Sources and References

• Nilsson and Riedel, Electric Circuits, 11th ed., Pearson.
• Irwin and Nelms, Basic Engineering Circuit Analysis, 12th ed., Wiley.
• Hayt, Kemmerly, Phillips, and Durbin, Engineering Circuit Analysis, 9th ed., McGraw-Hill.
• Sadiku, Elements of Electromagnetics, 7th ed., Oxford University Press.
• Fitzgerald, Kingsley, and Umans, Electric Machinery, 7th ed., McGraw-Hill.

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Chapter 1: Electromagnetic Foundations

Circuit analysis is an abstraction of electromagnetic theory in the limit where wavelengths of interest are far larger than circuit dimensions. Currents and voltages replace fields, and ideal lumped elements replace distributed regions of space. Understanding where the abstraction comes from, and where it breaks, is prerequisite to fluent circuit design.

1.1 Charge, Current, and Voltage

Electric charge obeys \( F = qE + qv \times B \). Current is the time rate of charge transfer through a cross-section,

\[ I = \frac{dq}{dt}. \]

Voltage is the line integral of electric field between two points; for a conservative field,

\[ V_{ab} = -\int_a^b \mathbf{E}\cdot d\mathbf{l}. \]

Power delivered to a two-terminal element is \( p = vi \), with the passive sign convention that current enters the positive terminal when the element absorbs energy.

1.2 Electromagnetic Laws, Circuit Form

Maxwell’s equations in the low-frequency limit produce Kirchhoff’s laws. Kirchhoff’s current law (KCL) at a node asserts that \( \sum I = 0 \); Kirchhoff’s voltage law (KVL) around a closed loop asserts that \( \sum V = 0 \). Both are consequences of conservation of charge and the conservative (or quasi-conservative) nature of the low-frequency electric field.

1.3 Magnetic Circuits

Magnetic circuits provide a parallel formalism. Magnetic flux \( \Phi \) plays the role of current; magnetomotive force (MMF) \( F = NI \) plays the role of voltage; reluctance \( \mathcal{R} = \ell/(\mu A) \) plays the role of resistance. The magnetic Ohm’s law is \( F = \Phi \mathcal{R} \), and KVL-like and KCL-like laws apply to flux circuits. Iron-cored inductors and transformers are analyzed by this formalism.

\[ \oint \mathbf{H}\cdot d\mathbf{l} = N I,\qquad B = \mu H. \]

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Chapter 2: Circuit Elements

2.1 Passive Elements

Resistors, capacitors, and inductors follow

\[ v = Ri, \qquad i = C \frac{dv}{dt}, \qquad v = L \frac{di}{dt}. \]

Resistors dissipate energy as heat; capacitors store energy \( \frac{1}{2} C V^2 \) in the electric field of the dielectric; inductors store energy \( \frac{1}{2} L I^2 \) in the magnetic field of the windings. All three are idealizations — real resistors have parasitic inductance, real capacitors have equivalent series resistance and inductance, real inductors have winding resistance.

2.2 Sources

Independent voltage and current sources produce prescribed terminal conditions regardless of the attached circuit. Dependent sources (voltage-controlled, current-controlled, of voltage or current) model amplifiers and transistors in small-signal analysis. Source conversions allow transformation between Thevenin and Norton equivalents, \( V_{th} = I_n R \).

2.3 Operational Amplifiers

The op-amp is an idealized dependent-voltage source with infinite gain, infinite input impedance, zero output impedance, and infinite bandwidth. Real op-amps approximate these to varying degrees. With negative feedback, the virtual-short principle (inverting and non-inverting inputs at the same potential) allows quick analysis of inverting, non-inverting, summing, integrator, and differentiator configurations.

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Chapter 3: DC Circuit Analysis

3.1 Node and Mesh Analysis

Nodal analysis writes KCL at each non-reference node; the unknowns are node voltages. Mesh analysis writes KVL around each independent loop; unknowns are mesh currents. Both yield linear systems whose solution is obtained by Gaussian elimination or matrix methods.

3.2 Theorems

Superposition states that in a linear circuit the response to multiple independent sources equals the sum of responses to each source acting alone. Thevenin’s and Norton’s theorems replace a linear network at a pair of terminals by a source and series/parallel impedance. Maximum power transfer to a resistive load \( R_L \) occurs when \( R_L = R_{th} \); the maximum power is \( V_{th}^2/(4 R_{th}) \).

3.3 Nonlinear Elements

Real diodes, transistors, and lamps are nonlinear. A piecewise-linear model linearizes about an operating point and applies linear analysis to small-signal perturbations; iterative solution recovers the nonlinear operating point when closed-form methods fail.

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Chapter 4: First-Order Transients

4.1 RC and RL Natural Response

With no forcing, an RC circuit discharges through a capacitor with time constant \( \tau = RC \),

\[ v(t) = V_0 e^{-t/\tau}. \]

An RL circuit with an initial current and no source decays as

\[ i(t) = I_0 e^{-t/\tau}, \qquad \tau = L/R. \]

Time constants quantify how long transients persist; practically, five time constants is commonly taken as “steady state.”

4.2 Step Response

Applying a constant forcing produces an exponential approach to a new steady state,

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

Boundary conditions at \( t = 0^+ \) are determined from the continuity of capacitor voltage and inductor current (they cannot change instantaneously).

4.3 Energy and Power

Stored energy is released or absorbed during transients. Power dissipated in the resistor equals the rate of loss of stored energy; for the RC discharge, the total energy dissipated equals \( \frac{1}{2} C V_0^2 \).

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Chapter 5: AC Circuit Analysis

5.1 Phasors and Impedance

Sinusoidal steady-state analysis represents voltages and currents as complex phasors. For angular frequency \( \omega \), impedances are \( Z_R = R \), \( Z_L = j\omega L \), \( Z_C = 1/(j\omega C) \). KVL and KCL then apply to phasors exactly as to DC quantities, with complex arithmetic replacing real.

5.2 Power in AC Circuits

Complex power \( S = V I^* = P + j Q \) combines real power \( P \) (watts) and reactive power \( Q \) (vars). Apparent power \( |S| \) is in volt-amperes. Power factor is \( \cos\phi = P/|S| \). Industrial loads with lagging power factor are corrected by shunt capacitors sized to deliver the required \( Q \), reducing current in upstream conductors.

5.3 Frequency Response and Resonance

The transfer function \( H(j\omega) = V_{out}/V_{in} \) describes frequency-dependent behaviour. RLC circuits exhibit resonance at \( \omega_0 = 1/\sqrt{LC} \); series resonance maximizes current, parallel resonance maximizes voltage. Quality factor \( Q = \omega_0 L / R \) relates stored to dissipated energy per cycle and sets bandwidth \( \Delta\omega = \omega_0/Q \).

Bode plots present magnitude (in dB) and phase against log frequency. A first-order low-pass has −20 dB/decade roll-off beyond the corner; a second-order response varies from −40 dB/decade with or without peaking depending on \( Q \).

5.4 Three-Phase and Transformers

Three-phase systems deliver constant instantaneous total power under balanced conditions and efficiently transmit bulk power. Y- and Δ-connections, balanced loads, and per-phase analysis are the standard tools. Ideal transformers scale voltage by turn ratio \( a = N_1/N_2 \) and current by \( 1/a \); they scale impedance by \( a^2 \). Practical transformers depart through winding resistance, leakage inductance, and core losses modelled by an equivalent circuit.