Network Analysis (EC-305) Unit 3 Notes - RGPV

UNIT 3: Network Analysis - Short Notes

I. Fundamental Circuit Laws and Analysis Methods

Kirchhoff's Current Law (KCL) & Voltage Law (KVL)

KCL: At any node, the algebraic sum of currents leaving the node is zero.

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

> [!TIP] Currents entering are negative, leaving are positive (or vice-versa).

KVL: Around any closed loop, the algebraic sum of voltage drops is zero.

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

Example: For a simple two-loop circuit, apply KCL at a shared node and KVL around each independent loop to form simultaneous equations.

Mesh Analysis (Mesh Current Method)

Applies to planar circuits (no crossing branches).
Assign a mesh current to each independent window (loop that contains no other loop).
Write KVL equations for each mesh. Use supermesh if a current source lies on a common branch.
Solve the linear system for mesh currents. Branch currents are combinations of mesh currents.

Nodal Analysis (Node Voltage Method)

More efficient for circuits with many parallel branches.
Select a reference node (ground). Assign voltages $V_1, V_2, ...$ to other $n-1$ nodes.
Apply KCL at each non-reference node. Express branch currents in terms of node voltages using Ohm's law.
Solve the resulting $(n-1)$ equations.

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.
Procedure:
1. Deactivate all but one independent source: voltage sources → short circuit, current sources → open circuit.
2. Solve for the branch response due to the active source.
3. Repeat for each source.
4. Sum all individual responses algebraically.
⚠️ Caution: Power cannot be found by superposition (non-linear). Dependent sources must remain active.

II. Graph Theory and Network Topology

Basic Definitions

Term	Definition
Graph	Set of branches (edges) connecting nodes (vertices). Represents circuit topology ignoring element values.
Oriented/Directed Graph	Graph where each branch is assigned a direction (arrow). Used for analysis.
Tree	A connected subgraph containing all nodes of the graph but no loops. Has $b = n - 1$ branches (twigs).
Co-tree	Branches not in the tree. Contains $l = b - (n-1)$ branches (links).
Twigs	Branches belonging to the tree.
Links	Branches belonging to the co-tree.

Fundamental Sets

Basic Tie Set (Fundamental Loop): Formed by adding one link to the tree. Each link creates one unique fundamental loop. Number of loops = $l$.
Basic Cut Set (Fundamental Cut Set): A set of branches whose removal disconnects the graph into two parts, containing exactly one tree branch. Number of cut sets = $n-1$.

Matrices

Incidence Matrix (A)
- Complete Incidence Matrix: An $n \times b$ matrix.
- Entry $a_{ij}$: +1 if branch $j$ leaves node $i$, -1 if it enters, 0 if not incident.
- Property: Sum of any column = 0. Rank = $n-1$.
[!TIP] For a given graph, list nodes as rows, branches as columns. Assign arbitrary orientations to branches.
Tie Set Matrix (B)
- Fundamental Loop Matrix: An $l \times b$ matrix.
- Rows correspond to fundamental loops (one per link). Columns correspond to branches.
- Entry $b_{ij}$: +1 if branch $j$ is in loop $i$ and same direction as loop current, -1 if opposite, 0 if not in loop.
- Property: $A \cdot B^T = 0$ (Orthogonality).
Cut Set Matrix (Q)
- Fundamental Cut Set Matrix: An $(n-1) \times b$ matrix.
- Rows correspond to fundamental cut sets (one per tree branch). Columns correspond to branches.
- Entry $q_{ij}$: +1 if branch $j$ is in cut set $i$ and goes from +ve to -ve side, -1 if opposite, 0 if not in cut set.
- Property: $Q \cdot B^T = 0$ (Orthogonality).

Network Topology Overview

Graph theory provides a systematic, algebraic method for circuit analysis (e.g., using $A$, $B$, $Q$ matrices to write KVL/KCL).
Key Relationships: $b = n - 1 + l$ (branches = tree branches + links).

III. Network Theorems

Thevenin's Theorem

Statement: Any linear bilateral network can be replaced by an equivalent circuit consisting of a voltage source $V_{TH}$ in series with an impedance $Z_{TH}$.
$V_{TH}$: Open-circuit voltage across the load terminals.
$Z_{TH}$: Impedance seen across the load terminals with all independent sources deactivated (voltage sources shorted, current sources opened).
AC Version: Same procedure, but $V_{TH}$ and $Z_{TH}$ are phasors (complex).

Norton's Theorem

Statement: Equivalent to a current source $I_N$ in parallel with an admittance $Y_N$ (or impedance $Z_N$).
$I_N$: Short-circuit current through the load terminals.
$Y_N = 1/Z_{TH}$: Same as Thevenin impedance.
Relation: $I_N = V_{TH} / Z_{TH}$.

Maximum Power Transfer Theorem

For a Thevenin equivalent driving a load $Z_L = R_L + jX_L$:
- Condition for Max Power: $Z_L = Z_{TH}^*$ (Complex conjugate match).
- For purely resistive load ($X_L=0$): $R_L = |Z_{TH}|$ only if $Z_{TH}$ is resistive.
- Max Power Delivered: $P_{max} = \frac{|V_{TH}|^2}{4 R_L}$ (for resistive case).
- Efficiency: $\eta = \frac{P_{load}}{P_{source}} = \frac{1}{2} = 50\%$ at maximum power condition.

Millman's Theorem

For parallel voltage sources: Multiple voltage sources $V_k$ with series impedances $Z_k$ connected in parallel.

Equivalent voltage $V_{eq}$ and impedance $Z_{eq}$:

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

Simplifies to a single equivalent voltage source in series with equivalent impedance.

Tellegen's Theorem

General Power Theorem: For any network (linear/non-linear, passive/active, time-variant/invariant) satisfying KCL and KVL:

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

where the sum is over all **branches**. \(v_k, i_k\) are branch voltage and current with consistent sign convention.

Interpretation: Total instantaneous power delivered by all sources equals total power absorbed by all elements (conservation of energy).
Verification: Compute $\sum v_k i_k$ for all branches; result must be zero.

Compensation Theorem

Statement: If the impedance of a branch carrying current $I$ is changed from $Z$ to $Z + \Delta Z$, the resulting changes in all other branch currents and voltages are the same as those produced by injecting a compensating source $V_c = I \cdot \Delta Z$ in series with the modified branch (with original $Z$ still present).
Use: Analyzes effect of small impedance changes.

Substitution Theorem

Statement: Any branch in a network satisfying KCL/KVL can be replaced by any combination of sources and impedances that maintains the same voltage and current across that branch. The rest of the network remains unaffected.
Example: A resistor with voltage $V$ and current $I$ can be replaced by a voltage source $V$ in series with a short, or a current source $I$ in parallel with an open.

IV. Resonance in AC Circuits

Series Resonance

Circuit: $R$, $L$, $C$ in series with source $V$.
Resonant Frequency ($f_r$ or $\omega_r$): Condition where reactive power cancels.

$$X_L = X_C \implies \omega_r L = \frac{1}{\omega_r C} \implies \boxed{\omega_r = \frac{1}{\sqrt{LC}}, \quad f_r = \frac{1}{2\pi\sqrt{LC}}}$$

Characteristics at Resonance:
- Impedance $Z = R$ (minimum, purely resistive).
- Current $I = V/R$ (maximum).
- Voltage Magnification: $V_L = V_C = Q \cdot V$, where $Q = \frac{\omega_r L}{R} = \frac{1}{\omega_r R C}$ (Quality Factor).
- Bandwidth ($\Delta \omega$): $\Delta \omega = \frac{\omega_r}{Q}$ (between half-power points).

Parallel Resonance

Ideal Parallel LC ($R=\infty$): Impedance $\infty$ at $\omega_r = 1/\sqrt{LC}$.
Practical Parallel RLC (with resistance in series with L or C):
- Resonant frequency slightly shifted. For parallel RLC with L having series R:

$$\omega_r \approx \frac{1}{\sqrt{LC}} \sqrt{1 - \frac{R^2 C}{L}} \quad (\text{if } R \text{ small})$$

*   **Characteristics**:

    *   Impedance **maximum** (high).

    *   **Current Magnification**: Input current \(I\) minimum; branch currents \(I_L, I_C\) are large and opposite, \(I_L \approx I_C = Q \cdot I\).

    *   \(Q = R \sqrt{\frac{C}{L}}\) (for parallel RLC with shunt R).

V. Transient Analysis Using Laplace Transform

s-Domain Equivalents

Element	Time Domain	s-Domain (Zero Initial Conditions)
Resistor	$v(t) = Ri(t)$	$V(s) = RI(s)$
Inductor	$v(t) = L\frac{di}{dt}$	$V(s) = L[sI(s) - i(0^-)]$
Capacitor	$i(t) = C\frac{dv}{dt}$	$I(s) = C[sV(s) - v(0^-)]$

First-Order Circuits (RC/RL)

General Form: $\tau \frac{dy}{dt} + y = K$ (for step input), where $\tau = RC$ or $L/R$.
RC Series Step Response (switch closes at $t=0$, $v_C(0^-)=0$):

$$v_C(t) = V(1 - e^{-t/RC}), \quad i(t) = \frac{V}{R} e^{-t/RC}$$

*Graph*: \(v_C\) rises exponentially from 0 to \(V\); \(i\) decays exponentially from \(V/R\) to 0.

RL Series Step Response ($i_L(0^-)=0$):

$$i_L(t) = \frac{V}{R}(1 - e^{-tL/R}), \quad v_L(t) = V e^{-tL/R}$$

Second-Order Circuits (RLC)

Series RLC with initial capacitor voltage $V_0$ and zero initial current:

KVL: $L\frac{di}{dt} + Ri + \frac{1}{C}\int i dt = 0$

Differential Equation: $L\frac{d^2i}{dt^2} + R\frac{di}{dt} + \frac{1}{C}i = 0$
Characteristic Equation: $Ls^2 + Rs + \frac{1}{C} = 0$

Roots: $s_{1,2} = -\alpha \pm \sqrt{\alpha^2 - \omega_0^2}$

where $\alpha = \frac{R}{2L}$ (damping factor), $\omega_0 = \frac{1}{\sqrt{LC}}$ (undamped natural freq).
Damping Cases:
1. Overdamped ($\alpha > \omega_0$): Two real distinct roots. $i(t) = A_1 e^{s_1 t} + A_2 e^{s_2 t}$.
2. Critically Damped ($\alpha = \omega_0$): Repeated real root. $i(t) = (A_1 + A_2 t) e^{-\alpha t}$.
3. Underdamped ($\alpha < \omega_0$): Complex conjugate roots $s = -\alpha \pm j\omega_d$, $\omega_d = \sqrt{\omega_0^2 - \alpha^2}$.

$$i(t) = e^{-\alpha t} (A_1 \cos \omega_d t + A_2 \sin \omega_d t) = I_m e^{-\alpha t} \sin(\omega_d t + \phi)$$

Initial & Final Value Theorems

Initial Value Theorem (if $sF(s)$ has no poles in Re[s] ≥ 0):

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

Final Value Theorem (if $sF(s)$ has poles only in Re[s] < 0):

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

> [!TIP] **Condition**: System must be **stable** (all poles of \(sF(s)\) in LHP). Fails for unstable or marginally stable systems.

Inverse Laplace & Pole-Zero Analysis

Partial Fraction Expansion: Express $F(s)$ as sum of simpler terms (e.g., $\frac{A}{s+a}$, $\frac{Bs+C}{s^2+as+b}$).
Pole-Zero Plot:
- Zeros: Roots of numerator $N(s)=0$.
- Poles: Roots of denominator $D(s)=0$.
- Time Response: Dominated by poles closest to imaginary axis. Real poles → exponential decay/growth. Complex poles → damped sinusoids.
Obtaining $f(t)$: Use standard Laplace pairs after partial fractions.

Laplace Transform of Waveforms

Express piecewise function $f(t)$ using unit step function $u(t-a)$ or ramp $r(t-a)$.

Example: $f(t) = \begin{cases} 0, & t<0 \\ A, & 0<t<t_1 \\ 0, & t>t_1 \end{cases} = A[u(t) - u(t-t_1)]$
Laplace: $\mathcal{L}\{f(t)\} = A\left(\frac{1}{s} - \frac{e^{-t_1 s}}{s}\right)$.

VI. Fourier Series

Trigonometric Form

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

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

Coefficients:

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

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

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

Exponential (Euler's) Form

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

Complex Coefficients:

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

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

Common Waveforms

Square Wave (odd, amplitude $A$, 50% duty): Only odd harmonics ($n=1,3,5...$), $a_n=0$, $b_n = \frac{4A}{n\pi}$.
Triangular Wave (odd): Only odd harmonics, amplitude $\propto 1/n^2$.
Sawtooth Wave (odd): All harmonics, amplitude $\propto 1/n$.

[!TIP] From a given waveform figure:

Determine period $T$ and symmetry (even/odd/half-wave).

This tells which coefficients are zero.

Integrate over one period (often simpler over half-period using symmetry).

VII. Two-Port Networks

A two-port has input port (1-1') and output port (2-2'). Four variables: $V_1, I_1, V_2, I_2$ (currents entering port 1).

Parameter Sets (All linear)

Impedance Parameters (Z)

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

*   \(Z_{11} = \left.\frac{V_1}{I_1}\right|_{I_2=0}\) (Input impedance, output open).

*   \(Z_{12} = \left.\frac{V_1}{I_2}\right|_{I_1=0}\) (Reverse transfer impedance).

*   \(Z_{21} = \left.\frac{V_2}{I_1}\right|_{I_2=0}\) (Forward transfer impedance).

*   \(Z_{22} = \left.\frac{V_2}{I_2}\right|_{I_1=0}\) (Output impedance, input open).

Admittance Parameters (Y)

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

*   \(Y_{ij}\) found by **short-circuiting** the opposite port.

Hybrid Parameters (h)

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

*   \(h_{11} = \left.\frac{V_1}{I_1}\right|_{V_2=0}\) (Input impedance, output short) → **h-parameter unit: Ω**.

*   \(h_{12} = \left.\frac{V_1}{V_2}\right|_{I_1=0}\) (Reverse voltage gain, dimensionless).

*   \(h_{21} = \left.\frac{I_2}{I_1}\right|_{V_2=0}\) (Forward current gain, dimensionless).

*   \(h_{22} = \left.\frac{I_2}{V_2}\right|_{I_1=0}\) (Output admittance, input open) → **h-parameter unit: S**.

Transmission Parameters (ABCD)

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

*   **Cascade Property**: For two networks in cascade, overall **ABCD matrix = product** of individual ABCD matrices (in order).

*   \(A = \left.\frac{V_1}{V_2}\right|_{I_2=0}\), \(B = \left.\frac{V_1}{-I_2}\right|_{V_2=0}\) (transfer impedance).

*   \(C = \left.\frac{I_1}{V_2}\right|_{I_2=0}\) (transfer admittance), \(D = \left.\frac{I_1}{-I_2}\right|_{V_2=0}\).

Inverse Hybrid (g-parameters)

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

Parameter Conversions

General Approach: Write defining equations in matrix form and invert/substitute.
Key Relations:
- $Y = Z^{-1}$ (if $Z$ non-singular).
- From Z to h:

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

*   From **ABCD to Y**:

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

    (assuming \(A \neq 0\)).

Terminated Two-Port Network

Load $Z_L$ connected to output port.
Input Impedance:

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

Voltage Gain $G_v = V_2/V_1$ and Current Gain $G_i = I_2/I_1$ derived from parameter equations and load condition.

VIII. Special Topics and Additional Concepts

Dual Networks

Duality Principle: For any theorem/equation in network theory, a dual theorem/equation exists by replacing:
- Voltage ↔ Current
- Series ↔ Parallel
- Resistance ↔ Conductance
- Inductance ↔ Capacitance
- Short circuit ↔ Open circuit
- Node ↔ Mesh
Dual Network: A network where all elements and connections are replaced by their duals. The dual of a dual is the original network.

Controlled Sources (Dependent Sources)

Type	Symbol	Controlling Variable	Output Variable
VCVS	Voltage-Controlled Voltage Source	Voltage $v_x$	Voltage $\mu v_x$
VCCS	Voltage-Controlled Current Source	Voltage $v_x$	Current $g v_x$
CCVS	Current-Controlled Voltage Source	Current $i_x$	Voltage $r i_x$
CCCS	Current-Controlled Current Source	Current $i_x$	Current $\beta i_x$

Mutual Inductance & Coupling

Mutual Inductance $M$: Voltage induced in one coil due to current change in another: $v_1 = M \frac{di_2}{dt}$ (dot convention determines sign).
Dot Convention: Dots indicate polarity of induced voltage. If currents enter dotted terminals, mutual voltage adds to self-induced voltage.
Coefficient of Coupling $k$:

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

*   \(k=1\): Perfect coupling (all flux links both coils).

*   \(k<1\): Partial coupling.

Series Combination:
- Aiding (dots together): $L_{eq} = L_1 + L_2 + 2M$
- Opposing (dots opposite): $L_{eq} = L_1 + L_2 - 2M$
- From given $L_{eq,aiding}$ and $L_{eq,opposing}$, solve for $M$ and $k$.

\boxed{\text{End of Unit 3 Short Notes}}

Element	Time Domain	s-Domain (Zero Initial Conditions)
Resistor	\(v(t) = Ri(t)\)	\(V(s) = RI(s)\)
Inductor	\(v(t) = L\frac{di}{dt}\)	\(V(s) = L[sI(s) - i(0^-)]\)
Capacitor	\(i(t) = C\frac{dv}{dt}\)	\(I(s) = C[sV(s) - v(0^-)]\)

Type	Symbol	Controlling Variable	Output Variable
VCVS	Voltage-Controlled Voltage Source	Voltage \(v_x\)	Voltage \(\mu v_x\)
VCCS	Voltage-Controlled Current Source	Voltage \(v_x\)	Current \(g v_x\)
CCVS	Current-Controlled Voltage Source	Current \(i_x\)	Voltage \(r i_x\)
CCCS	Current-Controlled Current Source	Current \(i_x\)	Current \(\beta i_x\)