Chapter 7 — Alternating Current

Fig 7.1 — Educational Diagram: Alternating current and voltage

Definition: An alternating current (AC) is an electric current that periodically reverses direction. Alternating voltage similarly changes polarity periodically. Most power distribution uses AC because it is easy to transform between voltages and to transmit efficiently.

Mathematical form: A sinusoidal AC voltage (or current) is written as
v(t) = V_m sin(ωt + φ_v)
i(t) = I_m sin(ωt + φ_i)

Here V_m and I_m are peak (maximum) values, ω = 2πf is the angular frequency (f is frequency in Hz), T = 1/f is the period, and φ_v, φ_i are phase angles. The phase difference between voltage and current is φ = φ_v − φ_i.

Key properties:

• Instantaneous value changes with time and is positive and negative over a cycle; the average over one full cycle for pure sinusoidal v or i is zero.
• Root-mean-square (RMS) values give equivalent DC heating effect: V_rms = V_m/√2, I_rms = I_m/√2 for sinusoidal quantities.
• For purely resistive circuits, voltage and current are in phase (φ = 0). For an inductor voltage leads current by 90° (v leads i). For a capacitor current leads voltage by 90° (i leads v).

Impedance and phasors: In AC analysis we use phasors (complex amplitudes). The impedance Z relates phasor voltage and current: V = I Z. For circuit elements,

• Z_R = R (resistor)
• Z_L = jωL (inductor)
• Z_C = 1/(jωC) = −j/(ωC) (capacitor)

For series RLC, Z = R + j(ωL − 1/(ωC)). Resonance occurs when ωL = 1/(ωC) so the reactive part is zero and current is maximum.

Power in AC circuits: Instantaneous power p(t) = v(t) i(t). For v = V_m sin ωt and i = I_m sin(ωt + φ):
p(t) = V_m I_m sin ωt sin(ωt + φ) = (V_m I_m/2) cos φ − (V_m I_m/2) cos(2ωt + φ).

The average (real) power delivered over a cycle is P_avg = V_rms I_rms cos φ. Here cos φ is the power factor. Reactive power Q = V_rms I_rms sin φ. Apparent power S = V_rms I_rms (in VA), and complex power S_complex = P + jQ.

Fig 7.2 — Educational Diagram: Mathematical representation of AC

Alternating current (AC) is an electric current (or voltage) that varies periodically with time, usually in a sinusoidal manner. The most general sinusoidal instantaneous quantity (voltage or current) is written as:

Instantaneous value: v(t) = V_m sin(ωt + φ)

• V_m = peak (maximum) amplitude
• ω = angular frequency = 2πf (rad/s)
• f = frequency (Hz), T = period = 1/f
• φ = phase constant (rad) — shifts the waveform in time

Key derived quantities:

• RMS (root-mean-square) value — equivalent DC value delivering same average power: V_rms = V_m/√2, I_rms = I_m/√2.
• Average over a full cycle of v(t) or i(t) is zero for a pure sinusoid (because positive and negative halves cancel).

When two sinusoidal quantities of same frequency differ in phase, write them as: v(t) = V_m sin(ωt) and i(t) = I_m sin(ωt + φ). The phase difference φ determines power transfer and instantaneous waveforms.

Phasor representation: Represent a sinusoid by a rotating vector (phasor) in complex plane: Ṽ = V_rms ∠φ. Phasors simplify algebra by replacing derivatives/integrals with multiplication/division by jω (where j = √-1).

Impedance (AC analogue of resistance): For circuit elements at angular frequency ω:

• Resistor: Z_R = R (real)
• Inductor: Z_L = jωL (positive imaginary, voltage leads current by 90°)
• Capacitor: Z_C = 1/(jωC) = -j/(ωC) (negative imaginary, current leads voltage by 90°)

Power in AC circuits:

• Instantaneous power: p(t) = v(t) i(t)
• Average (real) power: P = V_rms I_rms cos φ (cos φ = power factor)
• Reactive power: Q = V_rms I_rms sin φ (stored and returned each cycle)
• Apparent power: S = V_rms I_rms, with |S| = √(P^2 + Q^2)

Using the mathematical forms and phasors simplifies analysis of AC circuits, resonance (when ωL = 1/ωC), and frequency-dependent behaviour in real electrical systems.

Fig 7.3 — Educational Diagram: Waveform characteristics: average, RMS and factors

Overview
Waveform characteristics describe useful numerical measures of a periodic waveform (voltage or current). The main quantities are:

• Average (mean) value — the arithmetic mean of the waveform over one period. For a pure sinusoidal alternating quantity the average over a full cycle is zero.
• RMS (root-mean-square, effective) value — the equivalent DC value that would deliver the same power to a resistive load. It is especially important for power calculations.
• Form factor — ratio Vrms / Vavg_rectified (i.e. Vrms divided by the mean of the absolute value); it indicates how peaky a waveform is relative to its mean magnitude.
• Peak (crest) factor — ratio Vpeak / Vrms; it measures how large the peaks are relative to the effective value.

Definitions (general periodic function f(t) with period T)

Average value: Vavg = (1/T) ∫_0^T f(t) dt

Mean of absolute value (often used when average of signed waveform is zero): Vavg_abs = (1/T) ∫_0^T |f(t)| dt

RMS value: Vrms = sqrt{ (1/T) ∫_0^T [f(t)]^2 dt }

Instantaneous power in a resistor R: p(t) = [v(t)]^2 / R. Average power over one period Pavg = Vrms^2 / R.

Common results for sinusoidal wave v(t) = Vp sin(ωt)

• Average over full cycle: 0
• Average of absolute (mean rectified) for full-wave rectified sine: Vavg_abs = 2 Vp / π
• RMS: Vrms = Vp / √2
• Form factor = Vrms / Vavg_abs = π / (2 √2) ≈ 1.1107
• Peak (crest) factor = Vp / Vrms = √2 ≈ 1.414

Rectified waveforms (useful when diodes are present)

• Half-wave rectified sine (positive half only):
• Vavg (over full period) = Vp / π
• Vrms = Vp / 2
• Peak factor = Vp / (Vp/2) = 2
• Full-wave rectified sine (absolute value of sine):
• Vavg = 2 Vp / π
• Vrms = Vp / √2 (same as unrectified sine because squaring removes sign)

Why RMS matters (physical meaning)
Power dissipated in a resistor depends on the square of instantaneous current or voltage. Vrms condenses the time-varying signal to a DC-equivalent that produces the same heating. For example, when you say household mains is "230 V", you mean 230 V (RMS), not the peak (~325 V).

Practical notes
Different wave shapes give different relationships. A square wave with peak Vp has Vrms = Vp (so crest factor = 1), while a triangular wave has higher form/crest factors. Instruments and standards usually refer to RMS values for power calculations and ratings.

Fig 7.4 — Educational Diagram: AC generator: emf induced

Basic idea
A simple AC generator (alternator) converts mechanical rotation into an alternating electromotive force (emf) using electromagnetic induction. A coil of N turns and area A rotates with constant angular speed ω in a uniform magnetic field B. As the coil rotates, the magnetic flux through the coil changes with time and an emf is induced according to Faraday's law.

Derivation (single coil)
Let the normal to the coil make an angle θ = ωt with the magnetic field. Magnetic flux through the coil: Φ(t) = N B A cos(θ) = N B A cos(ωt). By Faraday's law, induced emf is
E(t) = −dΦ/dt = N B A ω sin(ωt).

Define the maximum (peak) emf E_max = N B A ω. Then the instantaneous emf is
E(t) = E_max sin(ωt).

Key features
- The emf is sinusoidal and alternates sign periodically (hence alternating current when a closed circuit is connected).
- Frequency f = ω / (2π) and period T = 1 / f.
- The average emf over one complete cycle is zero (positive and negative halves cancel).
- RMS (effective) value for a sinusoidal emf: E_rms = E_max / √2.

Direction and Lenz's law
The minus sign in Faraday's law expresses Lenz's law: the induced emf (and resulting current) is such that its magnetic effect opposes the change of flux that produced it. In an AC generator, the direction of induced emf (and current) reverses every half cycle.

Practical construction notes
- To extract AC without reversing connections, slip rings are used on the rotating shaft and brushes contact them. (If a commutator is used instead, the output is DC.)
- Real alternators often use many turns and multiple coils to increase voltage and produce near-sinusoidal output.

When connected to a resistive load R
Instantaneous current: i(t) = E(t)/R = (E_max/R) sin(ωt).
Average power delivered to R: P_avg = (E_rms)² / R = E_max² / (2R).

Summary of physical quantities
- Magnetic flux: Φ(t) = N B A cos(ωt).
- Induced emf: E(t) = −dΦ/dt = N B A ω sin(ωt) = E_max sin(ωt).
- Peak emf: E_max = N B A ω.
- Frequency: f = ω / (2π).
- RMS emf: E_rms = E_max / √2.

Fig 7.5 — Educational Diagram: AC through a resistor

Basic idea: When an alternating voltage v(t) = V_m sin(ωt) is applied across a pure resistor R, the resulting current i(t) is also sinusoidal and given by Ohm's law: i(t) = v(t)/R = I_m sin(ωt), where I_m = V_m/R. In a pure resistor voltage and current are in phase (phase difference φ = 0).

Phasor and impedance view: The impedance of a resistor is Z = R (a real number). Using phasors, Ṽ = Ĩ·R and both phasors point in the same direction (no phase shift).

RMS and peak values: For sinusoidal signals the root-mean-square (rms) values are V_rms = V_m/√2 and I_rms = I_m/√2. RMS values give the DC-equivalent heating effect.

Instantaneous and average power: Instantaneous electrical power absorbed by the resistor is p(t) = v(t)·i(t) = V_mI_m sin²(ωt) = (V_mI_m/2)[1 - cos(2ωt)]. This is always nonnegative for an ideal resistor and contains a steady (DC) part plus a component oscillating at double the supply frequency (2ω). The time-average (real) power over a cycle is P_avg = V_rmsI_rms = I_rms²R = V_rms²/R. For a resistor the power factor cosφ = 1.

Energy conversion: A resistor converts the electrical energy of the AC source into heat (Joule heating). Because voltage and current are in phase, all the delivered power is dissipative, with no reactive exchange of energy between source and resistor.

Notes: In real components (e.g., lamp filaments) resistance may vary with temperature during each cycle, slightly distorting the sinusoid and changing the simple ideal relations. But the ideal-resistor model is very useful and accurate for many practical resistive loads.

Fig 7.6 — Educational Diagram: AC through an inductor

Basic law: For an inductor of inductance L, the voltage across it is v(t) = L (di/dt). In steady-state sinusoidal AC, this determines the relation between v and i.

SINUSOIDAL EXCITATION — phasor result: Let i(t) = I_m sin(ωt). Then

v(t) = L (di/dt) = ωL I_m cos(ωt) = ωL I_m sin(ωt + 90°).

So the voltage across a pure inductor leads the current by 90° (or the current lags the voltage by 90°).

Inductive reactance and impedance: The magnitude of the opposition offered to AC is the inductive reactance

X_L = ωL = 2πfL.

In complex form (phasors) the impedance of a pure inductor is Z = jωL = jX_L, so V = Z I = jωL I.

RMS relations: If I_rms is the current rms and V_rms the voltage rms, then

V_rms = I_rms X_L (for a pure inductor).

Instantaneous and average power: Instantaneous power p(t) = v(t)i(t). For i(t) = I_m sin(ωt) and v(t) = V_m sin(ωt + 90°) with V_m = ωL I_m,

p(t) = V_m I_m sin(ωt + 90°) sin(ωt) = (V_m I_m / 2) sin(2ωt).

This oscillates at frequency 2ω and has zero average over a cycle: the inductor absorbs energy in one part of the cycle and returns it in the next, so the average (real) power P = 0 for a pure inductor. The energy stored at an instant is W = (1/2) L i^2.

Reactive power: The circuit exchanges reactive power (VAR). For a pure inductor the reactive power magnitude is

Q = V_rms I_rms = I_rms^2 X_L = V_rms^2 / X_L.

Physical picture: A changing current in the inductor produces a changing magnetic flux. The source does work to build the magnetic field when current increases, storing energy (1/2 L i^2). When current decreases, that field returns energy to the circuit (induced emf opposes change). Because energy is alternately stored and returned each half-cycle, no net energy is dissipated.

Important classroom points:

• Voltage leads current by 90° in a pure inductor.
• Inductive reactance increases linearly with frequency, so an inductor opposes high-frequency currents more.
• Pure inductors do not consume real power; they exchange reactive power with the source.

Suggested quick experiment: Connect a sinusoidal generator across a known inductor and observe on an oscilloscope the two traces for v(t) and i(t). You will see v(t) lead i(t) by ~90°. Measure amplitudes and verify V_rms = I_rms X_L.

Fig 7.7 — Educational Diagram: AC through a capacitor

Basic relation: For a capacitor of capacitance C, the instantaneous current i(t) is related to the time‑rate of change of voltage v(t) across it by

i(t) = C · dv(t)/dt

Sinusoidal excitation: If the applied voltage is sinusoidal, v(t) = V_m sin(ωt), then

i(t) = C · d/dt[V_m sin(ωt)] = ω C V_m cos(ωt) = I_m sin(ωt + π/2)

Thus the capacitor current leads the applied voltage by 90° (current is maximum when voltage is changing fastest).

RMS values and capacitive reactance: Using rms values V_rms = V_m/√2 and I_rms = I_m/√2, we get

I_rms = ω C V_rms = V_rms / X_C

X_C = 1 / (ω C) = 1 / (2π f C)

X_C (capacitive reactance) is the effective opposition to AC; it decreases with increasing frequency. At f = 0 (DC) X_C = ∞ so no steady DC current; at very high f X_C → 0 and the capacitor behaves like a short.

Impedance and phasor form: In phasor notation the capacitor has impedance

Z_C = 1 / (j ω C) = -j / (ω C)

On the complex plane Z_C lies on the negative imaginary axis. The current phasor I leads the voltage phasor V by 90°.

Power: Instantaneous power p(t) = v(t)i(t). For a pure capacitor with v = V_m sin(ωt) and i = I_m sin(ωt + π/2), p(t) is oscillatory with zero average over a cycle. Hence a pure capacitor consumes zero average (real) power; it exchanges reactive power with the source. Reactive power magnitude Q = V_rms I_rms.

Energy storage: Energy stored in the capacitor at any instant is

U(t) = 1/2 C [v(t)]²

During each cycle energy is alternately stored and returned to the circuit.

Physical insight: A capacitor resists changes in voltage by accumulating or releasing charge. Under AC, the changing voltage causes a displacement current even where no conduction current flows (important in analysis of circuits and electromagnetic theory).

Important practical consequences:

• Capacitors are used for coupling/decoupling in AC circuits: they pass AC (especially high frequencies) and block DC.
• They form frequency‑dependent elements in filters: high frequencies pass more easily (low X_C).
• In power systems they supply reactive power (power factor correction) to compensate inductive loads.
• At very high frequency a capacitor behaves like a short; at low frequency like an open circuit.

Fig 7.8 — Educational Diagram: Phase relationships and phasor diagrams

Overview
In alternating-current (AC) circuits the voltages and currents are sinusoidal functions of time. Phase relationship describes how one sinusoid is shifted in time relative to another. A phasor is a rotating vector representation of a sinusoid used to simplify analysis of steady-state AC by converting time dependence into vector algebra.

Phase and phase difference
A sinusoidal quantity can be written as x(t)=X_m sin(ωt+φ) where X_m is amplitude, ω angular frequency and φ the phase angle. If two quantities are x1(t)=A sin(ωt+φ1) and x2(t)=B sin(ωt+φ2), their phase difference is Δφ=φ2−φ1. If Δφ>0, x2 leads x1 by Δφ; if Δφ<0, x2 lags x1 by |Δφ|.

Phasor concept
Represent x(t)=X_m cos(ωt+φ) by a complex (or vector) phasor X̅ = X_m∠φ (or RMS value X_rms∠φ). The time function is the real part of the rotating phasor: x(t)=Re{X̅ e^{jωt}}. For steady-state analysis we drop the e^{jωt} factor and operate on phasors using complex algebra. Phasor diagrams are static drawings of these vectors showing magnitudes and phase angles.

Phase relations for basic circuit elements

• Pure resistor (R): voltage and current are in phase (φ_v−φ_i=0). Phasors V and I coincide directionally.
• Pure inductor (L): v_L = L di/dt ⇒ v_L leads i by 90° (v_L = jωL I in phasor form). On a phasor diagram V_L is 90° ahead of I.
• Pure capacitor (C): i_C = C dv/dt ⇒ i_C leads v_C by 90° (V_C = I/(jωC) = −j(1/ωC) I). On a phasor diagram I is 90° ahead of V_C.

Composite circuits (series RLC)
For a series combination, the total voltage phasor V̅ = I̅(R + j(ωL − 1/ωC)) = I̅ Z̅. The impedance Z̅ has real part R and imaginary part X = ωL − 1/ωC. The current lags or leads the applied voltage by angle φ = arg(Z̅) = tan^{-1}((ωL − 1/ωC)/R).

Using phasors

• Replace time-varying sinusoids by phasors (magnitudes and angles).
• Apply Ohm's law with complex impedances: V̅ = I̅ Z̅.
• Add voltages as vector (phasor) sums; use right-angle geometry for reactive drops.
• Convert back to time domain by multiplying phasor by cos(ωt) and taking the real part if needed.

Practical significance
Phasor diagrams give visual insight into phase relationships, power factor (cosφ), reactive vs real power, resonance (when ωL=1/ωC so X=0) and how inductive or capacitive behavior makes current lag or lead voltage.

Fig 7.9 — Educational Diagram: Impedance and reactance

Overview
In alternating current (AC) circuits, elements oppose current flow not only by resistance (R) but also by frequency-dependent opposition called reactance. The combined opposition of resistance and reactance is called impedance (Z). While resistance dissipates energy as heat, reactance stores and returns energy to the circuit (in magnetic fields for inductors and electric fields for capacitors).

Reactance
Reactance is the opposition offered by inductors and capacitors to the change of current in an AC circuit. It depends on angular frequency ω = 2πf (f = frequency in Hz).

• Inductive reactance (X_L): X_L = ωL = 2πfL. It increases linearly with frequency. It is represented as a positive imaginary quantity (+jX_L).
• Capacitive reactance (X_C): X_C = 1/(ωC) = 1/(2πfC). It decreases with increasing frequency. In complex notation it is −jX_C (i.e., negative imaginary).

Impedance (Z)
Impedance generalizes resistance to AC and is a complex quantity: Z = R + jX, where X is the net reactance (X = X_L − X_C). The magnitude |Z| gives the effective opposition to current and the argument (phase) gives the phase shift between voltage and current.

Key relations
For a series R-L-C circuit: Z = R + j(X_L − X_C). The magnitude and phase are given by:

• |Z| = sqrt(R² + (X_L − X_C)²)
• φ = arctan((X_L − X_C)/R) where φ > 0 means current lags voltage (inductive), φ < 0 means current leads voltage (capacitive).

Resonance
When X_L = X_C the net reactance is zero (X = 0). In a series RLC circuit this gives minimum impedance Z = R and maximum current at the resonant angular frequency ω₀ = 1/√(LC) (or f₀ = 1/(2π√(LC))). At resonance the phase angle φ = 0 (voltage and current in phase).

Power and power factor
Average (real) power delivered to the circuit: P = V_rmsI_rmscosφ, where cosφ is the power factor. Reactive power Q = V_rmsI_rmssinφ (measured in VAR).

Interpretation and sign convention
Reactances are frequency dependent and have sign in complex notation: inductive reactance contributes +jX, capacitive contributes −jX. Magnitudes X_L and X_C are positive numbers (units: ohm).

Why this matters (intuition)
- At low frequency an inductor behaves like short (low X_L)? actually at f→0 X_L→0 so it behaves like a short; a capacitor behaves like an open (X_C→∞).
- At high frequency an inductor behaves like an open (X_L→∞); a capacitor behaves like a short (X_C→0).
This frequency dependence is used in filters, tuning circuits, power factor correction and in transformers and motors.

Fig 7.10 — Educational Diagram: Complex representation and phasors as complex numbers

What is a phasor?
Any sinusoidal quantity v(t)=V_m cos(ωt+φ) can be represented by a rotating vector (phasor) in the complex plane. The phasor encodes the amplitude and phase but not the time variation. Mathematically,

v(t)=V_m cos(ωt+φ)=Re{V_m e^{j(ωt+φ)}}=Re{V˜ e^{jωt}}, where V˜=V_m e^{jφ} is the complex phasor (j = √-1).

Phasor as a complex number
A phasor is a complex number in polar form: V˜=V_m ∠φ = V_m(cosφ + j sinφ). In practice we often use RMS phasors: V˜_rms = (V_m/√2)∠φ. The physical instantaneous quantity equals the real part of the product of the phasor and e^{jωt}.

Why complex numbers?
Using complex algebra converts differential equations (in time) into algebraic equations (in phasor domain). Time differentiation corresponds to multiplication by jω and integration to division by jω. This makes circuit analysis with sinusoidal steady states simple.

Key operations
- Differentiation: if v(t) ↔ V˜, then dv/dt ↔ jωV˜.
- Integration: ∫v dt ↔ V˜/(jω).
- Addition/subtraction: phasors add as complex numbers (vector sum).

Impedance and Ohm's law (phasor form)
For sinusoidal steady state, Ohm's law becomes V˜ = I˜ Z, where Z is complex impedance:
Z_R = R (real), Z_L = jωL (inductive, positive imaginary), Z_C = 1/(jωC) = -j/(ωC) (capacitive, negative imaginary).

Phase relationships
- Across a resistor, voltage and current are in phase.
- Across an inductor, voltage leads current by 90° (V˜_L = jωL I˜).
- Across a capacitor, voltage lags current by 90° (V˜_C = (1/jωC) I˜).

Power in phasor form
With RMS phasors V˜_rms and I˜_rms, the complex power is S = V˜_rms I˜*_rms (where * denotes complex conjugate). Real (average) power P = Re{S} = V_rms I_rms cosφ and reactive power Q = Im{S} = V_rms I_rms sinφ, where φ is the phase difference between V and I.

How to use phasors to solve circuits (procedure)
1. Express sources and initial quantities as phasors (usually RMS).
2. Replace circuit elements with their impedances Z.
3. Use algebraic circuit laws (KCL, KVL, Ohm’s law) with complex numbers to find phasors of currents/voltages.
4. Convert phasor results back to time domain by multiplying by e^{jωt} and taking the real part, or express final answers as magnitude ∠ phase.

Fig 7.11 — Educational Diagram: Series RLC circuit

Definition and circuit: A series RLC circuit consists of a resistor R, an inductor L and a capacitor C connected in series to an alternating voltage source V(t)=V_m sin(ωt). In steady-state AC analysis we study sinusoidal response at angular frequency ω.

Phasor and impedance approach: Use phasors and complex impedances. Impedance of elements: Z_R = R (real), Z_L = jX_L = jωL, Z_C = -jX_C = -j/(ωC). Total impedance Z = R + j(ωL - 1/(ωC)). The magnitude of impedance is |Z| = sqrt(R^2 + (X_L - X_C)^2) where X_L = ωL and X_C = 1/(ωC). The steady-state current (rms) is I = V_rms / |Z| and the phase angle between supply voltage and current is φ = arctan((X_L - X_C)/R).

Voltage drops and phasor diagram: Current I is common to all components. V_R = I R is in phase with I. V_L = I X_L leads I by 90 degrees; V_C = I X_C lags I by 90 degrees. On a phasor diagram take I along reference axis, plot V_R on same axis, V_L upward by 90°, V_C downward by 90°; resultant supply voltage phasor V is vector sum of V_R and (V_L - V_C).

Resonance: Resonance occurs when reactances cancel: X_L = X_C or ω_0 L = 1/(ω_0 C). Resonant angular frequency ω_0 = 1/sqrt(LC), resonant frequency f_0 = ω_0/(2π) = 1/(2π sqrt(LC)). At resonance impedance is minimum and equals R, so I_max = V_rms / R. Although the supply sees only R, magnitudes of V_L and V_C can be large and equal (but opposite in phase), leading to voltage magnification across L or C.

Quality factor and bandwidth: The quality factor Q measures sharpness of resonance. For series RLC, Q = ω_0 L / R = 1/(R) sqrt(L/C). Bandwidth Δω = ω_0 / Q = R / L (in rad/s). Half-power (–3 dB) angular frequencies satisfy ω_2 - ω_1 = Δω.

Power: Average (real) power absorbed by the circuit is P = V_rms I_rms cosφ = I_rms^2 R. Reactive power (Q_reactive) is I_rms^2 (X_L - X_C). At resonance cosφ = 1 and reactive power net is zero.

Transient behavior: If switched, the circuit shows damped oscillations. Relevant parameters: damping coefficient α = R/(2L), natural (undamped) frequency ω_0 = 1/sqrt(LC), and damped angular frequency ω_d = sqrt(ω_0^2 - α^2). For α < ω_0 the response is underdamped oscillatory; for α > ω_0 it is overdamped.

Practical notes: Series RLC circuits are used for frequency selection (band-pass behavior), tuning radio receivers, impedance matching and measuring component values. At resonance large voltages across L and C require care in high-Q circuits.

Fig 7.12 — Educational Diagram: Parallel RLC circuit

Definition: A parallel RLC circuit (also called a tank circuit) consists of a resistor (R), an inductor (L) and a capacitor (C) connected in parallel across an alternating voltage source. Each branch carries its own current; the total supply current is the phasor sum of branch currents.

Branch currents and admittance: If the applied voltage is V (phasor), branch currents are

• I_R = V/R (in phase with V)
• I_C = V · ωC (leads V by 90°)
• I_L = V / (ωL) (lags V by 90°)

Y = 1/R + j( ωC - 1/(ωL) ) = G + jB

Resonance in parallel RLC: Resonance occurs when the net susceptance is zero (B = 0):

ω_0 C - 1/(ω_0 L) = 0 ⇒ ω_0 = 1/√(LC), f_0 = 1/(2π√(LC)).

At resonance the inductive and capacitive branch currents are equal in magnitude and opposite in phase (I_L = I_C), so they cancel in the supply current. The total admittance reduces to G = 1/R, thus the circuit presents a maximum impedance to the source equal to R and the supply current is minimum and in phase with V.

Impedance and phase:

• Y(ω) = 1/R + j(ωC - 1/(ωL))
• |Z(ω)| = 1/|Y| = R / √(1 + [R(ωC - 1/(ωL))]^2)
• Phase angle (current relative to voltage): φ = arctan[ R(ωC - 1/(ωL)) ]. Positive φ means current leads voltage (capacitive), negative means current lags (inductive).

Quality factor and bandwidth:

• Resonant angular frequency: ω_0 = 1/√(LC) (as above).
• Quality factor (parallel): Q_p = R · √(C/L) = R / (ω_0 L) = ω_0 R C.
• Bandwidth (angular): Δω = ω_0 / Q_p. In frequency: Δf = f_0 / Q_p. The half-power frequencies f1 and f2 are where |Z| falls to 1/√2 of its maximum at resonance; Δf = f2 - f1.

Important physical points:

• At resonance the circuit behaves as a pure resistance R (maximum impedance), opposite to series resonance where impedance is minimum.
• Branch currents I_L and I_C can be much larger than the supply current (energy oscillates between L and C), especially for high Q.
• Parallel resonance is used for frequency selection (high impedance at unwanted frequencies) and in oscillators and filters.

Fig 7.13 — Educational Diagram: Resonance in RLC circuits

What is resonance?
Resonance in an RLC circuit occurs when the inductive reactance (X_L = ωL) equals the capacitive reactance (X_C = 1/(ωC)). At this angular frequency (ω0) the reactive parts cancel each other and the circuit behaves as a purely resistive circuit. Energy oscillates between the inductor and the capacitor with minimum net reactive energy exchange with the source.

Resonance condition and frequency
Resonance occurs when ωL = 1/(ωC), giving the resonant angular frequency ω0 = 1/√(LC). The resonant (ordinary) frequency is f0 = ω0/(2π) = 1/(2π√(LC)).

Series vs Parallel resonance — key differences

• Series RLC: R, L and C are in series. At resonance the total impedance is minimum (Z_min = R), so the current is maximum. The circuit acts like a low-impedance path.
• Parallel RLC: R, L and C are in parallel. At resonance the total impedance is maximum (for a high-Q circuit) so the supply current is minimum and the branch currents can be large and nearly cancel. The circuit behaves like a high-impedance path.

Why resonance matters physically
At resonance the inductor and capacitor repeatedly exchange equal amounts of magnetic and electric energy. Because their voltages/currents are equal and opposite, the net reactive effect on the source is zero. This leads to extreme behavior: large currents (series) or very high voltages across components (if Q is high), sharp frequency selectivity, and strong amplification of response near f0.

Phase behavior
The phase angle between supply voltage and current φ is given by φ = arctan((ωL - 1/(ωC))/R). At resonance ω = ω0 the phase angle is zero (voltage and current in phase) in a series circuit. Below resonance the circuit is capacitive (current leads), above resonance it is inductive (current lags).

Quality factor and bandwidth
The quality factor Q measures sharpness of the resonance. For a series RLC, Q = ω0 L / R = (1/R)√(L/C). A higher Q means a narrower and taller resonance peak. The bandwidth (angular) is Δω = ω0/Q; for a series circuit this reduces to Δω = R/L. In ordinary frequency units, Δf = Δω/(2π) and Q = f0/Δf.

Power at resonance
Average power delivered to a series RLC at resonance is P = V_rms I_rms cosφ = V_rms^2 / R because cosφ = 1. Although branch voltages across L and C can be large, net reactive power exchanged with the source is zero.

Fig 7.14 — Educational Diagram: Quality factor and bandwidth

Overview
Quality factor (Q) measures the sharpness of resonance of an RLC circuit. Bandwidth (Δω or Δf) is the range of frequencies around the resonant frequency where the circuit response is appreciable (commonly defined between the half-power points).

Resonance (series RLC)
Resonant angular frequency: ω₀ = 1/√(LC). At resonance the inductive and capacitive reactances cancel and the circuit current is maximum.

Definition of Quality factor
Q quantifies how underdamped or sharp the resonance is. Two equivalent practical definitions:

• Frequency definition: Q = ω₀ / Δω = f₀ / Δf, where Δω (Δf) is the bandwidth between the two half-power (−3 dB) frequencies.
• Energy definition: Q = 2π × (maximum energy stored in reactive elements) / (energy dissipated per cycle in the resistor).

Series RLC formulas (common)
For a series RLC circuit at resonance:

• Q = (ω₀ L) / R = (1/R) × √(L/C).
• Bandwidth (angular) Δω = ω₂ − ω₁ = R / L.
• Thus Q = ω₀ / Δω = (ω₀ L) / R.
• Half-power condition: I(ω₁) = I(ω₂) = I_max/√2 (power ∝ I²).

Parallel RLC
For a parallel RLC circuit (high-Q assumption): Q ≈ R / (ω₀ L) = R × √(C/L). Bandwidth relation still Δω = ω₀/Q.

Physical meaning
High Q: narrow, sharp resonance (small bandwidth), stores energy many cycles before dissipation (useful in tuned circuits and filters). Low Q: broad resonance (large bandwidth), energy dissipates quickly (useful for damping).

Typical classroom derivation sketch (series)
Start from impedance Z(ω) = R + j(ωL − 1/ωC). Current amplitude I(ω) = V/Z. At resonance ω0 = 1/√(LC) current is maximum. Solve |I(ω)| = |I(max)|/√2 for ω1 and ω2 to get Δω = ω2 − ω1 = R/L.

Practical notes
Q depends on losses: conductor resistance, dielectric loss, radiation, loading by measurement devices. Frequency Q (f0) used in communications: Q = f0/Δf.

Fig 7.15 — Educational Diagram: Power in AC circuits

Introduction
In AC circuits voltage and current vary sinusoidally with time. Power in AC circuits has several components: instantaneous power p(t), average (real) power P, reactive power Q, apparent power S, and the power factor (pf). These quantities describe how energy is transferred, stored temporarily, and returned in circuits containing resistors, inductors and capacitors.

Basic sinusoidal expressions
Let v(t) = Vm cos(ωt) and i(t) = Im cos(ωt − φ), where φ is the phase difference between voltage and current (φ = φv − φi). RMS values are Vrms = Vm/√2 and Irms = Im/√2.

Instantaneous power
p(t) = v(t)·i(t)
     = Vm Im cos(ωt) cos(ωt − φ)
Using identity 2 cos A cos B = cos(A+B)+cos(A−B),
 p(t) = (Vm Im /2)[cos φ + cos(2ωt − φ)].

Thus instantaneous power has two parts: a constant term (Vm Im /2) cos φ and a time-varying term at frequency 2ω. The constant part represents net energy converted to heat or work per unit time; the oscillating part represents energy exchanged between source and reactive elements (stored and returned each cycle).

Average (Real) Power
Average over a cycle eliminates the 2ω term. Average (real) power P is
P = (Vm Im /2) cos φ = Vrms Irms cos φ.
Units: watts (W). This is the useful power converted to heat, mechanical work, etc.

Reactive Power
Reactive power Q measures the power alternately stored and returned by inductors and capacitors. It is defined as
Q = Vrms Irms sin φ.
Units: volt–ampere reactive (VAR). Q is positive for net inductive behavior (current lags voltage) and negative for net capacitive behavior (current leads).

Apparent Power and Complex Power
Apparent power S is the product of RMS voltage and RMS current magnitude:
S = Vrms Irms (units: VA).
Complex power is S_complex = P + jQ. Its magnitude |S_complex| = S. In phasor notation S = V_rms · I*_rms (conjugate of I phasor), and the phase of S equals φ (so cos φ is the power factor).

Power Factor
Power factor (pf) = cos φ = P / S. It indicates the fraction of apparent power doing real work. pf = 1 for purely resistive loads, pf = 0 for purely reactive loads. A low pf increases current for the same P, causing higher losses in supply lines; hence industries use power-factor correction (capacitors or synchronous condensers).

Special cases

• Pure resistor (φ = 0): i in phase with v. p(t) = Vm^2/(2R)(1 + cos 2ωt). Average P = Vrms^2 / R = Vrms Irms.
• Pure inductor (φ = +90°): current lags by 90°. p(t) = (Vm Im /2) sin 2ωt, average P = 0 (no net energy transfer, only exchange).
• Pure capacitor (φ = −90°): current leads by 90°. p(t) = −(Vm Im /2) sin 2ωt, average P = 0.

Series RLC circuit
For a series circuit with impedance Z = R + j(XL − XC) and φ = arctan((XL − XC)/R):
P = Vrms^2 (R / |Z|^2) = Vrms Irms cos φ,
Q = Vrms^2 ((XL − XC) / |Z|^2) = Vrms Irms sin φ,
S = Vrms^2 / |Z| = Vrms Irms.

Practical notes
- Power meters measure real power P. Utility bills charge for real energy (kWh), but industrial customers may be penalized for low power factor.
- Power factor correction uses capacitors (to cancel inductive Q) so that current and voltage are more in phase, reducing losses and required line capacity.
- Reactive power is essential for maintaining voltage levels in power systems; transmission and generation dispatch consider P and Q separately.

Fig 7.16 — Educational Diagram: Power factor and correction

Definition: Power factor (pf) of an AC circuit is the cosine of the phase angle (φ) between the supply voltage and the supply current: pf = cos φ. It is the ratio of real (useful) power to apparent power.

Why it matters: A low power factor means larger current is required to deliver the same real power. Larger current -> higher I2R losses in lines and transformers, larger conductor sizes, higher electricity bills (penalties) for industrial consumers. Improving (correcting) pf reduces current, losses and cost.

Types:

• pf = 1 (unity): current and voltage in phase (purely resistive load).
• pf < 1, lagging: current lags voltage (inductive loads such as motors, transformers, coils). Most industrial loads are lagging.
• pf < 1, leading: current leads voltage (capacitive loads such as large capacitor banks, some power electronic supplies).

Power triangle (visual concept): Represent apparent power S (VA) as the hypotenuse, real power P (W) as the adjacent side, and reactive power Q (VAR) as the opposite side. Angle between S and P is φ. pf = P/S = cos φ.

Reactive power and its role: Reactive power Q (measured in VAR) does no net work but is needed to sustain magnetic and electric fields in inductive/capacitive elements. Inductive loads absorb positive Q (lagging); capacitors supply negative Q (leading). By adding capacitors, we supply some of the reactive power locally and reduce the reactive demand from the source — thus improving pf.

Correction (how it’s done): Power factor correction is usually done by connecting capacitors (fixed or switched banks) in parallel with the load. For varying loads, automatic switched capacitor banks or synchronous condensers are used. The goal may be to reach a target pf (e.g., 0.95) or unity, without overcorrection (leading pf) which can cause problems.

Steps to size a capacitor for pf correction (single-phase or per-phase balanced three-phase):

1. Calculate initial reactive power Q1 = P tan φ1, where φ1 = arccos(pf_initial).
2. Decide desired pf (pf2) and compute Q2 = P tan φ2, where φ2 = arccos(pf2).
3. Required capacitor reactive power Qc = Q1 − Q2. (For correction to unity pf, Q2 = 0 so Qc = Q1.)
4. For single-phase (voltage V rms, frequency f): C = Qc / (V^2 ω), where ω = 2πf.

Practical considerations and limits:

• Avoid overcorrection (leading pf) unless intended—can cause resonance with system inductance and amplify harmonics.
• Use step-switched capacitor banks or automatic PF controllers for variable loads.
• Harmonics from power electronics may require tuned or detuned filters (with reactors) rather than plain capacitors.
• Utilities often charge penalties for low pf; correcting reduces demand charges and losses.

Measurement and devices: Power factor meters, three-phase power analysers and energy meters with pf measurement are used. Correction equipment includes capacitor banks (fixed/switched), synchronous condensers and power electronic PF correctors.

Fig 7.17 — Educational Diagram: Series and parallel resonance practical consequences

Overview
Resonance in RLC circuits occurs when the inductive reactance equals the capacitive reactance (ωL = 1/ωC). At this resonant angular frequency ω0 = 1/√(LC) (f0 = 1/(2π√(LC))), the circuit shows extreme behaviour: series resonance gives minimum impedance and maximum current; parallel (anti-)resonance gives maximum impedance and minimum source current. Both types produce large circulating reactive voltages/currents inside the circuit while the net reactive component seen by the source cancels.

Practical consequences — what changes at resonance

• Series resonance: circuit impedance Z is minimum (≈ R), so source current I = V/Z is maximum. Though source sees only R, the inductor and capacitor individually can have voltages much larger than the supply (voltage magnification).
• Parallel resonance: input impedance Z is maximum, so source current is minimum. Branch currents in L and C are large and nearly equal/opposite, producing circulating currents. The voltage across the parallel combination can be much larger than the applied current would suggest.
• Quality factor Q controls sharpness: a high Q gives a sharp, narrow resonance peak (good selectivity) but large internal voltages/currents; a low Q gives broader response and less magnification.
• Bandwidth and selectivity: bandwidth Δf = f0/Q. Narrow bandwidth means better frequency selectivity (useful in tuning), but slower response and higher sensitivity to component variations.
• Power and heating: large circulating currents at resonance cause power dissipation in resistances (heating) and can damage components if not limited.
• Unwanted resonance in power systems: capacitor banks interacting with line inductances can create harmful overvoltages or oscillations — requires detuning or damping.

Practical roles and implications

• Tuning and filtering: Resonant circuits are used in radio/TV front-ends and RF filters to select desired frequencies (parallel or series tuned circuits depending on design).
• Voltage/current magnification: Used in RF amplifiers and oscillators where high voltages or currents at a particular frequency are needed (resonant transformers, tank circuits).
• Wireless power transfer: Resonant inductive coupling increases power transfer efficiency at the resonant frequency of transmitter and receiver coils.
• Protection and design trade-offs: Designers add damping resistors, detune capacitor banks, or use series reactors to avoid destructive resonance in power distribution.

Summary
Series resonance → minimum impedance, maximum current, voltage magnification across L or C, used in band-pass type functions. Parallel resonance → maximum impedance, minimum source current, large internal branch currents and voltage, used in frequency-selective high-impedance networks. Both require attention to Q, bandwidth and safety measures to prevent damage from large internal magnitudes.

Fig 7.18 — Educational Diagram: Measurement of AC quantities

What we measure in AC: An alternating quantity (voltage or current) changes with time. Important numbers used to describe AC are instantaneous value v(t), peak (maximum) value Vm, peak-to-peak value Vpp, average value over a period, and root-mean-square (RMS) value. For sinusoidal AC the waveform is v(t)=Vm sin(ωt).

Instantaneous, peak and peak-to-peak: Instantaneous value is v(t). Peak value Vm is the maximum magnitude of v(t). Peak-to-peak Vpp = 2Vm.

Average value: The algebraic average of a pure sine over a full period is zero. For practical measurement (when using rectifiers) we often use the average of the absolute (rectified) waveform. For a full-wave rectified sinusoid the average (mean of absolute value) is Vavg = 2Vm/π.

RMS value (most important): The RMS value of a periodic signal is the equivalent DC value that would deliver the same average power to a resistive load. Mathematically, Vrms = sqrt((1/T) ∫0^T [v(t)]^2 dt). For a sinusoid this evaluates to Vrms = Vm/√2. Similarly Irms = Im/√2. RMS is the value used in ratings: e.g. household supply 230 V is Vrms; its peak is ≈ 230×√2 ≈ 325 V.

Form factor and peak factor: Form factor = Vrms / Vavg(rectified). For a sine: form factor = π/(2√2) ≈ 1.11. Peak factor (crest factor) = Vm / Vrms = √2 ≈ 1.414 for a sine wave.

Power quantities: For sinusoidal voltages and currents with phase difference φ: apparent power S = Vrms·Irms (in VA), real (average) power P = Vrms·Irms·cosφ (in W), reactive power Q = Vrms·Irms·sinφ (in VAR). Power factor = cosφ.

How instruments measure AC: - Moving-coil (D’Arsonval) meter responds to average DC; to read AC it is used with a rectifier. Rectifier-type meters read the average of the rectified waveform and are calibrated to give correct RMS only for pure sine waves. They are not true-RMS for non-sinusoidal signals.
- Moving-iron meters respond to magnetic force ∝ (i^2) averaged, so they indicate a value proportional to RMS for both sinusoidal and many non-sinusoidal waveforms (more nearly true-RMS than rectifier types for many cases).
- True-RMS digital meters compute Vrms by sampling and numerical computation (sqrt of mean of squares) and give correct RMS for any waveform within instrument bandwidth.
- Wattmeter and energy (rotating disc) meters measure power and energy respectively; the energy meter integrates instantaneous power over time (correct for sinusoidal loads and many practical loads).

Using an oscilloscope (CRO): The CRO directly displays waveforms. From the time-base you measure period T and compute frequency f = 1/T. Amplitude (peak or peak-to-peak) is read from the vertical scale. Phase difference between two signals is measured from time shift Δt: φ = 2π(Δt/T) radians = 360°(Δt/T). Alternatively use Lissajous figures (X–Y mode) to measure frequency ratio and phase: an ellipse corresponds to same frequency and the phase is related to ellipse intercepts.

Practical notes: Always know whether your instrument reads RMS (true-RMS), average-rectified (calibrated for sine), or peak. For non-sinusoidal signals (e.g. square, distorted, pulses, inverter output) only true-RMS instruments give correct power-related measurements. Use proper ranges and safety practices for mains measurements.

Fig 7.19 — Educational Diagram: Transient response in AC circuits (brief)

Transient response in AC circuits is the short-term behaviour that appears immediately after a change (for example, when an AC source is switched on or when circuit connections change). The total response of a linear R, L, C circuit to an AC excitation is the sum of a steady-state (particular) solution at the driving frequency and a transient (homogeneous) solution set by initial conditions. The transient terms contain decaying exponentials (for RL/RC) or exponentially-damped oscillations (for RLC) and die out after a few time constants, leaving only the steady sinusoidal response.

Key points: the transient depends on initial current/voltage and circuit parameters; for RL and RC circuits the decay time is characterized by time constant τ (τ = L/R for RL and τ = RC for RC); for series RLC the transient may be underdamped, overdamped or critically damped depending on damping α = R/(2L) relative to natural frequency ω0 = 1/√(LC). After t ≫ a few τ (or after several damping times 1/α), only the forced AC steady-state remains.