Alternating currents (A-level only)

AQA·Physics·3.7.5.5

Sinusoidal alternating waveforms

An alternating current repeatedly changes direction. An alternating voltage likewise repeatedly reverses its polarity. On a graph of voltage or current against time, positive and negative values represent opposite directions or polarities relative to a chosen reference.

A sinusoidal waveform has the shape of a sine wave. It varies smoothly between equal positive and negative extremes about zero, then repeats. For example, a cycle can be followed from a zero crossing through the positive and negative extremes to the next equivalent zero crossing.

Period and frequency

The period, $$T$$, is the time taken for one complete cycle. The frequency, $$f$$, is the number of complete cycles per second. They are related by:

f=1Tf = \frac{1}{T}

When $$T$$ is measured in seconds, $$f$$ is measured in hertz ($$\mathrm{Hz}$$). A shorter period means a higher frequency.

Peak and peak-to-peak values

The peak value is the maximum magnitude measured from zero. For a sinusoidal voltage it is written $$V_0$$, so the waveform reaches $$+V_0$$ and $$-V_0$$. For a sinusoidal current, the corresponding extremes are $$+I_0$$ and $$-I_0$$.

The peak-to-peak value is the difference between the positive and negative extremes. It spans both sides of zero, so it is $$2V_0$$ for voltage and $$2I_0$$ for current.

RMS values of sinusoidal waveforms

An alternating voltage or current changes with time and reverses direction. Its root mean square (RMS) value is the value of a steady DC voltage or current that would produce the same average power in the same resistor.

Power in a resistor depends on the square of the current or voltage: $$P=I^2R$$ or $$P=V^2/R$$. Reversing the direction therefore does not make the power negative. The RMS value is found by squaring the changing value, taking the mean over a complete cycle, and then taking the square root.

RMS and peak values

For a sinusoidal waveform, the RMS value is related to the peak value by:

Irms=I02,Vrms=V02I_{\mathrm{rms}}=\frac{I_0}{\sqrt{2}},\qquad V_{\mathrm{rms}}=\frac{V_0}{\sqrt{2}}

Here, $$I_0$$ and $$V_0$$ are the peak current and peak voltage: the greatest magnitudes reached by the waveform. These relationships apply to sinusoidal waveforms.

Sinusoidal mains voltage

A mains voltage is usually stated as an RMS value. For a sinusoidal mains voltage, multiply the RMS value by $$\sqrt{2}$$ to find the peak voltage:

V0=2VrmsV_0=\sqrt{2}V_{\mathrm{rms}}

The voltage varies from $$-V_0$$ to $$+V_0$$. The peak-to-peak voltage is the difference between these extremes, so it is $$2V_0$$.

For example, a sinusoidal mains voltage of $$230\ \mathrm{V}$$ RMS has a peak voltage of approximately $$325\ \mathrm{V}$$ and a peak-to-peak voltage of approximately $$650\ \mathrm{V}$$.

Oscilloscope measurements

An oscilloscope displays a voltage as a trace on a screen. The trace’s vertical displacement represents voltage, while its horizontal position represents time. The screen is marked with divisions so that both can be measured.

The vertical sensitivity, or volts-per-division setting, gives the voltage represented by each vertical division. The time-base setting gives the time represented by each horizontal division. Position controls move the trace on the screen; they help place it where measurements can be made clearly. A zero-voltage reference identifies the level from which voltage is measured.

Measuring voltage on an oscilloscope

A steady DC voltage produces a horizontal trace because its voltage does not change with time. Measure its vertical displacement from the zero-voltage reference:

voltage=vertical displacement in divisions×volts per division\text{voltage} = \text{vertical displacement in divisions} \times \text{volts per division}

The direction of the displacement indicates whether the voltage is positive or negative relative to the zero reference.

An AC voltage changes with time, so it produces a varying trace. To measure a peak voltage, use the vertical distance from the zero-voltage reference to the peak. The peak-to-peak voltage is found from the full vertical distance between the highest and lowest points of the trace, multiplied by the vertical scale.

For a waveform centred on zero, the peak-to-peak voltage is twice the peak voltage. If the waveform has a steady offset, its centre is not at zero, so this relationship does not generally apply to a peak voltage measured from zero.

DC coupling displays the voltage including any steady offset. AC coupling suppresses the steady component, allowing the varying part to be viewed without that offset.

Measuring time and frequency on an oscilloscope

Measure the horizontal distance between two features of the trace to find the time between them:

time interval=horizontal distance in divisions×time per division\text{time interval} = \text{horizontal distance in divisions} \times \text{time per division}

For a repeating waveform, measure between corresponding points on successive cycles, such as two successive peaks. This interval is the period, $$T$$, the time for one complete cycle. The frequency, $$f$$, is the number of cycles per second:

f=1Tf = \frac{1}{T}

Use the period in seconds to obtain the frequency in hertz. For example, if one cycle spans 4 divisions at $$0.5\ \text{ms}$$ per division, its period is $$2\ \text{ms} = 0.002\ \text{s}$$ and its frequency is $$500\ \text{Hz}$$.

The trigger control can be adjusted so that repeated sweeps begin at the same point in the waveform. This makes a repeating trace appear stable and easier to measure.