Electric Circuits & Networks – Complete Revision Notes

A single-page, exam-ready reference for Electric Circuits and Networks covering every core topic from Kirchhoff's laws through network synthesis and state-space analysis. Formulas, concepts, derivations and worked examples are organised in IEEE-standard notation for fast revision before GATE and university examinations.

Author: Dr. Mithun Mondal EEE · Network Theory Quick Reference · Exam Ready

Concept map linking Kirchhoff's and Ohm's laws to elements, sources, network theorems, nodal and mesh analysis, transients, AC phasors, Laplace, two-port networks, Fourier analysis and network synthesis. — Course concept map showing how the foundation of Kirchhoff's and Ohm's laws connects to circuit elements, sources, network theorems, systematic analysis, transient and AC behaviour, and the advanced s-domain, two-port and synthesis topics that build on them.

SECTION 01

Basic Concepts, Sources & Elements

Charge, Current, Voltage & Power

Fundamental Quantities

The defining relations between charge, current, voltage, power and energy are:

\[ i(t) = \frac{dq}{dt}\ \text{(A)} \qquad q = \int i\,dt \] \[ v(t) = \frac{dw}{dq}\ \text{(V)} \qquad p(t) = v(t)\,i(t)\ \text{(W)} \] \[ W = \int_{t_0}^{t_1} p(t)\,dt\ \text{(J)} \]

Units: \(1\,\text{C} = 6.24\times10^{18}\) electrons, \(1\,\text{A}=1\,\text{C/s}\), \(1\,\text{V}=1\,\text{J/C}\), \(1\,\text{W}=1\,\text{J/s}\).

Passive Sign Convention (PSC)

When current \(i\) enters the + terminal of an element:

\[ p = +vi \;\Rightarrow\; \text{element absorbs power} \] \[ p = -vi \;\Rightarrow\; \text{element delivers power} \]

This single convention fixes the sign of every power calculation in the circuit.

Conservation Laws

\(\sum P_{\text{delivered}} = \sum P_{\text{absorbed}}\). Tellegen's theorem guarantees this holds for any network obeying Kirchhoff's laws.

SI Units, Prefixes & Engineering Notation

Electrical Quantities

Quantity	Symbol	Unit	SI base
Charge	\(q\)	C	A·s
Current	\(i\)	A	A
Voltage	\(v\)	V	kg·m\(^2\)/(A·s\(^3\))
Resistance	\(R\)	\(\Omega\)	V/A
Conductance	\(G\)	S	A/V
Capacitance	\(C\)	F	C/V
Inductance	\(L\)	H	V·s/A
Power	\(P\)	W	V·A
Energy	\(W\)	J	W·s
Frequency	\(f\)	Hz	1/s
Flux	\(\Phi\)	Wb	V·s

Engineering Prefixes

Prefix	Symbol	Factor
tera	T	\(10^{12}\)
giga	G	\(10^{9}\)
mega	M	\(10^{6}\)
kilo	k	\(10^{3}\)
milli	m	\(10^{-3}\)
micro	\(\mu\)	\(10^{-6}\)
nano	n	\(10^{-9}\)
pico	p	\(10^{-12}\)
femto	f	\(10^{-15}\)

Always Carry Units

A numeric answer without units is an incorrect answer. Use dimensional consistency as a routine sanity check.

Classification of Sources & Elements

Source Types

Type	Description
Independent V	\(v\) fixed, \(i\) depends on circuit
Independent I	\(i\) fixed, \(v\) depends on circuit
VCVS	\(v = \mu v_x\)
VCCS	\(i = g\,v_x\)
CCVS	\(v = r\,i_x\)
CCCS	\(i = \beta\,i_x\)

Ideal V-source: \(R_s = 0\) (short when off). Ideal I-source: \(R_s = \infty\) (open when off).

Element & Network Classification

Active vs Passive:

Active: supplies energy (sources, op-amps)
Passive: absorbs or stores energy (R, L, C)

Linear vs Nonlinear: does it obey superposition?
Bilateral vs Unilateral: same \(v\)–\(i\) both ways?
Lumped vs Distributed: \(\lambda \gg\) size \(\Rightarrow\) lumped.
Time-invariant vs Time-varying: are parameters constant?

Practical Sources

Practical voltage source: ideal V in series with small \(R_s\). Practical current source: ideal I in parallel with large \(R_s\).

Op-Amp as an Active Circuit Element

Ideal Op-Amp Assumptions

Open-loop gain \(A \to \infty\)
Input impedance \(\to \infty \Rightarrow i^+ = i^- = 0\)
Output impedance \(\to 0\)
Infinite bandwidth, zero offset

Virtual short (with negative feedback): \(v^+ = v^-\).

Standard Configurations

Inverting: \(v_o/v_i = -R_f/R_1\)
Non-inverting: \(v_o/v_i = 1 + R_f/R_1\)
Voltage follower: \(v_o = v_i,\ R_{in} = \infty\)
Summer: \(v_o = -R_f\sum(v_k/R_k)\)
Differentiator: \(v_o = -RC\,dv_i/dt\)
Integrator: \(v_o = -\tfrac{1}{RC}\!\int v_i\,dt\)

Non-idealities

Finite open-loop gain \(A\)
Finite gain–bandwidth: \(A(f)\cdot f_{-3\text{dB}}\) = const.
Slew-rate limit: \(|dv_o/dt|_{max}\)
Input-offset voltage and bias currents
CMRR, PSRR

In Network Theory

The op-amp behaves as a VCVS (\(\mu \to \infty\)). It enables active filters, gyrators and negative-impedance converters (NICs) that synthesise inductance from R–C–op-amp combinations.

SECTION 02

Fundamental Laws

Ohm's Law & Power Relations

Ohm's Law

\[ \boxed{V = IR} \qquad I = \frac{V}{R} \qquad R = \frac{V}{I} \] \[ \text{Conductance: } G = \frac{1}{R}\ \text{(S)} \]

Power & Energy

\[ P = VI = I^2R = \frac{V^2}{R} \] \[ W = Pt = VIt\ \text{(J)} \] \[ P_{\text{absorbed}} = +vi,\quad P_{\text{delivered}} = -vi \]

Schematic symbols for a resistor R, capacitor C and inductor L with labelled voltage and current. — Schematic representation of the three passive elements — resistor, capacitor and inductor — with the passive sign convention applied so that current enters the positive voltage terminal.

Sign Convention

Under the passive sign convention, current entering the + terminal means the element absorbs power.

Kirchhoff's Laws (KVL & KCL)

KCL — Current Law

At any node:

\[ \sum_k i_k = 0 \qquad \sum i_{\text{in}} = \sum i_{\text{out}} \]

Expresses conservation of charge.

KVL — Voltage Law

Around any loop:

\[ \sum_k v_k = 0 \qquad \sum v_{\text{rise}} = \sum v_{\text{drop}} \]

Expresses conservation of energy.

A single node with two currents entering and two currents leaving, illustrating that the sum of entering currents equals the sum of leaving currents. — Application of Kirchhoff's current law at a node: incoming currents \(i_1\) and \(i_2\) equal outgoing currents \(i_3\) and \(i_4\), so \(i_1+i_2=i_3+i_4\).

Systematic Analysis Steps

Assign branch currents and voltages.
Choose loop directions (clockwise positive).
Write the KVL and KCL equations.
Solve the resulting linear system.

Series & Parallel Combinations

Series Circuits

\[ \boxed{R_{eq}=R_1+R_2+\cdots+R_n} \] \[ \tfrac{1}{C_{eq}}=\tfrac{1}{C_1}+\tfrac{1}{C_2}+\cdots \qquad L_{eq}=\textstyle\sum L_k \]

Voltage divider:

\[ V_k = V_s\cdot\frac{R_k}{R_1+R_2+\cdots+R_n} \]

Parallel Circuits

\[ \boxed{\tfrac{1}{R_{eq}}=\tfrac{1}{R_1}+\tfrac{1}{R_2}+\cdots} \]

Two resistors: \(R_{eq}=R_1 R_2/(R_1+R_2)\)

\[ C_{eq}=\textstyle\sum C_k, \qquad \tfrac{1}{L_{eq}}=\textstyle\sum \tfrac{1}{L_k} \]

Current divider: \(I_k = I_s R_{eq}/R_k\)

Two resistors R1 and R2 shown first connected in series and then connected in parallel. — Series and parallel connection of two resistors, the building blocks for reducing any resistive ladder to a single equivalent resistance.

Star–Delta (Y–\(\Delta\)) Transformation

\(\Delta \rightarrow\) Y Conversion

\[ R_1=\frac{R_a R_b}{R_a+R_b+R_c} \] \[ R_2=\frac{R_b R_c}{R_a+R_b+R_c} \] \[ R_3=\frac{R_a R_c}{R_a+R_b+R_c} \]

Rule: product of the two adjacent \(\Delta\) arms divided by the sum of all three.

Y \(\rightarrow \Delta\) Conversion

\[ R_a=\frac{R_1R_2+R_2R_3+R_3R_1}{R_2} \] \[ R_b=\frac{R_1R_2+R_2R_3+R_3R_1}{R_3} \] \[ R_c=\frac{R_1R_2+R_2R_3+R_3R_1}{R_1} \]

Rule: sum of the pair-products divided by the opposite Y arm.

A three-terminal delta network of resistors Ra, Rb, Rc shown alongside its equivalent wye network of resistors R1, R2, R3. — Equivalence between the delta (\(\Delta\)) network with arms \(R_a, R_b, R_c\) and the wye (Y / star) network with arms \(R_1, R_2, R_3\), used to simplify bridge and three-phase circuits.

Pro Tip

If all arms are equal: \(R_Y = \tfrac{R_\Delta}{3}\) and \(R_\Delta = 3R_Y\). Especially useful for balanced bridge and three-phase circuits.

Duality of Networks

Dual Quantities

Original	Dual
Voltage \(v\)	Current \(i\)
Current \(i\)	Voltage \(v\)
Resistance \(R\)	Conductance \(G\)
Inductance \(L\)	Capacitance \(C\)
Impedance \(Z\)	Admittance \(Y\)
KVL (loop)	KCL (node)
Series	Parallel
Open circuit	Short circuit
Mesh	Node
Thévenin	Norton

Constructing a Dual Network

Rules (planar networks only):

Place a node inside each mesh of the original; add one outside as reference.
Draw a dual branch crossing each original branch, connecting adjacent mesh-nodes.
Replace each element with its dual (\(R\!\to\!G\), \(L\!\to\!C\), \(V\!\to\!I\)).

Why Duality?

Solve one circuit and the solution for its dual follows by symbol substitution, halving the workload across problem sets.

Memory Aids & Mnemonics for Quick Recall

ELI the ICE man (AC phase)

ELI: in an inductor (L), EMF (E) leads current (I).
ICE: in a capacitor (C), current (I) leads EMF (E).

\(\Rightarrow\) Inductor: \(V\) leads \(I\) by \(90^\circ\).
\(\Rightarrow\) Capacitor: \(I\) leads \(V\) by \(90^\circ\).

Initial Conditions

"Capacitor keeps its Voltage, Inductor keeps its current."

\[ v_C(0^+) = v_C(0^-) \qquad i_L(0^+) = i_L(0^-) \]

DC Steady State

"L is Lazy, C is Closed." In DC steady state, \(L \to\) short circuit (\(v_L = 0\)) and \(C \to\) open circuit (\(i_C = 0\)).

Kill the Source (Superposition)

V-source \(\to\) Vanish as a wire (short). I-source \(\to\) Invisible (open circuit).

Power Triangle (PAQ)

P: Active power (real, W) — useful work.
A / S: Apparent power (VA) — the hypotenuse.
Q: Reactive power (VAR) — shuttled back and forth.

\[ S^2 = P^2 + Q^2, \qquad \text{pf} = \cos\theta = P/S \]

SECTION 03

Network Theorems

Thévenin's & Norton's Theorems

Thévenin's Theorem

Any linear network seen from two terminals reduces to a single voltage source in series with a resistance:

\[ \boxed{V_{Th}\ \text{(series) with}\ R_{Th}} \]

\(V_{Th}\) = open-circuit voltage \(V_{oc}\)
\(R_{Th}\) = resistance at the terminals with all independent sources killed

Norton's Theorem

\[ \boxed{I_N = \frac{V_{oc}}{R_{Th}}}, \quad R_N = R_{Th} \]

\(I_N\) in parallel with \(R_N\)
\(I_N\) = short-circuit current \(I_{sc}\)

Thévenin equivalent of a voltage source in series with a resistance beside the Norton equivalent of a current source in parallel with a resistance, both feeding terminals A and B. — Thévenin equivalent (voltage source \(V_{Th}\) in series with \(R_{Th}\)) and Norton equivalent (current source \(I_N\) in parallel with \(R_N\)) presented at terminals A–B; the two are interchangeable by source transformation.

Source Transformation

\[ V_s\ \text{(series } R) \;\Longleftrightarrow\; I_s=\frac{V_s}{R}\ \text{(parallel } R) \]

Superposition, Maximum Power Transfer & Reciprocity

Superposition Theorem

Linear circuits only. The total response equals the algebraic sum of the responses to each source acting alone.

\[ v = \textstyle\sum_k v_k, \quad i = \textstyle\sum_k i_k \]

Activate one source, deactivate the others (V \(\to\) short, I \(\to\) open), and sum the contributions.

Maximum Power Transfer

\[ \boxed{R_L = R_{Th} \;\Rightarrow\; P_{max} = \frac{V_{Th}^2}{4R_{Th}}} \]

Peak occurs at \(R_L = R_{Th}\); the corresponding efficiency is 50%.

Curve of normalised load power versus the ratio of load resistance to Thévenin resistance, peaking at a ratio of one. — Normalised power delivered to the load \(P/P_{max}\) versus \(R_L/R_{Th}\); the transfer is maximum when the load matches the Thévenin resistance.

Reciprocity Theorem

In a linear bilateral network, \(v_1/i_2 = v_2/i_1\): excitation and response are interchangeable.

Millman's Theorem

\[ V_{AB} = \frac{\sum_k V_k/R_k}{\sum_k 1/R_k} \]

Advanced Theorems: Tellegen, Substitution & Compensation

Tellegen's Theorem

For any network (linear or nonlinear, time-varying or invariant) obeying KVL and KCL:

\[ \boxed{\sum_{k=1}^{b} v_k\,i_k = 0} \]

Holds for any \(\{v_k\}\) satisfying KVL
Holds for any \(\{i_k\}\) satisfying KCL
\(v_k, i_k\) may come from different circuits of the same topology
Implies conservation of power

Substitution Theorem

Any branch with known \(v_k, i_k\) may be replaced by a voltage source \(v_k\) or a current source \(i_k\) without disturbing the rest of the network — useful for staged, multi-block problems.

Compensation Theorem

If a branch impedance changes by \(\Delta Z\), the change in network response equals the response produced by an inserted series source

\[ \boxed{V_c = -I_0\,\Delta Z} \]

where \(I_0\) is the original branch current and all other sources are set to zero.

Why These Matter

Tellegen → energy audits and sensitivity; Substitution → staged analysis; Compensation → tolerance and small-change problems solved without re-analysing the whole circuit.

Nodal & Mesh Analysis

Nodal Analysis (KCL)

Choose a reference node (ground). For \(n\) nodes, write \(n-1\) KCL equations.

Conductance matrix form:

\[ \mathbf{G}\,\mathbf{V} = \mathbf{I} \] \[ G_{kk} = \sum \text{conductances at node } k \] \[ G_{km} = -\,\text{conductance between } k, m \]

Supernode: a voltage source between two non-reference nodes ⇒ combine the nodes and add a constraint.

Mesh Analysis (KVL)

Assign mesh currents \(\{i_1, i_2, \ldots\}\) and write KVL for each mesh.

Resistance matrix form:

\[ \mathbf{R}\,\mathbf{I} = \mathbf{V} \] \[ R_{kk} = \sum \text{resistances in mesh } k \] \[ R_{km} = -\,\text{shared resistance} \]

Supermesh: a current source shared between two meshes ⇒ combine the meshes and add a constraint.

Cramer's Rule

\(x_k = \dfrac{\det(\mathbf{A}_k)}{\det(\mathbf{A})}\), where \(\mathbf{A}_k\) replaces column \(k\) with the right-hand-side vector.

SECTION 04

Capacitors & Inductors

Capacitor & Inductor V–I Relations

Capacitor

\[ i_C = C\,\frac{dv}{dt}, \quad v_C(t) = \frac{1}{C}\!\int_0^t i\,d\tau + v_C(0) \] \[ W_C = \tfrac{1}{2}CV^2 \]

DC steady state	\(i_C = 0\) (open circuit)
Continuity	\(v_C(0^-) = v_C(0^+)\)

Inductor

\[ v_L = L\,\frac{di}{dt}, \quad i_L(t) = \frac{1}{L}\!\int_0^t v\,d\tau + i_L(0) \] \[ W_L = \tfrac{1}{2}LI^2 \]

DC steady state	\(v_L = 0\) (short circuit)
Continuity	\(i_L(0^-) = i_L(0^+)\)

Series / Parallel Summary

Element	Series	Parallel	Duality
\(R\)	\(\sum R_k\)	\(\left(\sum G_k\right)^{-1}\)	\(G\)
\(C\)	\(\left(\sum 1/C_k\right)^{-1}\)	\(\sum C_k\)	\(L\)
\(L\)	\(\sum L_k\)	\(\left(\sum 1/L_k\right)^{-1}\)	\(C\)

SECTION 05

First & Second Order Circuits

Initial Conditions & Switching Transients

Continuity Rules at \(t=0\)

When a switch acts at \(t=0\):

\[ \boxed{v_C(0^+) = v_C(0^-)} \;\text{(capacitor voltage continuous)} \] \[ \boxed{i_L(0^+) = i_L(0^-)} \;\text{(inductor current continuous)} \]

But \(i_C\) and \(v_L\) can jump:

\[ i_C(0^+) = C\,\frac{dv_C}{dt}\bigg|_{0^+},\quad v_L(0^+) = L\,\frac{di_L}{dt}\bigg|_{0^+} \]

Three-Step Procedure

Step 1 — at \(t=0^-\): DC steady state; \(L\) = short, \(C\) = open. Find \(v_C(0^-)\), \(i_L(0^-)\).

Step 2 — at \(t=0^+\): use continuity. Replace \(C\) by a V-source \(v_C(0^+)\) and \(L\) by an I-source \(i_L(0^+)\).

Step 3 — at \(t=\infty\): new DC steady state; again \(L\) = short, \(C\) = open. Find \(x(\infty)\).

Common Pitfall

Never assume \(v_C(0^-)=0\) or \(i_L(0^-)=0\). Always verify using the pre-switch circuit.

First-Order Circuits: RC & RL

Natural + Step Response

General form:

\[ \boxed{x(t) = x(\infty) + \bigl[x(0^+)-x(\infty)\bigr]e^{-t/\tau}} \]

Circuit	\(\tau\)	State variable
RC	\(RC\)	\(v_C\)
RL	\(L/R\)	\(i_L\)

\(x(0^+)\) = initial condition; \(x(\infty)\) = final DC steady-state value.

Time-Constant Meaning

At \(t=\tau\) the response has changed by about 63.2%, leaving \(\approx 36.8\%\) of the initial gap; at \(5\tau\) it is 99% settled.

Decaying exponential curve of normalised response versus time in units of the time constant, marking the 0.368 value at one time constant. — Normalised first-order decay \(e^{-t/\tau}\): the response falls to 0.368 of its initial value after one time constant and is essentially complete after five time constants.

Step-by-Step Method

(1) Find \(x(0^+)\) by continuity. (2) Find \(x(\infty)\) at DC steady state (\(L\to\) short, \(C\to\) open). (3) Find \(\tau = RC\) or \(L/R\). (4) Substitute into the general form.

Second-Order Circuits: RLC

Series RLC ODE

\[ L\frac{d^2i}{dt^2}+R\frac{di}{dt}+\frac{i}{C}=\frac{dv_s}{dt} \]

Characteristic equation:

\[ s^2 + 2\alpha s + \omega_0^2 = 0 \]

\(\alpha = R/(2L)\) (series), \(\alpha = 1/(2RC)\) (parallel); \(\omega_0 = 1/\sqrt{LC}\) (natural frequency).

\[ s_{1,2} = -\alpha \pm \sqrt{\alpha^2 - \omega_0^2} \]

Damping Cases

Case	Condition	Response
Over-	\(\alpha\gt\omega_0\)	\(A_1e^{s_1t}+A_2e^{s_2t}\)
Critical	\(\alpha=\omega_0\)	\((A_1+A_2t)e^{-\alpha t}\)
Under-	\(\alpha\lt\omega_0\)	\(e^{-\alpha t}(B_1\cos\omega_d t+B_2\sin\omega_d t)\)

\(\omega_d = \sqrt{\omega_0^2-\alpha^2}\), \(\;Q = \omega_0/(2\alpha)\).

Parallel RLC

\[ \frac{d^2v}{dt^2}+\frac{1}{RC}\frac{dv}{dt}+\frac{v}{LC}=\frac{i_s}{C} \]

Second-Order Response: Visual Comparison

Step responses of a second-order system showing an overdamped curve rising slowly, a critically damped curve, and an underdamped curve overshooting and oscillating toward the final value. — Step response of a second-order circuit for the three damping regimes — overdamped (\(\zeta\gt1\)), critically damped (\(\zeta=1\)) and underdamped (\(\zeta\lt1\)) — all settling to the same final value.

Damping Ratio \(\zeta\)

\[ \zeta = \frac{\alpha}{\omega_0} \]

\(\zeta \gt 1\)	Overdamped (slow)
\(\zeta = 1\)	Critical (fastest, no overshoot)
\(0\lt\zeta\lt1\)	Underdamped (oscillates)
\(\zeta = 0\)	Undamped

Key Response Metrics

\[ \%\text{OS} = 100\,e^{-\zeta\pi/\sqrt{1-\zeta^2}} \] \[ t_s \approx 4/(\zeta\omega_0)\ \text{(2% settling)} \] \[ t_p = \pi/\omega_d\ \text{(peak time)} \]

SECTION 06

AC Circuit Analysis

Phasors & Impedance

Phasor Representation

\[ v(t)=V_m\cos(\omega t+\phi)\;\xleftrightarrow{\text{phasor}}\;\mathbf{V}=V_m\angle\phi \] \[ \mathbf{V}=V_m e^{j\phi}=V_m(\cos\phi+j\sin\phi) \]

Time \(\to\) phasor: \(\cos(\omega t+\phi)\to\angle\phi\); \(\dfrac{d}{dt}\leftrightarrow j\omega\).

Impedance (Ohm's Law in AC)

\[ \mathbf{Z} = \mathbf{V}/\mathbf{I} = R+jX\ (\Omega) \]

Element	\(\mathbf{Z}\)	\(\mathbf{Y}=1/\mathbf{Z}\)
\(R\)	\(R\)	\(G\)
\(L\)	\(j\omega L\)	\(1/(j\omega L)\)
\(C\)	\(1/(j\omega C)\)	\(j\omega C\)

A phasor drawn from the origin in the complex plane at angle phi, with its real and imaginary projections marked. — Phasor diagram of \(\mathbf{V}=V_m\angle\phi\) in the complex plane, with real part \(V_m\cos\phi\) and imaginary part \(V_m\sin\phi\).

Phase Relations — "ELI the ICE man"

\(R\): \(V\) in phase with \(I\); \(L\): \(V\) leads \(I\) by \(90^\circ\); \(C\): \(V\) lags \(I\) by \(90^\circ\).

AC Power Analysis

AC Power Triangle

\[ \mathbf{S} = \mathbf{V}\mathbf{I}^* = P + jQ\ \text{(VA)} \] \[ P = V_{rms}I_{rms}\cos\theta\ \text{(W, active)} \] \[ Q = V_{rms}I_{rms}\sin\theta\ \text{(VAR, reactive)} \] \[ |\mathbf{S}| = V_{rms}I_{rms}\ \text{(VA, apparent)} \] \[ \text{pf} = \cos\theta = P/|\mathbf{S}| \]

RMS Values

\[ V_{rms} = V_m/\sqrt{2} \approx 0.707\,V_m \] \[ I_{rms} = I_m/\sqrt{2} \] \[ P_{avg} = \tfrac{V_m I_m}{2}\cos\theta = V_{rms}I_{rms}\cos\theta \]

Right triangle with horizontal side active power P, vertical side reactive power Q, and hypotenuse apparent power S at angle theta. — The AC power triangle relating active power \(P\), reactive power \(Q\) and apparent power \(|\mathbf{S}|\), with the power-factor angle \(\theta\) between \(P\) and \(|\mathbf{S}|\).

Power Factor Correction

Add a capacitor in parallel to improve a lagging power factor:

\[ C = \frac{P(\tan\theta_1-\tan\theta_2)}{\omega V_{rms}^2} \]

Goal: \(\theta_2 \lt \theta_1\), i.e. power factor closer to unity.

Maximum Power in AC

\(\mathbf{Z}_L = \mathbf{Z}_{Th}^* \;\Rightarrow\; P_{max}=|V_{Th}|^2/(8R_{Th})\).

Resonance

Series Resonance

\[ \omega_0 = \frac{1}{\sqrt{LC}}, \quad Z_{min}=R \] \[ Q_s = \frac{\omega_0 L}{R} = \frac{1}{\omega_0 CR} \] \[ BW = \frac{\omega_0}{Q_s} = \frac{R}{L}\ \text{(rad/s)} \]

At resonance \(V_L = V_C = Q_s\cdot V_s\) (voltage magnification).

Parallel Resonance

\[ \omega_0 = \frac{1}{\sqrt{LC}}, \quad Y_{min}=G,\;Z_{max}=R \] \[ Q_p = \frac{\omega_0 C}{G} = \frac{R}{\omega_0 L} \] \[ BW = \frac{G}{C} = \frac{\omega_0}{Q_p} \]

At resonance \(I_L = I_C = Q_p\cdot I_s\) (current magnification).

Resonance curve of impedance magnitude versus frequency, peaking sharply at the resonant frequency between the two half-power frequencies. — Impedance magnitude versus frequency for a resonant circuit, showing the resonant frequency \(\omega_0\) and the half-power frequencies \(\omega_1\) and \(\omega_2\) that bound the bandwidth.

Non-sinusoidal Periodic Sources

RMS of a Non-sinusoid

If \(v(t) = V_0 + \sum_{n=1}^{\infty} V_n\cos(n\omega_0 t + \phi_n)\):

\[ \boxed{V_{rms} = \sqrt{V_0^2 + \textstyle\sum_{n\ge 1}(V_n/\sqrt{2})^2}} \] \[ = \sqrt{V_{0}^2 + V_{1,rms}^2 + V_{2,rms}^2 + \cdots} \]

Average power into \(R\): \(P = V_{rms}^2/R\).

Average Power (general)

For \(v, i\) both non-sinusoidal of the same period:

\[ P_{avg} = V_0 I_0 + \textstyle\sum_{n\ge 1}\tfrac{V_n I_n}{2}\cos(\phi_n^V - \phi_n^I) \]

Each harmonic contributes independently; the DC terms add.

Distortion Metrics

Total harmonic distortion:

\[ \text{THD} = \frac{\sqrt{\sum_{n\ge 2}V_{n,rms}^2}}{V_{1,rms}} \]

Distortion factor: \(\text{DF} = V_{1,rms}/V_{rms}\). Form factor: \(V_{rms}/V_{avg}\). Crest factor: \(V_{pk}/V_{rms}\).

SECTION 07

Three-Phase Circuits

Balanced Three-Phase Systems

Balanced 3-\(\phi\) System

\[ V_{an}=V_p\angle 0^\circ,\; V_{bn}=V_p\angle{-120^\circ},\; V_{cn}=V_p\angle{-240^\circ} \]

Y-connection: \(V_L = \sqrt{3}\,V_p,\quad I_L = I_p\)

\(\Delta\)-connection: \(V_L = V_p,\quad I_L = \sqrt{3}\,I_p\)

Total power (balanced):

\[ \boxed{P_T = 3V_pI_p\cos\theta = \sqrt{3}V_LI_L\cos\theta} \] \[ Q_T = \sqrt{3}V_LI_L\sin\theta,\quad S_T=\sqrt{3}V_LI_L \]

Y–\(\Delta\) & \(\Delta\)–Y Conversion

\(\Delta\to\)Y: \(Z_Y = \dfrac{Z_\Delta}{3}\)

Y\(\to\Delta\): \(Z_\Delta = 3Z_Y\)

Per-phase analysis: convert everything to Y, analyse a single phase, then multiply by three.

Two-Wattmeter Method

\[ P_T = W_1 + W_2 \] \[ \tan\theta = \sqrt{3}\,\frac{W_1-W_2}{W_1+W_2}, \qquad \text{pf} = \cos\theta \]

Valid for any three-wire, three-phase load.

Unbalanced 3-\(\phi\) & Symmetrical Components

Fortescue's Theorem

Any unbalanced three-phase phasor set \(\{V_a, V_b, V_c\}\) decomposes into:

\[ V_a = V_{a0} + V_{a1} + V_{a2} \]

Zero-sequence: equal magnitude, in phase.
Positive-sequence: balanced, rotation \(abc\).
Negative-sequence: balanced, rotation \(acb\).

With \(a = 1\angle 120^\circ\):

\[ \begin{bmatrix}V_{a0}\\V_{a1}\\V_{a2}\end{bmatrix} = \tfrac{1}{3} \begin{bmatrix}1 & 1 & 1\\ 1 & a & a^2\\ 1 & a^2 & a\end{bmatrix} \begin{bmatrix}V_a\\V_b\\V_c\end{bmatrix} \]

Inverse (Reconstruction)

\[ \begin{bmatrix}V_a\\V_b\\V_c\end{bmatrix} = \begin{bmatrix}1 & 1 & 1\\ 1 & a^2 & a\\ 1 & a & a^2\end{bmatrix} \begin{bmatrix}V_{a0}\\V_{a1}\\V_{a2}\end{bmatrix} \]

Properties: \(1 + a + a^2 = 0\), \(\;a^3 = 1\).

Key Fact

In a balanced load with no neutral wire, \(V_{a0} = 0\) (no zero-sequence path). This is central to power-system fault analysis.

SECTION 08

Laplace Transform & s-Domain

Laplace Transform in Circuit Analysis

Key Laplace Pairs

\(f(t)\)	\(F(s)\)
\(\delta(t)\)	\(1\)
\(u(t)\)	\(\dfrac{1}{s}\)
\(e^{-at}u(t)\)	\(\dfrac{1}{s+a}\)
\(t^n u(t)\)	\(\dfrac{n!}{s^{n+1}}\)
\(\cos(\omega_0 t)u(t)\)	\(\dfrac{s}{s^2+\omega_0^2}\)
\(\sin(\omega_0 t)u(t)\)	\(\dfrac{\omega_0}{s^2+\omega_0^2}\)
\(te^{-at}u(t)\)	\(\dfrac{1}{(s+a)^2}\)

s-Domain Element Models

\(R\)	\(R\)
\(L\)	\(sL - Li(0^-)\)
\(C\)	\(\dfrac{1}{sC} + \dfrac{v(0^-)}{s}\)

\[ H(s) = \frac{Y(s)}{X(s)}\ \text{(transfer function)} \]

Laplace Theorems

Initial value	\(f(0^+) = \lim_{s\to\infty}sF(s)\)
Final value	\(f(\infty) = \lim_{s\to 0}sF(s)\)
Differentiation	\(\mathcal{L}\{f'\}=sF(s)-f(0)\)
Integration	\(\mathcal{L}\left\{\int f\right\}=\dfrac{F(s)}{s}\)

Partial Fraction Expansion & Inverse Laplace

Partial Fractions

\[ \frac{N(s)}{D(s)} = \frac{k_1}{s-p_1}+\frac{k_2}{s-p_2}+\cdots \]

Simple poles: \(k_i = (s-p_i)F(s)\big|_{s=p_i}\)

Repeated pole \(p_1\) (order \(m\)):

\[ k_{1,r} = \frac{1}{(m-r)!}\frac{d^{m-r}}{ds^{m-r}}\bigl[(s-p_1)^m F(s)\bigr]_{s=p_1} \]

Complex poles \(s=-\alpha\pm j\beta\), \(k = A+jB\):

\[ \to 2e^{-\alpha t}\bigl(A\cos\beta t - B\sin\beta t\bigr) \]

Transfer Function & Frequency

\[ H(j\omega) = H(s)\big|_{s=j\omega} \] \[ |H|_{\text{dB}} = 20\log_{10}|H(j\omega)| \] \[ \angle H = \arctan\!\left(\frac{\operatorname{Im}\{H\}}{\operatorname{Re}\{H\}}\right) \]

Convolution

\[ y(t) = h(t)*x(t) = \int_0^t h(\tau)x(t-\tau)\,d\tau \] \[ Y(s) = H(s)\cdot X(s) \]

Stability

Stable: poles in the left half-plane; marginal: poles on the \(j\omega\) axis; unstable: any pole in the right half-plane.

Convolution & Impulse Response

Convolution Integral

For an LTI system with impulse response \(h(t)\):

\[ \boxed{y(t) = h(t)*x(t) = \int_0^t h(\tau)\,x(t-\tau)\,d\tau} \]

Graphical steps (flip–shift–multiply–integrate):

Flip \(h(\tau)\to h(-\tau)\).
Shift by \(t\): \(h(t-\tau)\).
Multiply: \(x(\tau)\,h(t-\tau)\).
Integrate over the overlap region.

Properties of Convolution

Commutative: \(x*h = h*x\)
Associative: \((x*h)*g = x*(h*g)\)
Distributive: \(x*(h_1+h_2) = x*h_1 + x*h_2\)
Identity: \(x(t)*\delta(t) = x(t)\)

s-domain: \(Y(s) = H(s)\,X(s)\)

Impulse Response Role

\(h(t) = \mathcal{L}^{-1}\{H(s)\}\). Knowing \(h(t)\) completely characterises the LTI network's zero-state response to any input.

SECTION 09

Frequency Response & Filters

Bode Plots

Bode Plot Construction

\[ H(j\omega)=K\frac{\prod(1+j\omega/z_i)}{\prod(1+j\omega/p_k)} \]

Factor	Mag. slope	Phase
Gain \(K\)	\(20\log K\)	\(0^\circ\) or \(\pm180^\circ\)
Zero \((1+j\omega/z)\)	\(+20\) dB/dec	\(+90^\circ\) total
Pole \((1+j\omega/p)^{-1}\)	\(-20\) dB/dec	\(-90^\circ\) total
\(j\omega\) (origin zero)	\(+20\) dB/dec	\(+90^\circ\)
\(1/(j\omega)\) (origin pole)	\(-20\) dB/dec	\(-90^\circ\)

Phase corner: \(\pm45^\circ\) at the break frequency; \(\pm45^\circ\)/decade over two decades.

Standard Filter Responses

Filter	Pass band	\(H(j\omega)\)
LPF	\(\omega\lt\omega_c\)	\(\dfrac{1}{1+j\omega/\omega_c}\)
HPF	\(\omega\gt\omega_c\)	\(\dfrac{j\omega/\omega_c}{1+j\omega/\omega_c}\)
BPF	\(\omega_1\lt\omega\lt\omega_2\)	\(\dfrac{j\omega/\omega_c}{1-(\omega/\omega_c)^2+j\omega Q^{-1}}\)
BSF	outside BW	\(\dfrac{1-(\omega/\omega_c)^2}{1-(\omega/\omega_c)^2+j\omega Q^{-1}}\)

\(\omega_c = 1/(RC)\) for RC filters.

Key Points

The \(-3\) dB frequency is the half-power point and equals the cutoff; \(|H|=1/\sqrt{2}\) at \(\omega_c\).

Filter Approximations: Butterworth & Chebyshev

Butterworth (Maximally Flat)

\[ |H(j\omega)|^2 = \frac{1}{1+(\omega/\omega_c)^{2n}} \]

No ripple in either band; roll-off \(-20n\) dB/dec; poles on a circle of radius \(\omega_c\) in the LHP; monotonic ("smoothest") magnitude.

\[ s_k = \omega_c e^{j(\pi/2 + (2k-1)\pi/2n)} \]

Chebyshev Type-I

\[ |H(j\omega)|^2 = \frac{1}{1+\epsilon^2\,C_n^2(\omega/\omega_c)} \]

Ripple in the pass band, flat in the stop band; sharper roll-off than Butterworth for the same order \(n\); \(C_n\) is the Chebyshev polynomial of order \(n\).

Magnitude response curves comparing a maximally flat Butterworth low-pass filter with a Chebyshev Type-I filter showing pass-band ripple and steeper roll-off. — Magnitude response of fourth-order Butterworth and Chebyshev Type-I low-pass filters: the Butterworth response is monotonic while the Chebyshev trades pass-band ripple for a sharper transition near the cutoff.

Other Families

Chebyshev-II: ripple in the stop band.
Elliptic (Cauer): ripple in both bands, steepest transition.
Bessel: linear phase, gentle roll-off.

Magnitude & Frequency Scaling

Magnitude Scaling (\(K_m\))

Multiplies all impedance levels by \(K_m\):

\[ R' = K_m R,\; L' = K_m L,\; C' = C/K_m \]

\(H(s)\) is unchanged for voltage-ratio, current-ratio and transfer-admittance functions; the driving-point impedance scales by \(K_m\).

Frequency Scaling (\(K_f\))

Compresses or expands the frequency axis by \(K_f\):

\[ R' = R,\; L' = L/K_f,\; C' = C/K_f \]

Cutoff, resonance and bandwidth all scale as \(\omega' = K_f\,\omega\).

Combined Scaling

\[ R' = K_m R,\; L' = (K_m/K_f)L,\; C' = C/(K_m K_f) \]

Design a normalised prototype (\(\omega_c=1\), \(R=1\,\Omega\)), then scale to the target.

Example

Normalised LPF: \(R=1\,\Omega\), \(C=1\,\text{F}\), \(\omega_c=1\,\text{rad/s}\). Target \(R=10\,\text{k}\Omega\), \(\omega_c=10^4\,\text{rad/s}\) ⇒ \(K_m=10^4\), \(K_f=10^4\) ⇒ \(C = 10\,\text{nF}\).

SECTION 10

Two-Port Networks

Two-Port Network Parameters

Parameter Definitions

Param	Equations	Condition
\(z\) (impedance)	\(V_1=z_{11}I_1+z_{12}I_2\); \(V_2=z_{21}I_1+z_{22}I_2\)	Open-circuit
\(y\) (admittance)	\(I_1=y_{11}V_1+y_{12}V_2\); \(I_2=y_{21}V_1+y_{22}V_2\)	Short-circuit
\(h\) (hybrid)	\(V_1=h_{11}I_1+h_{12}V_2\); \(I_2=h_{21}I_1+h_{22}V_2\)	Hybrid
\(ABCD\) (chain)	\(V_1=AV_2-BI_2\); \(I_1=CV_2-DI_2\)	Transmission

Inter-conversions

\[ \mathbf{z} = \mathbf{y}^{-1} \] \[ \mathbf{h} = \begin{bmatrix}\Delta_z/z_{22} & z_{12}/z_{22}\\ -z_{21}/z_{22} & 1/z_{22}\end{bmatrix} \] \[ \mathbf{T}=\begin{bmatrix}A&B\\C&D\end{bmatrix}= \begin{bmatrix}-\Delta_h/h_{21}&-h_{11}/h_{21}\\ -h_{22}/h_{21}&-1/h_{21}\end{bmatrix} \]

Key Properties

Reciprocal: \(z_{12}=z_{21}\), \(y_{12}=y_{21}\), \(AD-BC=1\).
Symmetric: \(z_{11}=z_{22}\), \(A=D\).
Cascade: \([\mathbf{T}]_{tot}=[\mathbf{T}_1][\mathbf{T}_2]\).
Series: \([\mathbf{z}]_{tot}=[\mathbf{z}_1]+[\mathbf{z}_2]\).
Parallel: \([\mathbf{y}]_{tot}=[\mathbf{y}_1]+[\mathbf{y}_2]\).

Scattering (S) Parameters — High-Frequency Two-Ports

S-Parameter Definition

Based on travelling waves normalised to a reference impedance \(Z_0\):

\[ a_i = \frac{V_i + Z_0 I_i}{2\sqrt{Z_0}},\quad b_i = \frac{V_i - Z_0 I_i}{2\sqrt{Z_0}} \] \[ \begin{bmatrix}b_1\\b_2\end{bmatrix} = \begin{bmatrix}S_{11} & S_{12}\\ S_{21} & S_{22}\end{bmatrix} \begin{bmatrix}a_1\\a_2\end{bmatrix} \]

Measured with ports terminated in \(Z_0\) (no reflections).

Interpretation

\(S_{11}\)	Input reflection coefficient
\(S_{22}\)	Output reflection coefficient
\(S_{21}\)	Forward transmission (gain)
\(S_{12}\)	Reverse transmission (isolation)

Properties

Reciprocal: \(S_{12} = S_{21}\). Lossless: \(\mathbf{S}^\dagger\mathbf{S} = \mathbf{I}\) (unitary). Matched: \(S_{11} = S_{22} = 0\).

SECTION 11

Fourier Series & Transform

Fourier Series

Periodic \(f(t)\), period \(T=2\pi/\omega_0\):

\[ f(t)=a_0+\textstyle\sum_{n=1}^{\infty}(a_n\cos n\omega_0 t+b_n\sin n\omega_0 t) \] \[ a_0=\tfrac{1}{T}\!\int_0^T f(t)\,dt \] \[ a_n=\tfrac{2}{T}\!\int_0^T f(t)\cos n\omega_0 t\,dt \] \[ b_n=\tfrac{2}{T}\!\int_0^T f(t)\sin n\omega_0 t\,dt \]

Complex form:

\[ f(t)=\textstyle\sum_{n=-\infty}^{\infty}c_n e^{jn\omega_0 t},\; c_n=\tfrac{1}{T}\!\int_0^T f(t)e^{-jn\omega_0 t}dt \]

Symmetry Properties

Symmetry	Effect
Even: \(f(-t)=f(t)\)	\(b_n=0\)
Odd: \(f(-t)=-f(t)\)	\(a_n=0,\ a_0=0\)
Half-wave	Even harmonics = 0

Parseval's Theorem

\[ \tfrac{1}{T}\!\int_0^T |f(t)|^2 dt = \sum|c_n|^2 \]

Average power equals the sum of the harmonic powers.

Waveform DC Values

Square wave: \(a_0=0\); sawtooth: \(A/2\); rectified sine: \(2A/\pi\).

Fourier Transform

Fourier Transform Pair

\[ F(\omega)=\mathcal{F}\{f(t)\}=\textstyle\int_{-\infty}^{\infty}f(t)e^{-j\omega t}dt \] \[ f(t)=\tfrac{1}{2\pi}\textstyle\int_{-\infty}^{\infty}F(\omega)e^{j\omega t}d\omega \]

Key pairs:

\(f(t)\)	\(F(\omega)\)
\(\delta(t)\)	\(1\)
\(1\)	\(2\pi\delta(\omega)\)
\(u(t)\)	\(\pi\delta(\omega)+1/(j\omega)\)
\(e^{-at}u(t)\)	\(1/(a+j\omega)\)
\(\text{rect}(t/\tau)\)	\(\tau\,\text{sinc}(\omega\tau/2)\)
\(\cos\omega_0 t\)	\(\pi[\delta(\omega{-}\omega_0)+\delta(\omega{+}\omega_0)]\)

FT Properties

Linearity	\(aF_1+bF_2\)
Time shift	\(e^{-j\omega t_0}F(\omega)\)
Frequency shift	\(F(\omega-\omega_0)\)
Scaling	\(\tfrac{1}{\|a\|}F(\omega/a)\)
Differentiation	\(j\omega F(\omega)\)
Integration	\(F(\omega)/(j\omega)\)
Convolution	\(F_1\cdot F_2\)
Multiplication	\(\tfrac{1}{2\pi}F_1*F_2\)
Duality	\(F(t)\leftrightarrow 2\pi f(-\omega)\)

Parseval's (Energy)

\[ \int|f(t)|^2 dt = \tfrac{1}{2\pi}\int|F(\omega)|^2 d\omega \]

SECTION 12

Graph Theory & Network Topology

Network Topology

Graph Theory Basics

\(b\)	Number of branches
\(n\)	Number of nodes
\(l\)	Number of links (chords)
Tree branches	\(n-1\)
Links	\(l = b-(n-1)\)

Independent equations:

\[ \text{KCL}: n-1\ \text{(nodes)} \] \[ \text{KVL}: b-(n-1)\ \text{(loops)} \]

Total: \(b\) equations for \(b\) unknowns.

Incidence & Loop Matrices

Incidence matrix \(\mathbf{A}\): \(a_{ij}=+1\) (branch \(j\) leaves node \(i\)), \(-1\) (enters), \(0\) (unconnected).

Loop matrix \(\mathbf{B}\): \(b_{ij}=+1,-1,\) or \(0\).

Network equations:

\[ \mathbf{A}\,\mathbf{i}_b = 0\ \text{(KCL)} \] \[ \mathbf{B}\,\mathbf{v}_b = 0\ \text{(KVL)} \] \[ \mathbf{A}\cdot\mathbf{B}^T = \mathbf{0} \]

Euler's Formula

For a planar graph: \(n - b + \text{meshes} = 1\), hence \(\text{meshes} = b - n + 1\).

Cut-set & Tie-set (Fundamental) Matrices

Fundamental Cut-set Matrix \(\mathbf{Q}\)

For a chosen tree, each twig (tree branch) defines one fundamental cut-set (the twig plus the links crossing it).

\[ q_{ij} = \pm 1, 0 \] \[ \boxed{\mathbf{Q}\,\mathbf{i}_b = \mathbf{0}}\ \text{(KCL)} \]

Size: \((n-1)\times b\); one fundamental cut-set per twig.

Fundamental Tie-set Matrix \(\mathbf{B}\)

Each link (chord) defines one fundamental tie-set: the unique loop formed by that link plus tree branches.

\[ b_{ij} = \pm 1, 0 \] \[ \boxed{\mathbf{B}\,\mathbf{v}_b = \mathbf{0}}\ \text{(KVL)} \]

Size: \((b-n+1)\times b\); one fundamental tie-set per link.

Orthogonality

\[ \mathbf{Q}\,\mathbf{B}^T = \mathbf{0},\quad \mathbf{B}\,\mathbf{Q}^T = \mathbf{0},\quad \mathbf{A}\,\mathbf{B}^T = \mathbf{0} \]

Usage

Reduced KCL through \(\mathbf{Q}\) eliminates redundant node equations; reduced KVL through \(\mathbf{B}\) yields independent loop equations — together giving \(b\) independent equations for \(b\) branch unknowns.

SECTION 13

Coupled Circuits & Transformers

Magnetically Coupled Circuits & Transformers

Mutual Inductance

\[ v_1 = L_1\frac{di_1}{dt} \pm M\frac{di_2}{dt} \] \[ v_2 = \pm M\frac{di_1}{dt} + L_2\frac{di_2}{dt} \] \[ M = k\sqrt{L_1L_2}, \quad 0\le k\le 1 \]

In the frequency domain:

\[ V_1 = j\omega L_1 I_1 \pm j\omega M I_2 \] \[ V_2 = \pm j\omega M I_1 + j\omega L_2 I_2 \]

Dot convention: current entering the dotted terminal makes the mutual voltage aiding.

Ideal Transformer

\[ \frac{V_2}{V_1} = \frac{N_2}{N_1} = n \] \[ \frac{I_1}{I_2} = \frac{N_2}{N_1} = n \] \[ Z_{in} = \frac{Z_L}{n^2}\quad\text{(impedance reflection)} \] \[ S_1 = S_2\quad\text{(lossless)} \]

Energy in Coupled Inductors

\[ W = \tfrac{1}{2}L_1 i_1^2 \pm M i_1 i_2 + \tfrac{1}{2}L_2 i_2^2 \]

Maximum storable energy occurs at \(k=1\) (perfect coupling).

Practical Transformer Equivalent Circuits

T-Model (Coupled Coils)

Two magnetically coupled coils \((L_1, L_2, M)\) can be replaced by a T-network:

\[ L_a = L_1 - M,\; L_b = L_2 - M,\; L_c = M \]

Valid only when both coils share a common node and the dot polarities agree with the substitution.

\(\pi\)-Model (Dual)

An alternative arrangement places three coupled inductors in a \(\pi\) configuration, common in network analysers and SPICE models.

Real Transformer Model

Non-idealities to include:

\(R_1, R_2\): winding resistance (copper loss)
\(L_{l1}, L_{l2}\): leakage inductance
\(L_m\): magnetising inductance
\(R_c\): core loss (hysteresis + eddy currents)

Referred to the primary: \(R_2' = R_2/n^2,\; L_{l2}' = L_{l2}/n^2,\; Z_L' = Z_L/n^2\).

Efficiency & Regulation

\[ \eta = \frac{P_{out}}{P_{out} + P_{Cu} + P_{core}} \] \[ \text{Reg.} = \frac{V_{nl}-V_{fl}}{V_{fl}}\times 100\% \]

SECTION 14

Network Functions, Poles & Zeros

Network Function Classification

Function Classification

Function	Definition	Port
Driving-point Z	\(V_1/I_1\)	Same
Driving-point Y	\(I_1/V_1\)	Same
Voltage transfer	\(V_2/V_1\)	Different
Current transfer	\(I_2/I_1\)	Different
Transimpedance	\(V_2/I_1\)	Different
Transadmittance	\(I_2/V_1\)	Different

\[ H(s)=K\frac{(s-z_1)(s-z_2)\cdots}{(s-p_1)(s-p_2)\cdots} \]

Properties of \(H(s)\)

Poles: \(H(s)\to\infty\); zeros: \(H(s)=0\)
Complex poles always occur in conjugate pairs
\(n\) poles \(\Rightarrow\) an \(n\)th-order system

Pole-zero plot on the complex s-plane showing poles as crosses and zeros as circles, with the left half-plane shaded as the stable region. — Pole–zero map in the complex \(s\)-plane: poles (\(\times\)) and zeros (\(\circ\)) determine the network response, and poles confined to the shaded left half-plane indicate a stable system.

Signal Flow Graphs & Mason's Gain Formula

SFG Terminology

Node: a system variable. Branch: a one-way gain.
Forward path \(P_k\): input-to-output, touching each node once.
Loop \(L_i\): a closed path returning to its start.
Non-touching loops: share no node.

Mason's Gain Formula

\[ T = \frac{Y_{out}}{Y_{in}} = \frac{1}{\Delta}\sum_k P_k\,\Delta_k \]

\(\Delta = 1 - \sum L_i + \sum L_i L_j - \sum L_i L_j L_k + \cdots\) (sums over individual loops, non-touching pairs, triples, …).

\(\Delta_k\): the value of \(\Delta\) with all loops touching \(P_k\) removed.

Procedure

Identify all forward paths \(P_k\).
Identify all loops \(L_i\).
Find the non-touching loop combinations.
Compute \(\Delta\) and each \(\Delta_k\).
Apply Mason's formula.

Use

Block-diagram reduction becomes tedious for three or more feedback loops; a signal flow graph with Mason's formula returns the gain \(T\) in a single step.

SECTION 15

Network Synthesis

Positive Real Functions & Hurwitz Polynomials

Positive Real (PR) Function

A rational \(F(s)=N(s)/D(s)\) is PR if and only if:

\(F(s)\) is real when \(s\) is real
\(\operatorname{Re}\{F(s)\}\ge 0\) whenever \(\operatorname{Re}\{s\}\ge 0\)

Necessary & sufficient tests:

\(N(s)\) and \(D(s)\) are Hurwitz.
Poles and zeros on the \(j\omega\)-axis are simple with real, positive residues.
Degrees of \(N, D\) differ by at most 1.
\(\operatorname{Re}\{F(j\omega)\}\ge 0\) for all \(\omega\).

⇒ Every driving-point impedance or admittance of a passive RLC network is PR.

Hurwitz Polynomial

\(P(s)\) is Hurwitz if all roots lie in the closed LHP (\(\operatorname{Re}\{s\}\le 0\)); strictly Hurwitz if in the open LHP.

Necessary: all coefficients present (no missing terms) and of the same sign.

Continued-fraction test: form the ratio of even to odd parts,

\[ \frac{m(s)}{n(s)} = q_1 s + \cfrac{1}{q_2 s + \cfrac{1}{q_3 s + \cdots}} \]

If all \(q_i \gt 0\), the polynomial is strictly Hurwitz.

Role

PR + Hurwitz together form the realisability conditions for a passive, lumped, linear, time-invariant one-port impedance.

Routh–Hurwitz Stability Criterion

Routh Array

For \(P(s)=a_n s^n + a_{n-1}s^{n-1} + \cdots + a_0\):

\[ \begin{array}{c|cccc} s^n & a_n & a_{n-2} & a_{n-4} & \cdots\\ s^{n-1} & a_{n-1} & a_{n-3} & a_{n-5} & \cdots\\ s^{n-2} & b_1 & b_2 & b_3 & \\ s^{n-3} & c_1 & c_2 & & \\ \vdots & \vdots & & & \\ s^0 & \ast & & & \\ \end{array} \]

with \(b_1 = \dfrac{a_{n-1}a_{n-2}-a_n a_{n-3}}{a_{n-1}}\), \(c_1 = \dfrac{b_1 a_{n-3} - a_{n-1} b_2}{b_1}\), and so on.

Stability Conclusions

Stable (all roots in the LHP) if and only if:

All coefficients of \(P(s)\) are present with the same sign
All entries in the first column of the Routh array have the same sign

The number of sign changes in the first column equals the number of right-half-plane roots.

Special Cases

Zero in the first column: replace it by \(\epsilon\to 0^+\).
An entire row of zeros: form the auxiliary polynomial from the row above, differentiate it, and use the result as the new row.

Foster & Cauer Realisation Forms

Foster Forms (Partial Fractions)

Expand \(Z(s)\) or \(Y(s)\) into partial fractions about its poles.

Foster-I (on \(Z(s)\)): a series of parallel L–C tanks,

\[ Z(s)= \frac{k_0}{s} + k_\infty s + \textstyle\sum_i\frac{2k_i s}{s^2+\omega_i^2} \]

where \(k_0/s\) is a series \(C\), \(k_\infty s\) a series \(L\), and each \(2k_i s/(s^2+\omega_i^2)\) a parallel L–C in series.

Foster-II (on \(Y(s)\)): a parallel combination of series L–C branches (the dual).

Cauer Forms (Continued Fraction)

Expand \(Z(s)\) as a continued fraction.

Cauer-I (about \(s=\infty\)): a ladder of series \(L\), shunt \(C\),

\[ Z(s) = sL_1 + \cfrac{1}{sC_1 + \cfrac{1}{sL_2 + \cdots}} \]

Cauer-II (about \(s=0\)): a ladder of series \(C\), shunt \(L\). Both yield easily realised ladder networks.

LC / RC / RL Immittance

LC: poles and zeros on the \(j\omega\)-axis, alternating.
RC-\(Z\) / RL-\(Y\): poles on the negative-\(\sigma\) axis.
RL-\(Z\) / RC-\(Y\): the dual arrangement.

SECTION 16

Signal Waveforms & Phasor Operations

Important Waveforms & Phasor Operations

Sinusoidal Parameters

\[ v(t)=V_m\cos(\omega t+\phi) \]

\(V_m\)	peak (amplitude)
\(\omega=2\pi f\)	angular frequency (rad/s)
\(T=1/f\)	period (s)
\(\phi\)	phase angle (rad or \(^\circ\))
\(V_{rms}=V_m/\sqrt{2}\)	RMS value
\(V_{avg}=2V_m/\pi\)	half-wave rectified

Step & Ramp Signals

\[ u(t)=\begin{cases}0,&t\lt0\\1,&t\gt0\end{cases},\quad r(t)=tu(t),\quad \delta(t)=\frac{d}{dt}u(t) \]

Phasor Arithmetic

Add (rectangular): \((a_1+jb_1)+(a_2+jb_2)=(a_1+a_2)+j(b_1+b_2)\)

Multiply (polar): \(r_1\angle\theta_1 \cdot r_2\angle\theta_2 = r_1r_2\angle(\theta_1+\theta_2)\)

Divide: \((r_1/r_2)\angle(\theta_1-\theta_2)\)

Conjugate: \((a+jb)^*=a-jb\)

Euler's Identity

\[ e^{\pm j\theta}=\cos\theta\pm j\sin\theta \]

Standard Test Signals — Unified Reference

Test Signals at a Glance

Signal	Time-domain \(x(t)\)	Laplace \(X(s)\)	Fourier \(X(\omega)\)	Use
Impulse	\(\delta(t)\)	\(1\)	\(1\)	Impulse response \(h(t)\)
Unit step	\(u(t)\)	\(1/s\)	\(\pi\delta(\omega)+1/j\omega\)	Step response
Unit ramp	\(t\,u(t)\)	\(1/s^2\)	—	Tracking tests
Parabola	\(\tfrac{1}{2}t^2 u(t)\)	\(1/s^3\)	—	Type-3 inputs
Exponential	\(e^{-at}u(t)\)	\(1/(s+a)\)	\(1/(a+j\omega)\)	Transient probe
Sinusoid	\(\cos(\omega_0 t)u(t)\)	\(s/(s^2+\omega_0^2)\)	\(\pi[\delta(\omega{\mp}\omega_0)]\)	Frequency response
Damped sinusoid	\(e^{-at}\cos(\omega_0 t)u(t)\)	\((s{+}a)/((s{+}a)^2{+}\omega_0^2)\)	—	Underdamped 2nd-order
Rectangular	\(\text{rect}(t/\tau)\)	\((1-e^{-s\tau})/s\)	\(\tau\,\text{sinc}(\omega\tau/2)\)	Pulse response

Interpretation

The impulse response \(h(t)\) fully determines any LTI network; every other response follows by convolution or transform of the appropriate test signal.

SECTION 17

State-Space Analysis

State-Space Representation of Circuits

Standard Form

\[ \boxed{\dot{\mathbf{x}}(t) = \mathbf{A}\,\mathbf{x}(t) + \mathbf{B}\,\mathbf{u}(t)} \] \[ \boxed{\mathbf{y}(t) = \mathbf{C}\,\mathbf{x}(t) + \mathbf{D}\,\mathbf{u}(t)} \]

\(\mathbf{x}\)	state vector (\(n\times 1\))
\(\mathbf{u}\)	input vector (\(p\times 1\))
\(\mathbf{y}\)	output vector (\(q\times 1\))
\(\mathbf{A}\)	system matrix (\(n\times n\))
\(\mathbf{B}\)	input matrix (\(n\times p\))
\(\mathbf{C}\)	output matrix (\(q\times n\))
\(\mathbf{D}\)	feed-through (\(q\times p\))

Choosing State Variables

The natural choice is the energy-storage variables:

\(v_C\) for each capacitor
\(i_L\) for each inductor

The order \(n\) equals the number of independent storage elements.

Formulation steps:

Pick \(v_C, i_L\) as states.
Write \(i_C = C\dot v_C\) and \(v_L = L\,\dot i_L\) using KCL/KVL.
Express \(\dot v_C, \dot i_L\) in terms of the states and inputs.

s-Domain Relation

\[ \mathbf{Y}(s) = \big[\mathbf{C}(s\mathbf{I}-\mathbf{A})^{-1}\mathbf{B}+\mathbf{D}\big]\,\mathbf{U}(s) \]

The poles of \(H(s)\) are the eigenvalues of \(\mathbf{A}\).

SECTION 18

Worked Examples for Quick Revision

Worked Example 1 — Thévenin + Maximum Power Transfer

Problem

Find \(R_L\) for maximum power transfer and the value of \(P_{max}\), given \(V_s = 24\,\text{V}\), \(R_1 = 6\,\Omega\) in series, and \(R_2 = 12\,\Omega\) in parallel with the \(R_L\) terminal pair.

A 24 volt source in series with a 6 ohm resistor feeding a 12 ohm resistor across output terminals A and B, with load resistor R-L connected at A-B. — Circuit for Worked Example 1: a 24 V source with a 6 Ω series resistor and a 12 Ω resistor across terminals A–B, where the load \(R_L\) is connected for maximum power transfer analysis.

Solution

Step 1 — \(V_{Th}\): open-circuit voltage across A–B (voltage divider)

\[ V_{Th} = 24\cdot\frac{12}{6+12} = 16\,\text{V} \]

Step 2 — \(R_{Th}\): kill the source; \(R_1 \parallel R_2\)

\[ R_{Th} = \frac{6\cdot 12}{6+12} = 4\,\Omega \]

Step 3 — maximum power transfer:

\[ R_L = R_{Th} = 4\,\Omega \] \[ P_{max} = \frac{V_{Th}^2}{4 R_{Th}} = \frac{256}{16} = 16\,\text{W} \]

Worked Example 2 — RC Transient Response

Problem

A switch closes at \(t=0\), connecting a 10 V source to a series \(R=2\,\text{k}\Omega\) and \(C=5\,\mu\text{F}\). The capacitor is initially uncharged. Find \(v_C(t)\) for \(t\ge 0\) and the energy stored at \(t=\tau\).

Always use the general form \(x(t) = x(\infty) + [x(0^+)-x(\infty)]e^{-t/\tau}\).

Solution

(i) Initial: \(v_C(0^+) = v_C(0^-) = 0\)

(ii) Final: at \(t=\infty\), \(C\) is open \(\Rightarrow v_C(\infty) = 10\,\text{V}\)

(iii) Time constant:

\[ \tau = RC = 2000\times 5\times 10^{-6} = 10\,\text{ms} \]

(iv) Complete response:

\[ \boxed{v_C(t) = 10\bigl(1 - e^{-t/0.01}\bigr)\,\text{V}} \]

(v) Energy at \(t=\tau\):

\[ v_C(\tau) = 10(1-e^{-1}) = 6.32\,\text{V} \] \[ W = \tfrac{1}{2}Cv^2 = 0.5(5\,\mu\text{F})(6.32)^2 \approx 99.9\,\mu\text{J} \]

Worked Example 3 — AC Power & Power-Factor Correction

Problem

A load draws \(P = 2.4\,\text{kW}\) at \(V_{rms} = 240\,\text{V}\), 50 Hz, with a lagging power factor of 0.6.

Find \(|S|\), \(Q\), and the line current \(I\).
What parallel capacitor \(C\) corrects the power factor to 0.95 lagging?

Solution

(a) \(\theta_1 = \cos^{-1}(0.6) = 53.13^\circ\)

\[ |S| = P/\cos\theta_1 = 2400/0.6 = 4000\,\text{VA} \] \[ Q_1 = |S|\sin\theta_1 = 4000(0.8) = 3200\,\text{VAR} \] \[ I = |S|/V_{rms} = 4000/240 \approx 16.67\,\text{A} \]

(b) \(\theta_2 = \cos^{-1}(0.95)= 18.19^\circ\)

\[ Q_2 = P\tan\theta_2 = 2400(0.3287) = 789\,\text{VAR} \] \[ Q_C = Q_1 - Q_2 = 2411\,\text{VAR} \] \[ C = \frac{Q_C}{\omega V_{rms}^2} = \frac{2411}{2\pi(50)(240^2)} \approx 133\,\mu\text{F} \]

SECTION 19

Common Pitfalls & Exam Tips

Top 10 Exam Pitfalls to Avoid

Circuit-Analysis Traps

Sign convention: forgetting the passive sign convention gives the wrong power sign.
Unit mix-ups: \(\mu\text{F}\) vs pF, rad/s vs Hz — always write units.
Treating a dependent source as independent during Thévenin — keep it active and use a test source to find \(R_{Th}\).
Applying superposition to power (it is nonlinear) — superpose only currents and voltages.
Using peak values in \(P = VI\cos\theta\) instead of RMS.

Transient & AC Traps

Forgetting continuity: \(v_C\) and \(i_L\) cannot jump.
Using \(\tau = RC\) with the total \(R\) instead of the Thévenin \(R\) seen by the capacitor.
Omitting initial-condition sources in the s-domain (\(L i(0^-)\) and \(v(0^-)/s\)).
Mixing \(\omega\) and \(j\omega\) in impedance — \(Z_L = \omega L\) is wrong; use \(j\omega L\).
In three-phase circuits, using line current where phase current is needed (or vice versa).

Golden Rule

Draw the circuit, label polarities, write units on every answer, and sanity-check the sign of power. Most marks are lost to carelessness, not to a lack of knowledge.

Problem-Solving Strategy — One-Page Playbook

DC Circuit

Identify sources and unknowns.
Check for simple series/parallel reductions.
If unknowns \(\le 2\): use a voltage or current divider.
Otherwise decide: nodal (few V-sources) or mesh (planar, few I-sources).
Need only one quantity? ⇒ Thévenin / Norton.
Multiple independent sources? ⇒ superposition.

Transient (1st / 2nd order)

Find \(x(0^-)\) in the old steady state.
Apply continuity to get \(x(0^+)\).
Find \(x(\infty)\) in the new steady state.
Find \(\tau\) (or \(\alpha, \omega_0\) for second order).
Plug into the general solution form.

AC / Phasor

Convert sources to phasors (same \(\omega\)).
Replace \(R, L, C\) by \(Z_R, Z_L, Z_C\).
Apply any DC technique (nodal / mesh / Thévenin).
Solve in the phasor domain.
Convert back to the time domain if needed.

s-Domain

Transform sources and elements (include IC sources).
Solve algebraically for \(X(s)\).
Perform partial-fraction expansion.
Inverse-Laplace each term.
Verify with the initial- and final-value theorems.

SECTION 20

Quick Reference Card

Master Formula Reference Card

DC Circuits

\(V=IR\); \(P=VI=I^2R=V^2/R\)
KCL: \(\sum i=0\); KVL: \(\sum v=0\)
VD: \(V_k=V_s R_k/R_{eq}\); CD: \(I_k=I_s R_{eq}/R_k\)
\(V_{Th}=V_{oc}\); \(R_{Th}=V_{oc}/I_{sc}\)
MPT: \(R_L=R_{Th}\), \(P_{max}=V_{Th}^2/4R_{Th}\)

AC / Phasors

\(Z_R=R\); \(Z_L=j\omega L\); \(Z_C=1/j\omega C\)
\(\mathbf{S}=\mathbf{V}\mathbf{I}^*=P+jQ\); \(\text{pf}=\cos\theta=P/S\)
\(V_{rms}=V_m/\sqrt{2}\); \(P=V_{rms}I_{rms}\cos\theta\)
Resonance: \(\omega_0=1/\sqrt{LC}\); \(Q=\omega_0L/R\)

Transients

\[ x(t)=x(\infty)+[x(0^+)-x(\infty)]e^{-t/\tau} \]

RC: \(\tau=RC\); RL: \(\tau=L/R\)
\(\omega_0=1/\sqrt{LC}\); \(\alpha=R/2L\) (series)

Three-Phase

Y: \(V_L=\sqrt{3}V_p\), \(I_L=I_p\); \(\Delta\): \(V_L=V_p\), \(I_L=\sqrt{3}I_p\)
\(P_T=\sqrt{3}V_LI_L\cos\theta\); \(Z_Y=Z_\Delta/3\)

Transforms

\(\mathcal{L}\{e^{-at}\}=1/(s+a)\); \(\mathcal{L}\{u(t)\}=1/s\)
\(\mathcal{L}\{f'\}=sF-f(0)\)
IVT: \(\lim_{s\to\infty}sF(s)=f(0^+)\)
FVT: \(\lim_{s\to 0}sF(s)=f(\infty)\)
\(H(j\omega)=H(s)|_{s=j\omega}\)

Two-Port

\([z]\): O.C.; \([y]\): S.C.; \([h]\): hybrid
Cascade: \([T]=[T_1][T_2]\); Series: \([z]=[z_1]+[z_2]\)
Reciprocal: \(z_{12}=z_{21}\); Symmetric: \(z_{11}=z_{22}\)
Ideal transformer: \(V_2/V_1=n\), \(Z_{in}=Z_L/n^2\)

⚡

Closing Thought

Master the core, and everything else follows

Every circuit tells a story — read it with Kirchhoff's eyes. Draw it, label it, check the units and the sign of power, and the rest of network theory unfolds from these foundations.