Alternating Current — Study Notes

1. Introduction

Most electrical power is generated, transmitted, and distributed as alternating current (AC). An alternating current reverses its direction periodically, varying sinusoidally with time. This chapter studies the behaviour of resistors, inductors, and capacitors in AC circuits, which is fundamentally different from their behaviour in DC circuits.

We introduce the important concepts of peak and root-mean-square (RMS) values of alternating current and voltage, which describe the effective values of a sinusoidally varying quantity. We then study the phase relationships between voltage and current in circuits containing resistance, inductance, and capacitance, leading to the ideas of reactance and impedance.

The chapter culminates in the LCR series circuit, where resonance occurs at a particular frequency, and in the study of power in AC circuits through the power factor. Finally, we study the transformer, the device that makes efficient long-distance power transmission possible by stepping voltage up and down.

2. AC Voltage, Current, and RMS Values

An alternating voltage varies sinusoidally with time:

V = V0 sin(omega t)

where V0 is the peak voltage and omega = 2 pi f is the angular frequency. The corresponding current in a purely resistive circuit is I = I0 sin(omega t), where I0 = V0/R. The alternating current has a frequency f, typically 50 Hz in India.

Because the average of a sinusoid over a full cycle is zero, we use the root-mean-square (RMS) values to describe AC. The RMS value of an AC is the value of the steady DC that would produce the same heating effect. For a sinusoidal AC:

I_rms = I0 / sqrt(2), V_rms = V0 / sqrt(2)

The RMS values are also called the effective values. A 220 V household supply has V_rms = 220 V and peak value V0 = 220 sqrt(2) = 311 V. The RMS current is I_rms = I0/sqrt(2).

3. AC through a Resistor

When an AC voltage is applied across a resistor, the current is V/R = (V0/R) sin(omega t), so the current and voltage are in phase. The resistance R is the only quantity opposing the current, and the instantaneous power is:

P = V I = V0 I0 sin^2(omega t)

The average power over a cycle is:

P_avg = V_rms I_rms = I_rms^2 R = V_rms^2 / R

In a pure resistor, all the electrical energy is converted into heat, and the power is always positive. The resistor obeys Ohm's law at every instant, and the voltage and current reach their maxima and minima together.

4. AC through an Inductor

For a pure inductor, the induced EMF L dI/dt opposes the applied voltage. Solving the differential equation gives the current lagging behind the voltage by 90 degrees:

I = I0 sin(omega t - pi/2)

The current lags the voltage by a quarter cycle. The opposition of the inductor to AC is called inductive reactance:

X_L = omega L = 2 pi f L

The peak current is I0 = V0/X_L, and the unit of reactance is the ohm. Inductive reactance increases with frequency: a high-frequency current is opposed more strongly than a low-frequency one. In a pure inductor, the average power is zero; energy is alternately stored in the magnetic field and returned to the circuit.

5. AC through a Capacitor

For a pure capacitor, the current is the rate of change of charge, I = C dV/dt. This gives the current leading the voltage by 90 degrees:

I = I0 sin(omega t + pi/2)

The current leads the voltage by a quarter cycle, opposite to the inductor. The opposition of the capacitor to AC is capacitive reactance:

X_C = 1/(omega C) = 1/(2 pi f C)

The peak current is I0 = V0/X_C. Capacitive reactance decreases with frequency and capacitance: a capacitor offers high opposition to low-frequency currents and low opposition to high-frequency currents. At DC (f = 0), the capacitive reactance is infinite, so a capacitor blocks steady current. In a pure capacitor, the average power is again zero.

6. The LCR Series Circuit and Resonance

When a resistor, an inductor, and a capacitor are connected in series to an AC source, the voltage across each element has a different phase. The net opposition is the impedance:

Z = sqrt(R^2 + (X_L - X_C)^2)

The current in the circuit is I = V/Z, and the phase angle phi between the voltage and current is:

tan(phi) = (X_L - X_C) / R

The circuit is capacitive if X_C > X_L (current leads), inductive if X_L > X_C (current lags), and purely resistive if X_L = X_C.

Resonance occurs when the inductive and capacitive reactances are equal, X_L = X_C, which happens at the resonant frequency:

f0 = 1 / (2 pi sqrt(L C))

At resonance, the impedance is minimum and equal to R, so the current is maximum: I = V/R. The resonant frequency depends only on L and C. Resonance is used in radio tuning, where a circuit is tuned to the frequency of the desired station.

7. Power in AC Circuits and Power Factor

The average power in an AC circuit is given by:

P = V_rms I_rms cos(phi)

The factor cos(phi) is the power factor, which depends on the phase difference between voltage and current. For a purely resistive circuit, cos(phi) = 1 and power is maximum. For pure inductance or capacitance, cos(phi) = 0 and the average power is zero; the current is called wattless current.

The power factor can be expressed in terms of the circuit elements: cos(phi) = R/Z. Power loss in AC transmission is reduced by improving the power factor, for example by connecting a capacitor in parallel with inductive loads. The instantaneous power fluctuates, but only the average power does useful work.

8. The Transformer

A transformer is a device that changes the voltage of alternating current using mutual induction. It consists of two coils - primary and secondary - wound on a laminated iron core. The changing flux in the primary induces an EMF in the secondary. For an ideal transformer:

V_s / V_p = N_s / N_p = I_p / I_s

If N_s > N_p, it is a step-up transformer (voltage increases, current decreases); if N_s < N_p, it is a step-down transformer. For an ideal transformer, the input power equals the output power: V_p I_p = V_s I_s.

Real transformers have losses due to eddy currents (reduced by lamination), hysteresis, and the resistance of the windings. Transformers work only with AC, not DC, because they require a changing flux. They enable step-up of voltage for transmission, reducing I^2R losses, and step-down at the point of use.

Quick Revision Tables

Mind Map

Important Diagrams (SVG)

Diagram 1: Phase Relationships in LCR Circuits

Diagram 2: Transformer

Common Mistakes

1. Using the peak value where the RMS value is required; most formulas for power use RMS values.
2. Forgetting that the average value of AC over a full cycle is zero; RMS, not average, measures effective value.
3. Confusing which quantity leads: current lags in an inductor and leads in a capacitor.
4. Adding X_L and X_C arithmetically in impedance; they subtract because they are opposite in phase, Z = sqrt(R^2 + (XL-XC)^2).
5. Believing power is always VI; for AC it is V_rms I_rms cos phi, and pure L or C consume zero average power.
6. Applying transformer formulas to DC; transformers work only on AC because they require changing flux.
7. Forgetting the factor 1/(2 pi) in the resonant frequency f0 = 1/(2 pi sqrt(LC)).
8. Treating reactance as a constant like resistance; X_L increases and X_C decreases with frequency.

Exam Tips

1. Define peak and RMS values and write I_rms = I0/sqrt(2), V_rms = V0/sqrt(2).
2. State that a resistor keeps current and voltage in phase, and give P = V_rms I_rms.
3. Show that in a pure inductor the current lags by 90 degrees, with X_L = omega L.
4. Show that in a pure capacitor the current leads by 90 degrees, with X_C = 1/(omega C).
5. Derive the impedance Z = sqrt(R^2 + (XL - XC)^2) and phase tan phi = (XL-XC)/R for an LCR circuit.
6. Define resonance, give f0 = 1/(2 pi sqrt(LC)), and state that current is maximum at resonance.
7. Write the AC power P = V_rms I_rms cos phi and define the power factor cos phi = R/Z.
8. Describe the transformer, its ideal relation Vs/Vp = Ns/Np, and its use in power transmission.

Conclusion

This chapter dealt with alternating current and its circuits. RMS values I_rms = I0/sqrt(2) describe the effective magnitude of sinusoidal AC. In a resistor the current is in phase with the voltage; in an inductor it lags by 90 degrees with reactance X_L = omega L; in a capacitor it leads by 90 degrees with reactance X_C = 1/(omega C). The LCR series circuit combines these into impedance Z = sqrt(R^2 + (XL-XC)^2), with resonance at f0 = 1/(2 pi sqrt(LC)) when current is maximum. The average power P = V_rms I_rms cos phi introduced the power factor. Finally, the transformer, working by mutual induction, steps voltages up and down to make power transmission efficient. AC theory underlies the entire electrical power network.