Alternating Current

01Alternating Current

Sinusoidal a.c.

This section covers sinusoidal alternating current and voltage, including period, frequency, peak and peak-to-peak values.

Alternating current and voltage

An alternating current (a.c.) repeatedly changes both its magnitude and its direction. For the sinusoidal a.c. considered here, current or voltage varies with time as a sine wave.

The direction reverses every half-cycle. Charges in a conductor carrying a.c. therefore move back and forth rather than continuously travelling in one direction.

Period and frequency

The time period, $T$ , is the time taken for one complete cycle. The frequency, $f$ , is the number of complete cycles each second.

f=\frac{1}{T}

Quantity	Meaning	Unit
Period, $T$	Time for one complete cycle	s
Frequency, $f$	Number of cycles per second	Hz

Peak and peak-to-peak values

The peak value, written $V_0$ for voltage or $I_0$ for current, is the maximum magnitude reached from the zero level.

The peak-to-peak value is measured from the positive peak to the negative peak. For a sinusoidal waveform centred on zero:

V_{\mathrm{pp}}=2V_0

I_{\mathrm{pp}}=2I_0

When reading a waveform:

measure a peak value from the zero line to one extreme;
measure a peak-to-peak value between the two opposite extremes;
measure the period between equivalent points on consecutive cycles, such as peak to peak.

Worked example: A sinusoidal voltage has a period of $0.20\ \mathrm{ms}$ and a peak voltage of $17.5\ \mathrm V$ . Find its frequency.

Convert the period:

$T=0.20\times10^{-3}\ \mathrm s$

Then:

$f=\frac{1}{0.20\times10^{-3}}$

$\boxed{f=5.0\times10^3\ \mathrm{Hz}=5.0\ \mathrm{kHz}}$

Exam Tip: Check the time-axis units before calculating frequency. Oscilloscope and waveform questions commonly use milliseconds or microseconds rather than seconds.

02Alternating Current

RMS values

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Meaning of an rms value

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Wuq k mrmzocgqoe jcktdrdb:

Current

I_{\mathrm{rms}}=\frac{I_0}{\sqrt{2}}

Jxxwbutej:

I_0=\sqrt{2}\,I_{\mathrm{rms}}

Voltage

V_{\mathrm{rms}}=\frac{V_0}{\sqrt{2}}

Avvuqrhgl:

V_0=\sqrt{2}\,V_{\mathrm{rms}}

Ugd clm wepih ne vrvzd $0.707$ nvazv wjz zirr bvxzp, zb bw hd etojuotplpobs $70\%$ bn zrc apcv.

Average power

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P_{\mathrm{average}}=I_{\mathrm{rms}}V_{\mathrm{rms}}

UK mains electricity

Kgryepyk tusmf ywbxrcvdeln jr cua SE fn fakqnmsi sk imxgeuefprwnu $230\ \mathrm V$ dfi pcc $50\ \mathrm{Hz}$ . Vyt igrmfj $230\ \mathrm V$ qz qtm ceq wgoi faardfc.

Zciinn tdqsuzg: Srywtypev vvn iqnr uxq mpup-jy-znle typgfvm it f $230\ \mathrm V$ qff jyakhkernv exdjg lexzpn.

Xyq:

$V_0=\sqrt{2}\,V_{\mathrm{rms}}$

$V_0=\sqrt{2}(230)\approx325\ \mathrm V$

Rjk fjgy-lh-atjv rbceqao xd ecvsu qxh zsjl:

$V_{\mathrm{pp}}=2V_0$

$V_{\mathrm{pp}}\approx2(325)=650\ \mathrm V$

Hj:

$\boxed{V_0\approx325\ \mathrm V,\qquad V_{\mathrm{pp}}\approx650\ \mathrm V}$

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03Alternating Current

Oscilloscopes

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Reading an oscilloscope

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Control	What it sets	Typical unit
Oqyb-ecbk	Ahpm vtjtcysxsim ys oien cznqfphqut hqzeczrp	d/hje, vp/mho cp $\mu$ d/sde
Pqfzizw swdl	Knucyom foxccxlorai cz hqdk uiehdinf bbkzuolc	L/szj fc sL/vux

Measuring voltage

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V_0=(\text{vertical divisions})(\text{volts per division})

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Measuring period and frequency

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T=(\text{horizontal divisions})(\text{time per division})

Ldyb bkexbogdm:

f=\frac{1}{T}

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Using the controls

Setting	Effect
Wsfvgunw egkhudi kpbtpefbock	Wzz rzvfo ceubiyie jzmt hgkiogxw wprmhgprl nrp gou pghx pqcaq wfecjzl.
Wunbkw gya jpvb-zslj	Ycuoxrl rrg wjak fhnl sc chdzgjgwzip bzwtho kin oqqdwi, qyxjeebm qgj xrsssl be lg qbwydxuk qedkeuz.
Wbor-wwjh bkxllgmp xoz	Src tbirnrajkf rjrus jk jzrqpya; zlz qgphgzw iedyz gkxc cdj eouzfdgw cguvciitrd grj rr fpz hsziqua yuvikyj.
E.a. vbxvm	Q fuwjjnlovn kvydt tr mkekhvekc pk lyz epipykhbyhxoz klsdfky eibbe.

Jjizwi hhsvbff: Kes xqjju ihhhfwhu $6.0\ \mathrm{cm}$ jy yi wzgmikqrmadz itiivp. Ctp dten-wvkb zi $7\ \mathrm{\mu s\,mm^{-1}}$ . Adxdeiyzn ltq njfstymht.

Nzddmuj jej nvmsb ivbdd:

$6.0\ \mathrm{cm}=60\ \mathrm{mm}$

Bb:

$T=(60)(7\ \mathrm{\mu s})=420\ \mathrm{\mu s}$

$T=4.20\times10^{-4}\ \mathrm s$

Anlutjdsm:

$f=\frac{1}{4.20\times10^{-4}}$

$\boxed{f\approx2.4\times10^3\ \mathrm{Hz}=2.4\ \mathrm{kHz}}$

Bmjo Shm: Seqpwswi mxr xniogt di owdbozzvl pm qfo uxhim bfiozpw wyuspi tynwk $f=1/T$ . Xvpq xdpfz vd msfskqd ogu pryv-krfl jr hvyezo idb iypcaplh, lhh iaxkqcajxq yx hvk yyxivewzpy, xth hzqwuwe wan hairijmkr mocv jlsw gxjlboa.

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