Physics › Magnetic fields › Alternating currents
Alternating currents
The generator delivers a sine wave, so ac needs its own vocabulary: peak, peak-to-peak, and the rms value that lets an alternating supply be compared honestly with a steady one. The oscilloscope is the instrument that lays the whole waveform out to be measured.
Builds on Electromagnetic induction: Faraday and Lenz and Current, charge and the direction problem.
IN THIS TOPIC
- Read peak, peak-to-peak and rms values from sinusoidal waveforms.
- Use the rms relations and apply them to mains electricity.
- Use an oscilloscope to measure voltages, time intervals and frequencies.
WHAT YOU PROBABLY THINK
Mains is 230 volts at its peak.
Three sizes of one sine
An alternating supply swings between +V0 and −V0. The peak value V0 is the height of the swing; the peak-to-peak value is the full 2V0 span, the easiest thing to read off a screen. Neither is the fair average: the supply spends most of its time below peak, and its plain average is zero. The honest measure is the root mean square, the steady dc value that would deliver the same power in the same resistor. Squaring the sine, averaging, and rooting again gives, for sinusoidal waveforms only,
about 0.707 of the peak. Quoted ac values are rms unless stated otherwise, which alone corrects the claim above: UK mains is 230 V rms, so its peak is 230 × 1.414 ≈ 325 V and its peak-to-peak span is about 650 V. Insulation is engineered for the peak, not the label.
The oscilloscope
The oscilloscope plots voltage against time on a squared screen, and two settings translate the squares. The y-gain gives the volts per division vertically; the timebase gives the seconds per division horizontally. A dc input draws a level line whose height, times the y-gain, is the voltage: the scope as a dc voltmeter. An ac input draws its actual waveform: the amplitude times the y-gain gives the peak voltage, the width of one full cycle times the timebase gives the period, and f = 1/T finishes the job. Only its use is examined, never the instrument's internal workings.
THE EXAM BIT
- State the rms meaning, not only the formula: the dc value giving the same power in the same resistor. The definition is the mark.
- The root-two relations hold for sinusoidal waveforms only, and saying so costs one clause.
- Mains numbers are the stock application: 230 V rms, 325 V peak, 650 V peak-to-peak. Know the trio.
- Scope arithmetic is two multiplications: divisions × y-gain for volts, divisions × timebase for seconds. Write the readings in divisions first, then convert.
- Frequency comes only through the period: read one full cycle's width, convert to seconds, then f = 1/T. Never scale frequency straight off the screen.
CHECK YOURSELF
An oscilloscope shows a sine wave of amplitude 3.0 divisions with one cycle spanning 4.0 divisions. The y-gain is 5.0 V per division and the timebase 2.0 ms per division. Find the peak voltage, the rms voltage, and the frequency.
Show a hint
Squares to volts and seconds first; the physics is two multiplications and a reciprocal.
Show the answer
Peak: V0 = 3.0 × 5.0 = 15 V.
Vrms = V0/√2: 15 / 1.414 = 10.6 V.
Period T = 4.0 × 2.0 ms = 8.0 ms, so f = 1/T = 125 Hz.
Quoted ac is rms: the dc-equivalent for power, 0.707 of the peak.
On the scope, squares become volts and seconds; f comes from 1/T.
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