use equations of the form x = x0 sin ωt representing a sinusoidally alternating current or voltage

1. Introduction to Alternating Current (AC)

In alternating current the magnitude and direction of the current (or voltage) vary periodically with time. For most A‑Level examinations the sinusoidal form is assumed because it is produced by the most common source – an AC generator.

1.1 Sinusoidal representation

The instantaneous value of a sinusoidally varying quantity $x$ (current $i$, voltage $v$, etc.) is written as

$$x(t) = x_0 \sin(\omega t + \phi)$$

where:

• $x_0$ – maximum (peak) value (amplitude)
• $\omega$ – angular frequency in rad s\(^{-1}\)
• $t$ – time in seconds
• $\phi$ – phase angle (radians) relative to a chosen reference

1.2 Relationship between angular frequency, frequency and period

The three quantities are related by

$$\omega = 2\pi f = \frac{2\pi}{T}$$

where $f$ is the frequency in hertz (Hz) and $T$ is the period in seconds.

2. Key Quantities for a Sinusoidal AC

Quantity	Symbol	Expression (for $x = x_0\sin\omega t$)	Physical meaning
Peak (maximum) value	$x_0$	$x_0$	Maximum magnitude reached during a cycle
Peak‑to‑peak value	$x_{pp}$	$2x_0$	Difference between maximum positive and negative values
Root‑mean‑square (RMS) value	$x_{\rm rms}$	$\displaystyle \frac{x_0}{\sqrt{2}}$	Effective DC equivalent; gives same heating effect
Average (mean) value over a full cycle	$\overline{x}$	$0$	Because the positive and negative halves cancel
Period	$T$	$\displaystyle \frac{2\pi}{\omega}$	Time for one complete cycle
Frequency	$f$	$\displaystyle \frac{1}{T}= \frac{\omega}{2\pi}$	Number of cycles per second

2.1 Example calculation

For a mains supply described by $v(t)=230\sin(2\pi 50t)$ V:

1. Peak voltage $v_0 = 230\,$V.
2. RMS voltage $v_{\rm rms}=230/\sqrt{2}\approx 163\,$V.
3. Period $T = 1/50 = 0.02\,$s.

3. Phase Relationships

Two sinusoidal quantities can be written as

$$x(t)=x_0\sin(\omega t+\phi_x),\qquad y(t)=y_0\sin(\omega t+\phi_y)$$

The phase difference is $\Delta\phi = \phi_y-\phi_x$. Common cases:

• In‑phase: $\Delta\phi = 0^\circ$ (or $0$ rad) – the waveforms reach maxima and zeroes together.
• Out‑of‑phase by $180^\circ$ (or $\pi$ rad) – one is the negative of the other.
• Quarter‑cycle shift: $\Delta\phi = 90^\circ$ (or $\pi/2$ rad) – the peak of one coincides with the zero crossing of the other.

3.1 Phasor representation

It is convenient to represent a sinusoid as a rotating vector (phasor) of magnitude $x_0$ making an angle $\phi$ with the real axis. The time‑dependent sinusoid is obtained by projecting the phasor onto the real axis.

4. AC in Basic Circuit Elements

4.1 Resistor (R)

Ohm’s law applies instantaneously:

$$v_R(t)=i_R(t)R$$

Voltage and current are in phase ($\Delta\phi=0$). The RMS relationship is $V_{\rm rms}=I_{\rm rms}R$.

4.2 Inductor (L)

For a sinusoidal current $i_L(t)=I_0\sin\omega t$, the induced emf is

$$v_L(t)=L\frac{di}{dt}=L\omega I_0\cos\omega t = \omega L I_0\sin\!\left(\omega t+\frac{\pi}{2}\right)$$

Thus the voltage leads the current by $90^\circ$ and the inductive reactance is

$$X_L = \omega L$$

RMS form: $V_{\rm rms}=I_{\rm rms}X_L$.

4.3 Capacitor (C)

For a sinusoidal voltage $v_C(t)=V_0\sin\omega t$, the current is

$$i_C(t)=C\frac{dv}{dt}=C\omega V_0\cos\omega t = \omega C V_0\sin\!\left(\omega t-\frac{\pi}{2}\right)$$

Current leads voltage by $90^\circ$ and the capacitive reactance is

$$X_C = \frac{1}{\omega C}$$

RMS form: $I_{\rm rms}=V_{\rm rms}/X_C$.

4.4 Summary of phase and reactance

Element	Voltage–Current Phase	Reactance	Impedance (Z)
Resistor (R)	In phase (0°)	—	$Z=R$
Inductor (L)	Voltage leads current (+90°)	$X_L=\omega L$	$Z=jX_L$
Capacitor (C)	Current leads voltage (‑90°)	$X_C=1/(\omega C)$	$Z=-jX_C$

5. Power in Sinusoidal AC Circuits

The instantaneous power is the product of instantaneous voltage and current:

$$p(t)=v(t)i(t)$$

For a single sinusoid with a phase difference $\Delta\phi$ between voltage and current, the average (real) power over a cycle is

$$P_{\rm av}=V_{\rm rms}I_{\rm rms}\cos\Delta\phi$$

The factor $\cos\Delta\phi$ is called the power factor.

5.1 Apparent, real and reactive power

• Apparent power $S = V_{\rm rms}I_{\rm rms}$ (unit \cdot A).
• Real (active) power $P = S\cos\Delta\phi$ (unit W).
• Reactive power $Q = S\sin\Delta\phi$ (unit \cdot AR).

5.2 Power triangle

6. Summary Checklist for the Examination

1. Write the sinusoidal form $x=x_0\sin(\omega t+\phi)$ and identify each symbol.
2. Convert between $\omega$, $f$, and $T$ using $\omega=2\pi f=2\pi/T$.
3. Calculate RMS values: $x_{\rm rms}=x_0/\sqrt{2}$.
4. State the phase relationship for R, L, and C and give the corresponding reactances $X_L$ and $X_C$.
5. Use $P_{\rm av}=V_{\rm rms}I_{\rm rms}\cos\Delta\phi$ to find real power and identify the power factor.
6. Draw phasor diagrams and power triangles when asked.