Alternating Currents

SYLLABUS

Peak and rms value of AC current and voltage, AC through a resistor, a capacitor and an inductor, Phasor diagram, Series circuits containing combination of resistance, capacitance and inductance, Series resonance, quality factor, Power in AC circuits: power factor

What is This Chapter About?

The electrical power delivered to every home, school, and factory in Nepal — and almost everywhere in the world — is alternating current (AC). Unlike direct current (DC), which flows steadily in one direction, AC reverses direction sinusoidally at a fixed frequency (50 Hz in Nepal, 60 Hz in North America). This choice is not arbitrary — AC has profound practical advantages over DC for power transmission and distribution, and its behaviour in circuits containing capacitors and inductors gives rise to rich and beautiful phenomena including resonance and power factor.

Chapter 19 of NEB Grade 12 Physics develops the complete theory of AC circuits, carrying 6 teaching hours. It covers peak and RMS values, the behaviour of resistors, capacitors, and inductors in AC circuits, phasor diagrams, series RLC circuits, resonance, and power in AC circuits. This chapter has direct applications in power engineering, radio and television technology, audio amplifiers, and signal processing.

Key Concepts You Will Learn

Peak and RMS Values

An AC voltage varies sinusoidally with time: [latex]V = V_0\sin(\omega t)[/latex], where [latex]V_0[/latex] is the peak voltage and [latex]\omega = 2\pi f[/latex] is the angular frequency. The corresponding current is [latex]I = I_0\sin(\omega t)[/latex] for a purely resistive circuit.

Since AC continuously alternates, its average value over a complete cycle is zero — not useful for calculating power. Instead, the root mean square (RMS) values are used:

[latex]V_{rms} = \frac{V_0}{\sqrt{2}} \approx 0.707 V_0[/latex]

[latex]I_{rms} = \frac{I_0}{\sqrt{2}} \approx 0.707 I_0[/latex]

RMS values are the AC equivalents of DC values for power calculations — an AC supply of [latex]V_{rms}[/latex] delivers the same average power to a resistor as a DC supply of the same voltage. The standard 230 V supply in Nepal is an RMS value; the peak voltage is actually [latex]230\sqrt{2} \approx 325[/latex] V.

AC Through a Resistor

For a pure resistor R, voltage and current are always in phase — they reach their peaks and zero crossings simultaneously. The relationship [latex]V = IR[/latex] holds at every instant. Power dissipated: [latex]P = I_{rms}^2 R = \frac{V_{rms}^2}{R}[/latex].

AC Through a Capacitor

For a pure capacitor C, the current leads the voltage by 90° (or [latex]\frac{\pi}{2}[/latex] radians) — when voltage is at zero and just starting to rise, current is already at its maximum. The opposition offered by a capacitor to AC is called capacitive reactance:

[latex]X_C = \frac{1}{\omega C} = \frac{1}{2\pi f C}[/latex]

Capacitive reactance decreases with increasing frequency — a capacitor offers high opposition to low-frequency (slowly changing) currents and low opposition to high-frequency currents. At DC (f = 0), [latex]X_C = \infty[/latex] — a capacitor completely blocks DC. A capacitor dissipates no average power — energy is stored in the electric field during one quarter cycle and returned during the next.

AC Through an Inductor

For a pure inductor L, the current lags the voltage by 90° — when voltage is at maximum, current is zero and just starting to rise. The opposition offered by an inductor is called inductive reactance:

[latex]X_L = \omega L = 2\pi f L[/latex]

Inductive reactance increases with frequency — an inductor offers low opposition to DC (it is just a wire) and high opposition to high-frequency currents. Like a capacitor, an ideal inductor dissipates no average power — energy is stored in the magnetic field and returned.

Phasor Diagrams

The phase relationships between voltage and current in AC circuits are most elegantly represented using phasor diagrams. A phasor is a rotating vector whose length represents the amplitude of a sinusoidal quantity and whose angle represents its phase. The projections of phasors onto a reference axis give the instantaneous values of the quantities.

In phasor diagrams: resistor voltage phasor [latex]V_R[/latex] is in phase with current I; capacitor voltage phasor [latex]V_C[/latex] lags current by 90°; inductor voltage phasor [latex]V_L[/latex] leads current by 90°. Phasor diagrams allow complex AC circuits to be analysed geometrically.

Series RLC Circuit

When a resistor R, inductor L, and capacitor C are connected in series with an AC source, the total voltage is the phasor sum of the individual voltage drops. Since [latex]V_L[/latex] and [latex]V_C[/latex] are in opposite directions (one leads current by 90° and the other lags by 90°), they partially cancel. The resultant voltage phasor V makes an angle [latex]\phi[/latex] with the current phasor:

[latex]V = \sqrt{V_R^2 + (V_L – V_C)^2} = I\sqrt{R^2 + (X_L – X_C)^2}[/latex]

The total opposition to AC in the circuit is the impedance Z:

[latex]Z = \sqrt{R^2 + (X_L – X_C)^2}[/latex]

The phase angle between voltage and current is:

[latex]\tan\phi = \frac{X_L – X_C}{R}[/latex]

If [latex]X_L > X_C[/latex], the circuit is inductive (current lags voltage). If [latex]X_L < X_C[/latex], it is capacitive (current leads voltage). If [latex]X_L = X_C[/latex], the circuit is at resonance.

Series Resonance

Resonance in a series RLC circuit occurs when the inductive and capacitive reactances are equal: [latex]X_L = X_C[/latex], giving:

[latex]\omega_0 L = \frac{1}{\omega_0 C} \Rightarrow \omega_0 = \frac{1}{\sqrt{LC}}[/latex]

The resonant frequency is [latex]f_0 = \frac{1}{2\pi\sqrt{LC}}[/latex].

At resonance, the impedance reduces to just R (the purely resistive component) — minimum impedance — and the current is at its maximum value: [latex]I_{max} = \frac{V}{R}[/latex]. The voltages across L and C can individually be much larger than the source voltage — a phenomenon called voltage magnification or Q-factor amplification.

Resonance has crucial practical applications. Radio tuning works by varying C (or L) to bring the circuit resonant frequency to match the frequency of the desired radio station — at resonance, that station’s signal drives maximum current through the circuit while other frequencies are suppressed. Filters use resonance to select or reject specific frequency ranges.

Quality Factor

The quality factor (Q-factor) of a series RLC circuit measures the sharpness of the resonance peak — how selectively the circuit responds to frequencies near resonance:

[latex]Q = \frac{\omega_0 L}{R} = \frac{1}{\omega_0 CR} = \frac{1}{R}\sqrt{\frac{L}{C}}[/latex]

A high Q-factor means a sharp, narrow resonance peak — the circuit is highly selective, responding strongly to frequencies very close to [latex]f_0[/latex] and weakly to others. A low Q-factor gives a broad resonance — less selective. High-Q circuits are used in radio receivers and oscillators where precise frequency selection is needed.

Power in AC Circuits

The instantaneous power in an AC circuit is [latex]p = vi[/latex]. For a circuit with impedance Z and phase angle [latex]\phi[/latex] between voltage and current, the average power dissipated is:

[latex]P = V_{rms}I_{rms}\cos\phi[/latex]

The factor [latex]\cos\phi[/latex] is called the power factor. For a purely resistive circuit, [latex]\phi = 0[/latex] and [latex]\cos\phi = 1[/latex] — all power is dissipated. For purely reactive circuits (pure L or pure C), [latex]\phi = 90°[/latex] and [latex]\cos\phi = 0[/latex] — no average power is dissipated despite significant current flowing. In practical AC circuits, power factor is always between 0 and 1.

Apparent power is [latex]S = V_{rms}I_{rms}[/latex] (measured in volt-amperes, VA), and real power (actual power dissipated) is [latex]P = S\cos\phi[/latex] (measured in watts). Electrical utilities charge for real power but must supply apparent power — a low power factor means more current (and therefore more transmission losses) for the same real power delivered, which is why industries are penalised for low power factors.

Why This Chapter Matters for NEB Exams

Alternating Currents is one of the most mathematically rich and thoroughly examined chapters in NEB Grade 12 Physics. RMS values, impedance of RLC circuits, derivation of resonant frequency, the Q-factor, and power factor are all standard examination topics. Phasor diagrams — drawing them for RC, RL, and RLC circuits — appear regularly as both standalone questions and as part of numerical problems. Series resonance — explaining the condition, its significance, and its applications — is a classic long-answer question. Numerical problems on impedance, phase angle, resonant frequency, and power factor appear in virtually every NEB board paper.

Important Topics to Focus On

• Peak and RMS values: [latex]V_{rms} = \frac{V_0}{\sqrt{2}}[/latex], [latex]I_{rms} = \frac{I_0}{\sqrt{2}}[/latex]
• AC through resistor — in phase, power [latex]P = I_{rms}^2 R[/latex]
• AC through capacitor — current leads by 90°, capacitive reactance [latex]X_C = \frac{1}{\omega C}[/latex]
• AC through inductor — current lags by 90°, inductive reactance [latex]X_L = \omega L[/latex]
• Phasor diagrams for R, L, C, RC, RL, and RLC circuits
• Series RLC circuit — impedance [latex]Z = \sqrt{R^2 + (X_L – X_C)^2}[/latex]
• Phase angle: [latex]\tan\phi = \frac{X_L – X_C}{R}[/latex]
• Series resonance condition: [latex]\omega_0 = \frac{1}{\sqrt{LC}}[/latex]
• Q-factor: [latex]Q = \frac{1}{R}\sqrt{\frac{L}{C}}[/latex] and its physical significance
• Power in AC circuits: [latex]P = V_{rms}I_{rms}\cos\phi[/latex]
• Power factor — definition, significance, and distinction from apparent power
• Numerical problems on all of the above

Connection to Other Chapters

Alternating Currents builds directly on Chapter 18 (Electromagnetic Induction) — the AC generator produces the sinusoidal EMF studied here, and self-inductance L is the key parameter of inductors in AC circuits. Chapter 22 (Capacitor) from Grade 11 provides the capacitance C used in AC circuit analysis. The resonance phenomenon in series RLC circuits is directly analogous to mechanical resonance in Chapter 2 (Periodic Motion) — the resonant frequency [latex]\frac{1}{\sqrt{LC}}[/latex] mirrors [latex]\frac{1}{\sqrt{m/k}} = \sqrt{\frac{k}{m}}[/latex] for a mass-spring system, with L playing the role of mass (inertia) and C playing the role of spring compliance. The power factor connects to energy conservation from Grade 11 Chapter 5 (Work, Energy and Power) — reactive power is stored and returned rather than dissipated, while real power is irreversibly converted to heat. Chapter 19 also connects forward to Chapter 22 (Semiconductor Devices) — diodes and transistors are used in rectifier circuits that convert AC to DC, and in amplifiers that process AC signals.