series for high } Q)$$

• At resonance: $$\displaystyle B=0 $$, $$\displaystyle Y = G $$ (minimum), impedance $$\displaystyle Z = 1/G $$ (maximum), current $$\displaystyle I = V \cdot G $$ (minimum from source).

• Q Factor:

$$Q = R \sqrt{\frac{C}{L}} = \frac{R}{\omega_0 L} \quad (\text{for parallel } R \text{ with } L \text{ and } C)$$

Coupled Inductors in Series/Parallel

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

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

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

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

[!TIP]

Derivation Tip: For parallel resonance, derive from admittance $$\displaystyle Y = j\omega C + \frac{1}{j\omega L + R_p} $$? Actually for high Q, approximate $$\displaystyle \omega_0 \approx 1/\sqrt{LC} $$. Always check if resistance is in series or parallel with L.

6. Laplace Transform Analysis

Laplace Transforms of Standard Waveforms

Inverse Laplace Transform

• Use partial fraction expansion and transform tables.

• For complex poles: complete the square, use $$\displaystyle e^{-\sigma t}\sin(\omega_d t) $$ form.

Solving Circuits with Initial Conditions

1. Replace $L$ with $$\displaystyle sL - L i(0^-) $$, $C$ with $$\displaystyle \frac{1}{sC} + \frac{v(0^-)}{s} $$.

2. Write circuit equations in $s$-domain.

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

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

Initial Value Theorem (IVT)

• Statement: If $F(s)$ is proper rational function and $sF(s)$ has no poles in Re$s \ge 0$, then:

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

• Condition: $f(t)$ has no impulse at $$\displaystyle t=0^+ $$.

Final Value Theorem (FVT)

• Statement: If $sF(s)$ has no poles in Re$s \ge 0$ (except possibly at $$\displaystyle s=0 $$), then:

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

• Condition: System stable (all poles in Re$$\displaystyle s < 0 $$).

[!TIP]

Verify Conditions First: For FVT, check poles of $sF(s)$. If pole at $$\displaystyle s=0 $$ (integrator), FVT fails. Example: $$\displaystyle F(s)=1/s $$ → $$\displaystyle f(t)=1 $$, but $$\displaystyle \lim_{s\to0} s \cdot (1/s) = 1 $$ works? Actually $$\displaystyle sF(s)=1 $$, no pole at $$\displaystyle s=0 $$? Wait: $$\displaystyle F(s)=1/s $$ → $$\displaystyle sF(s)=1 $$, pole canceled? FVT gives $$\displaystyle f(\infty)=1 $$, correct. But for $$\displaystyle F(s)=1/s^2 $$, $$\displaystyle sF(s)=1/s $$ has pole at $$\displaystyle s=0 $$ → FVT invalid (ramp grows to $\infty$).

Network 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 phasor/Laplace variable (e.g., $$\displaystyle H(s) = V_2(s)/V_1(s) $$).

• Pole-Zero Plot: Poles (denominator roots) → stability; zeros (numerator roots) → frequency response.

Switched Circuits

• At $$\displaystyle t=0 $$, switch action: use initial conditions $$\displaystyle i_L(0^-) $$, $$\displaystyle v_C(0^-) $$.

• For $$\displaystyle t>0 $$, analyze new circuit configuration in $s$-domain.

• For $$\displaystyle t<0 $$, assume steady state (DC: $L$ short, $C$ open).

7. 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

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

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

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

Exponential Fourier Series (EFS)

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

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

Relation: $$\displaystyle C_0 = a_0 $$, $$\displaystyle C_n = \frac{1}{2}(a_n - jb_n) $$, $$\displaystyle C_{-n} = \frac{1}{2}(a_n + jb_n) $$.

Symmetry Properties

Standard Waveforms

• Square Wave (odd, amplitude $A$, duty 50%):

$$f(t) = \frac{4A}{\pi} \sum_{n=1,3,5,\dots}^{\infty} \frac{1}{n} \sin n\omega_0 t$$

• Triangular Wave (odd, even harmonics decay as $$\displaystyle 1/n^2 $$):

$$f(t) = \frac{8A}{\pi^2} \sum_{n=1,3,5,\dots}^{\infty} \frac{(-1)^{(n-1)/2}}{n^2} \sin n\omega_0 t$$

• Sawtooth Wave (odd, all harmonics decay as $1/n$):

$$f(t) = \frac{2A}{\pi} \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} \sin n\omega_0 t$$

Parseval's Theorem (Power)

Average power over period $T$:

$$P = \frac{1}{T}\int_{0}^{T} [f(t)]^2 dt = a_0^2 + \frac{1}{2}\sum_{n=1}^{\infty} (a_n^2 + b_n^2) = \sum_{n=-\infty}^{\infty} |C_n|^2$$

8. Two-Port Network Parameters

Z-Parameters (Impedance)

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

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

• $$\displaystyle Z_{12} = \left.\frac{V_1}{I_2}\right|_{I_1=0} $$ (reverse transfer impedance)

• $$\displaystyle Z_{21} = \left.\frac{V_2}{I_1}\right|_{I_2=0} $$ (forward transfer impedance)

• $$\displaystyle Z_{22} = \left.\frac{V_2}{I_2}\right|_{I_1=0} $$ (open-circuit output impedance)

• Reciprocal Condition: $$\displaystyle Z_{12} = Z_{21} $$ if network is reciprocal.

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 input admittance)

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

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

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

• Reciprocal: $$\displaystyle Y_{12} = Y_{21} $$.

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

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 transfer impedance)

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

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

• Cascade Property: For two cascaded networks, overall $$\displaystyle [ABCD] = [ABCD]_1 \cdot [ABCD]_2 $$.

• Reciprocal: $$\displaystyle AD - BC = 1 $$.

Interconversion of Parameters

where $$\displaystyle \Delta = AD - BC $$, $$\displaystyle \Delta Z = Z_{11}Z_{22} - Z_{12}Z_{21} $$, $$\displaystyle \Delta Y = Y_{11}Y_{22} - Y_{12}Y_{21} $$, $$\displaystyle \Delta h = h_{11}h_{22} - h_{12}h_{21} $$.

Terminated Two-Port Network

• Input Impedance ($$\displaystyle Z_{in} $$) with load $$\displaystyle Z_L $$:

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

• Voltage Gain ($$\displaystyle G_v = V_2/V_1 $$):

$$G_v = \frac{Z_{21}}{Z_{11} + Z_s} \cdot \frac{Z_L}{Z_{22} + Z_L} \quad (\text{with source impedance } Z_s)$$

• Current Gain ($$\displaystyle G_i = I_2/I_1 $$): Similar expressions using Y-params.

9. Additional and Special Topics (Short Notes)

Dual Networks and Duality Principle

• Dual: Replace every element with its dual: resistance ↔ conductance ($$\displaystyle R \leftrightarrow G $$), series ↔ parallel, voltage source ↔ current source, node ↔ mesh.

• Duality Theorem: Equations of dual networks are identical if dual variables are substituted.

• Example: Series RLC circuit ↔ Parallel GLC circuit.

Network Functions and Properties

• Driving Point Function: Impedance/admittance at a port.

• Transfer Function: Ratio of output to input.

• Properties:

  • Rational: Ratio of polynomials in $s$.

  • Proper: Degree numerator ≤ degree denominator.

  • Realizable: Positive real function (PRF) for impedance: Re$[Z(s)] \ge 0$ for Re$s \ge 0$.

Relationship between h and ABCD

From conversion table:

$$A = \frac{h_{11}}{h_{21}}, \quad B = \frac{\Delta h}{h_{21}}, \quad C = \frac{1}{h_{21}}, \quad D = \frac{h_{22}}{h_{21}}$$

where $$\displaystyle \Delta h = h_{11}h_{22} - h_{12}h_{21} $$.

Controlled Sources (Dependent Sources)

• VCVS: Voltage-controlled voltage source (e.g., op-amp).

• VCCS: Voltage-controlled current source (e.g., FET).

• CCVS: Current-controlled voltage source (e.g., gyrator).

• CCCS: Current-controlled current source (e.g., BJT in current mirror).

• Key: Control variable is elsewhere in circuit; source is active (can deliver power).

Millman's Theorem

• Use Case: Multiple parallel voltage sources with series resistances.

• Formula:

$$V_{eq} = \frac{\sum_{k=1}^{n} \frac{V_k}{R_k}}{\sum_{k=1}^{n} \frac{1}{R_k}}, \quad R_{eq} = \left( \sum_{k=1}^{n} \frac{1}{R_k} \right)^{-1}$$

• Tip: Also applies to parallel current sources with parallel conductances (Norton form).

Y-Parameters (Short Note)

• Also called admittance parameters.

• Defined with all ports voltage-controlled: $$\displaystyle I_1 = Y_{11}V_1 + Y_{12}V_2 $$, $$\displaystyle I_2 = Y_{21}V_1 + Y_{22}V_2 $$.

• Measurement: Apply $$\displaystyle V_1 $$ with $$\displaystyle V_2=0 $$ (port 2 shorted) to get $$\displaystyle Y_{11}=I_1/V_1 $$, $$\displaystyle Y_{21}=I_2/V_1 $$; then apply $$\displaystyle V_2 $$ with $$\displaystyle V_1=0 $$ to get $$\displaystyle Y_{22} $$, $$\displaystyle Y_{12} $$.

• Reciprocal: $$\displaystyle Y_{12}=Y_{21} $$ for reciprocal networks.

Cut Set Matrix (Short Note)

• Fundamental Cut Set: Partition graph into two parts by removing one twig and necessary links.

• Matrix [Q]: Rows = cut sets, columns = branches.

  • Entry $$\displaystyle q_{ij} = +1 $$ if branch $j$ leaves cut set $i$, $-1$ if enters, $0$ if not in cut set.

• Property: $$\displaystyle [Q][B] = 0 $$ (orthogonal to tie set matrix).

• Use: Write KCL equations compactly: $$\displaystyle [Q][i] = 0 $$ (branch currents).

Compensation Theorem (Short Note)

• If impedance $Z$ in a branch has current $I$, replace $Z$ by a voltage source $-IZ$ (in series) or current source $-I/Z$ (in parallel) → no change in rest of circuit.

• Application: Study effect of small change in $Z$ (e.g., $\Delta Z$) by compensating with source $-I \Delta Z$.

Terminated Two-Port Network (Short Note)

• Two-port with load $$\displaystyle Z_L $$ at output and source $$\displaystyle Z_s $$ at input.

• Input Impedance seen by source: $$\displaystyle Z_{in} = Z_{11} - \frac{Z_{12}Z_{21}}{Z_{22}+Z_L} $$ (Z-params).

• Voltage Gain: $$\displaystyle G_v = \frac{V_2}{V_s} = \frac{Z_{21}}{(Z_{11}+Z_s)(1 + \frac{Z_{22}}{Z_L}) + Z_{12}Z_{21}} $$? Simpler: $$\displaystyle G_v = \frac{V_2}{V_1} \cdot \frac{V_1}{V_s} $$.

• Key: Use parameter equations with termination conditions ($$\displaystyle I_2 = -V_2/Z_L $$, $$\displaystyle V_1 = V_s - I_1 Z_s $$).

Network Topology (Short Note)

• Study of circuit arrangement without regard to element values.

• Graph: Skeleton showing connectivity.

• Tree: Loop-free connected subgraph covering all nodes.

• Fundamental Loops (Tie Sets): Independent loops formed by adding one link to tree.

• Fundamental Cut Sets: Independent cuts formed by removing one twig.

• Significance: Enables systematic equation formulation (KVL via tie sets, KCL via cut sets) independent of element values.

Tellegen's Theorem (Short Note)

• Statement: $$\displaystyle \sum_{k=1}^{b} v_k i_k = 0 $$ for any network, where $$\displaystyle v_k, i_k $$ satisfy KVL and KCL with passive sign convention.

• Proof: Based on orthogonality of incidence matrix and loop matrix.

• Applications:

  1. Power Balance: Total power absorbed = total power delivered.

  2. Reciprocity Check: For two different excitations, $$\displaystyle \sum v_k i_k' = \sum v_k' i_k $$.

  3. Network Equivalence: If two networks have same $$\displaystyle v_k, i_k $$ for all branches, they are equivalent.

10. Key Formulas at a Glance

[!TIP]

Exam Strategy: For two-port conversions, derive from definitions rather than memorizing all formulas. For example, to get $$\displaystyle Y_{11} $$ from $Z$: $$\displaystyle Y_{11} = Z_{22}/\Delta Z $$. For Laplace switched circuits, always draw $$\displaystyle t<0 $$ steady state first (DC: $L$ short, $C$ open) to find initial conditions.

11. Common Past Paper Patterns

1. Graph Theory: Given a circuit → draw graph → choose a tree → list twigs/links → write complete incidence matrix or cut set matrix.

2. Theorems: Find voltage/current using superposition, Thevenin/Norton, max power (find $$\displaystyle R_L $$ and $$\displaystyle P_{max} $$).

3. Laplace: Solve switched RL/RC/RLC circuit → find $i(t)$ or $v(t)$; compute IVT/FVT; verify FVT conditions.

4. Fourier: Find trigonometric/exponential series for square/triangular/sawtooth wave; use symmetry to simplify.

5. Two-Port: Given circuit → find Z/Y/h/ABCD; convert between parameters; find $$\displaystyle Z_{in} $$ for terminated network.

6. Resonance: Derive $$\displaystyle \omega_0 $$ for series/parallel RLC; calculate $Q$, voltage magnification.

Final Note: Always define variables and state assumptions (e.g., "assuming ideal op-amp", "for reciprocal network"). For proofs (efficiency, cascade), show key steps concisely. Use boxed for final results.