Comprehensive theory, key formulas, diagrams, and memory aids for Alternating Currents.
An alternating current (AC) is one that periodically reverses direction. It is generated by rotating coils in magnetic fields (generators) and is the standard form of electrical power supplied to homes and industry. In contrast, direct current (DC) flows only in one direction (batteries, solar cells).
The voltage of an AC supply varies sinusoidally with time: $$V = V_0 \sin(\omega t)$$
And the resulting current: $$I = I_0 \sin(\omega t)$$
Where: - $V_0$, $I_0$ = peak (maximum) voltage and current - $\omega = 2\pi f$ = angular frequency (rad sโปยน) - $f$ = frequency. In the UK: 50 Hz; USA: 60 Hz. - $T = 1/f$ = period
On an oscilloscope, AC appears as a sinusoidal trace. The time base setting determines the x-axis scale (ms/div) and the Y-gain determines the y-axis scale (V/div), allowing both period and peak voltage to be measured.
Since AC continuously changes, we cannot simply use peak values for power calculations. The root-mean-square (r.m.s.) values represent the equivalent DC values that would deliver the same average power to a resistive load.
Derivation: The average power in a resistor $R$ with AC: $$\bar{P} = \frac{1}{T}\int_0^T \frac{V^2}{R}dt = \frac{V_0^2}{2R}$$
Comparing with DC power $P = V_{dc}^2/R$: the equivalent DC voltage is $V_{dc} = V_0/\sqrt{2}$.
$$V_{rms} = \frac{V_0}{\sqrt{2}} \approx 0.707 V_0$$ $$I_{rms} = \frac{I_0}{\sqrt{2}} \approx 0.707 I_0$$
Average power: $$\bar{P} = V_{rms} I_{rms} = \frac{1}{2}V_0 I_0 = \frac{V_{rms}^2}{R} = I_{rms}^2 R$$
In the UK, the 230 V mains supply is an r.m.s. value. The peak voltage is: $$V_0 = 230\sqrt{2} \approx 325 \text{ V}$$
This is why electrical insulation must be rated well above 230 V.
Rectification converts AC to DC. It is essential for powering electronic circuits, charging batteries, and producing the DC supply needed by most electronic devices.
A single diode passes only the positive half-cycles of the AC waveform. The negative half-cycles are blocked. The output is a pulsating DC โ current only flows in one direction, but there are gaps.
Four diodes arranged in a bridge circuit pass both positive and negative half-cycles, converting both to the same direction. The output is a full-wave rectified signal โ a series of positive humps with no gaps, but still varying.
graph LR
A[AC Input] --> B[Diode Bridge: 4 diodes]
B --> C[Full-wave rectified DC: pulsating]
C --> D[Smoothing capacitor]
D --> E[Smooth DC output]
A capacitor connected in parallel with the load smooths the rectified output. The capacitor charges up to the peak voltage and slowly discharges through the load during the troughs, filling in the gaps.
The amount of ripple (residual variation) depends on: - Capacitance ($C$): Larger $C$ โ smaller ripple (charges/discharges more slowly) - Load resistance ($R$): Larger $R$ โ smaller ripple (slower discharge) - Frequency ($f$): Higher $f$ โ smaller ripple (less time to discharge between peaks)
The time constant $\tau = RC$ should be much larger than the period $T$ for good smoothing.
The cathode-ray oscilloscope (CRO) is an essential instrument for observing and measuring electrical signals.
Key controls: - Y-gain (V/div): Sets the vertical scale (volts per division). Allows measurement of peak voltage. - Time base (ms/div): Sets the horizontal scale. Allows measurement of period โ frequency.
Measurements from a CRO trace: 1. Peak voltage $V_0$ = (amplitude in divisions) ร (Y-gain setting) 2. Period $T$ = (horizontal length of one cycle in divisions) ร (time base setting) 3. Frequency $f = 1/T$ 4. r.m.s. voltage = $V_0 / \sqrt{2}$
Lissajous figures: If two AC signals are connected to X and Y inputs of a CRO, the resulting pattern gives information about the ratio of frequencies and their phase difference.
In DC circuits, resistance is the only opposition to current. In AC circuits, capacitors and inductors also oppose current โ their opposition is called reactance, and the total opposition is impedance ($Z$).
For a series RC circuit: $$Z = \sqrt{R^2 + X_C^2} \quad \text{and} \quad I_{rms} = \frac{V_{rms}}{Z}$$
Phase difference: In a purely capacitive circuit, current leads voltage by 90ยฐ. In a purely inductive circuit, current lags voltage by 90ยฐ. In a purely resistive circuit, current and voltage are in phase.
In AC circuits containing reactance, the power factor ($\cos\phi$) accounts for the phase difference $\phi$ between voltage and current:
$$\bar{P} = V_{rms} I_{rms} \cos\phi$$
For a purely resistive load: $\phi = 0$, $\cos\phi = 1$ โ maximum power. For a purely reactive load: $\phi = 90ยฐ$, $\cos\phi = 0$ โ zero average power (energy is stored and returned, not dissipated).
Industrial loads (motors, transformers) are partly inductive, resulting in a lagging power factor. Power companies supply capacitor banks to correct this and maximise power delivery efficiency.