Chapter 8: Alternating Current

One terminal of the source stays positive

A DC source keeps its polarity: the same terminal is always the positive one, so the current flows one way only. Its size may still vary: the fixed polarity, not a steady voltage, keeps the current one way.

Alternating current and direct current

AC is alternating current, which reverses its direction at regular intervals, and DC is direct current, which always flows the same way. Both name a current, not a charge.

Yes, as on a winding marked '12 V AC'

Although AC strictly means alternating current, the abbreviation is used for voltages that alternate too, as in '12 V AC' or 'an AC voltage'. Keeping it for currents only is not how the term is used.

The point's height above or below the axle

The height a of the rim point above or below the axle rises and falls as the sine of the angle, one cycle per full turn of 360°. The angle itself grows steadily: it is the graph's horizontal axis.

a = r sin Ø

From sin Ø = a ÷ r, the height is a = r sin Ø, which is why the curve is called a sine function. r cos Ø is the sideways distance of the point from the axle, not its height.

8 V

V = VPeak × sin Ø, VPeak being the peak, the highest value reached: 16 × 0,5 = 8 V. Using cos 30° = 0,866 by mistake gives 16 × 0,866 = 13,9 V.

40°

The sine peaks at 90° and falls back as a mirror image of its rise, so 140° matches 180° − 140° = 40°. 50° is only how far 140° lies past the peak, and at 220° the value is negative.

1 ms

The period is 1 ÷ 250 = 4 ms, and the peak comes a quarter cycle (90°) after the rising zero crossing: 1 ms. At 2 ms, half a cycle, it is back at zero; 0,64 ms comes from 2πft = 1.

No, a sine curve is not made of half-circles

The manual warns that a sine is often wrongly drawn as two semicircles. Its true shape follows a = r sin Ø: it crosses the line at a slant, where half-circles would stand straight up.

Any periodic waveform can be built from sines

Fourier showed that any periodic signal is a sum of sines of different frequencies, so a circuit's response to sines gives its response to any waveform (Fourier analysis), not only to the mains.

A sine signal

Only a sine signal is pure, a single frequency: the fundamental. A triangular signal looks smooth but still holds harmonics, whole-number multiples of the fundamental, as square, pulse and sawtooth signals do.

Whole-number multiples of the fundamental

Harmonics lie at whole-number multiples of the fundamental f: 2f, 3f, 4f and so on. Fractions such as f ÷ 2 are not harmonics, and a pure sine signal has no harmonics at all.

11,1 MHz

The third harmonic is three times the fundamental: 3 × 3,7 = 11,1 MHz. 7,4 MHz is only 2 × 3,7, the second harmonic, and dividing by three gives no harmonic at all.

The time, from a rising zero crossing

In V = VPeak sin (2πft), t is the time since the sine passed through zero going positive, and f is the frequency. It is not the period, which the manual also writes as t.

Identical cycles, repeated again and again

An AC signal is a string of identical cycles, each a copy of the one before. That is why it has one period and one frequency: every cycle takes the same time.

The positive half-cycle

Each cycle has a positive half-cycle, above zero, and a negative half-cycle below it; a whole cycle holds both. The positive peak is only the highest point of the positive half-cycle.

Half of each cycle

A sine signal is positive for the first half of each cycle, 0° to 180°, and negative for the second half. A quarter of a cycle only takes it from zero up to its positive peak.

Twice

A sine crosses zero going down at 180°, where the positive half-cycle ends, and going up at 360°, where the next begins: twice per cycle. At the peaks, 90° and 270°, it turns back without reaching zero.

One cycle per second

Frequency counts the cycles completed each second, and its unit, the hertz, is one cycle per second. One second per cycle describes a period, the time of each cycle, which is measured in seconds.

The time taken by one complete cycle

The period is the time of one complete cycle, from any point to the matching point of the next cycle, given in s, ms or µs. The signal is positive for only half of that time.

1,8 kHz

Frequency is cycles per second: 720 ÷ 0,4 = 1800 Hz = 1,8 kHz. Multiplying, 720 × 0,4 = 288, means nothing, and 0,4 ÷ 720 = 556 µs is the time of one cycle, the period.

At any point, ending at the next matching point

A cycle may start at any point, a zero crossing, a peak or a value in between, and runs to the corresponding point of the next cycle. Starting at a zero crossing is merely convenient.

62,5 ms

t = 1 ÷ f = 1 ÷ 16 = 0,0625 s = 62,5 ms. 6,25 ms slips a power of ten: 1 s is 1000 ms, so 0,0625 s is 62,5 ms.

25 kHz

f = 1 ÷ t = 1 ÷ 0,000 04 s = 25 000 Hz = 25 kHz. 2,5 kHz is a power of ten out: it is the frequency of a signal whose period is 400 µs.

0,25 µs

t = 1 ÷ f = 1 ÷ 4 000 000 = 0,000 000 25 s = 0,25 µs, or 250 ns. 2,5 µs is ten times too long: it is the period of a 400 kHz signal.

1,4 MHz

f = 1 ÷ t = 1 ÷ 0,000 000 7 s = 1 428 571 Hz, about 1,43 MHz, nearest 1,4 MHz. 14 MHz is a power of ten out: it needs a period of only 0,07 µs.

1,25 ms

t = 1 ÷ f = 1 ÷ 800 = 0,001 25 s = 1,25 ms. 12,5 ms is ten times too long: it is the period of an 80 Hz tone, a tenth of the frequency.

50 Hz

South African mains runs at 50 Hz, so each cycle lasts 1 ÷ 50 = 20 ms. 60 Hz is the mains frequency of North America and some other countries, not of South Africa.

20 µs

t = 1 ÷ f = 1 ÷ 50 000 = 0,000 02 s = 20 µs. 2 µs is ten times too short: it is the period of a 500 kHz signal.

The distance it travels during one cycle

The wavelength is the distance the wave covers in one complete cycle, so λ = c × t, the speed of light times the period. From a positive peak to the next negative peak is only half a wavelength.

1200 m

λ = c × t = 3 × 10⁸ m/s × 4 × 10⁻⁶ s = 1200 m: the wave covers one wavelength in each period. Taking c as 3 × 10⁷ m/s, a power of ten short, gives 120 m.

It avoids handling powers of ten

With F in megahertz the zeros cancel: c = 300 000 000 m/s, so λ = c ÷ f becomes λ = 300 ÷ F. The two give exactly the same answer; neither is more accurate.

5 m

λ = 300 ÷ F = 300 ÷ 60 = 5 m, with F in MHz. 50 m is a power of ten out: it is the wavelength of a 6 MHz signal.

42,3 m

λ = 300 ÷ F = 300 ÷ 7,1 = 42,3 m, which is why 7,1 MHz lies in the 40 m band. 4,23 m has the decimal comma one place out, a power of ten.

69 cm

λ = 300 ÷ F = 300 ÷ 433 = 0,693 m, nearest 69 cm, inside the 70 cm band. 6,9 cm is a power of ten out, and 433 ÷ 300 = 1,44 turns the formula upside down.

3,75 MHz

F = 300 ÷ λ = 300 ÷ 80 = 3,75 MHz, in the 80 m band. 37,5 MHz is a power of ten out: its wavelength would be 8 m.

427 m

λ = c ÷ f = 300 000 000 ÷ 702 000 = 427 m. Putting 702 into λ = 300 ÷ F gives 0,427 m, but the shortcut needs F in MHz, here 0,702 MHz.

17 m

λ = 300 ÷ F = 300 ÷ 18,1 = 16,6 m, nearest 17 m, inside the 17 m band. 1,7 m has the decimal comma one place out, a power of ten.

2,3 GHz

F = 300 ÷ λ = 300 ÷ 0,13 = 2308 MHz, nearest 2,3 GHz, with λ in metres. Taking 13 cm as 1,3 m gives 230 MHz, a power of ten too low.

250 m

λ = 300 ÷ F = 300 ÷ 1,2 = 250 m. 25 m is ten times too short: it is the wavelength of a 12 MHz signal. 360 m multiplies instead of dividing.

20 MHz

F = 300 ÷ λ = 300 ÷ 15 = 20 MHz, with λ in metres. 2 MHz is a power of ten out: its wavelength is 150 m. 300 × 15 = 4500 MHz multiplies instead of dividing.

1500 km

Radio waves travel at the speed of light, c = 3 × 10⁸ m/s, so in 0,005 s they cover 1 500 000 m = 1500 km. 1500 m is the distance for 5 µs, a thousand times shorter.

The name is a handy label, not an exact wavelength

At 144 MHz, λ = 300 ÷ 144 = 2,08 m, so 'two metres' is a convenient name rather than a precise wavelength. No rule rounds band names; the name is simply shorter than the exact figure.

144 and 146 MHz

In South Africa the two-metre band runs from 144 to 146 MHz, where λ is about 2,07 m. 144 to 148 MHz is the wider band of the Americas, not the South African one.

At its positive peak

180° is half a cycle, so the two signals are mirror images: when one is at its negative peak the other is at its positive peak. Passing through zero would mean they were 90° apart.

A leads B by 120°

Signals of one frequency whose cycles begin at different moments have a phase difference. The one that gets there first leads, so A leads B; a third of a cycle is 360° ÷ 3 = 120°, not 33°.

B peaks a twelfth of a cycle after A

Lagging means arriving later: B reaches its peak, and every other point of its cycle, after A; 30° is 30 ÷ 360 = a twelfth of a cycle. Peaking earlier would make B the leading signal.

Any point, if it is the matching point on both

Any point can be used, a peak, a zero crossing or a value in between, as long as it is the corresponding point on both signals. Peaks and zero crossings are merely the easiest to see.

45°

One cycle is 360°, so one-eighth of a cycle is 360 ÷ 8 = 45°. Taking a whole cycle as 180° gives 22,5°, but 180° is only half a cycle.

72°

The period is 1 ÷ 2000 = 0,5 ms, so B is 0,1 ÷ 0,5 = 0,2 of a cycle later: 0,2 × 360° = 72°. Multiplying 0,1 by 360 without dividing by the period gives 36°.

60°

Both phases are measured from the same reference, so subtract: 85° − 25° = 60°. Adding them, 110°, would be right only if Y lagged the reference by 25° instead of leading it.

65°

The period is 1 ÷ 1200 = 0,833 ms, so B is 0,15 ÷ 0,833 = 0,18 of a cycle later: 0,18 × 360° = 64,8°, nearest 65°. Multiplying 0,15 by 360 without the period gives 54°.

10 µs

One cycle lasts 1 ÷ 5000 = 0,2 ms = 200 µs, and 18° is 18 ÷ 360 = 0,05 of a cycle: 0,05 × 200 µs = 10 µs. 100 µs is a power of ten out.

At both the positive and the negative peaks

P = V² ÷ R, and V² is largest at either peak, since squaring a negative voltage gives a positive number. At the zero crossings the power falls to zero.

V² is positive even when V is negative

The power is P = V² ÷ R, and squaring a negative voltage gives a positive result, so the resistor is heated just as much while V is negative. No stored heat is needed to explain it.

Root mean square

RMS means root mean square: the square root of the mean (average) of the squared values over a cycle. It is named after squaring the values, not after the sine.

The steady voltage that heats a resistor equally

The RMS value is the steady (DC) voltage that would make a resistance dissipate the same power. The average while the voltage is positive is another, smaller figure: 0,637 of the peak against 0,707.

Its RMS value

By definition the RMS value heats a resistance just as a DC voltage of the same size would. The average of a sine over a cycle is zero, and its peak is 1,414 times the RMS value.

0,707

VRMS = VPeak ÷ √2 = 0,707 × VPeak, since 1 ÷ √2 = 0,707. Multiplying by 1,414, which is √2, goes the other way: from the RMS value up to the peak.

35,4 V

VRMS = 0,707 × VPeak = 0,707 × 50 = 35,4 V. Multiplying by 1,414 gives 70,7 V, but that turns an RMS value into a peak, the wrong way round here.

25 V

VPeak = √2 × VRMS = 1,414 × 18 = 25,45 V, nearest 25 V. Multiplying by 0,707 instead gives 12,7 V, less than the RMS value, which a peak can never be.

Sine signals only

The 0,707 factor holds only for a sine signal; other waveforms have other ratios. A square signal, for one, has an RMS value equal to its peak value.

1,7 A

IRMS = √(P ÷ R) = √(72 ÷ 50) = √1,44 = 1,2 A, and IPeak = 1,414 × 1,2 = 1,697 A, nearest 1,7 A. 1,2 A is the RMS current, which the power formula gives, not the peak.

RMS

Any AC figure without a label, voltage or current, is taken to be an RMS value, so 24 V means 24 V RMS. A peak value has to be labelled as one.

15 A RMS

I = P ÷ V = 3450 ÷ 230 = 15 A. The 230 V is an RMS value, as AC figures are unless stated otherwise, so the 15 A is RMS too; its peak is 1,414 × 15 = 21,2 A.

8 W

With RMS values the DC formula holds: P = V² ÷ R = 40² ÷ 200 = 1600 ÷ 200 = 8 W. Using the peak voltage, 56,6 V, doubles the answer to a wrong 16 W.

20 V

V = √(P × R) = √(25 × 16) = √400 = 20 V, an RMS value, as the power formulas work in RMS. √(P ÷ R) = 1,25 gives the current in amperes, not the voltage.

4,8 W

P = V² ÷ R = 12,6² ÷ 33 = 158,8 ÷ 33 = 4,81 W, nearest 4,8 W. Squaring the peak voltage, 17,8 V, instead of the RMS value doubles it to about 9,6 W.

2,5 A

Ohm's Law holds with RMS values: I = V ÷ R = 30 ÷ 12 = 2,5 A RMS. 12 ÷ 30 = 0,4 divides the wrong way round, and 30² ÷ 12 = 75 is the power in watts, not a current.

0,75 A

I = P ÷ V = 36 ÷ 48 = 0,75 A, an RMS current, as the power formulas work with RMS values. 1,06 A is the peak current, 1,414 times larger; 48 ÷ 36 = 1,33 divides the wrong way.

3 MHz to 30 MHz

HF runs from 3 to 30 MHz, wavelengths from 100 m down to 10 m, and holds the amateur bands from 80 m to 10 m. 30 to 300 MHz is VHF, the next range up.

30 MHz to 300 MHz

VHF runs from 30 to 300 MHz, wavelengths 10 m down to 1 m, and includes the 6 m and 2 m bands. 300 MHz to 3 GHz is the next range up, UHF.

300 MHz to 3 GHz

UHF runs from 300 MHz to 3 GHz, wavelengths 1 m down to 100 mm, and includes the 70 cm and 23 cm bands. 30 to 300 MHz is VHF, the range below it.

VHF

The 2 m band, about 300 ÷ 2 = 150 MHz, lies in VHF, 30 to 300 MHz, as does the 6 m band near 50 MHz. UHF begins only at 300 MHz.

MF

300 ÷ 160 = 1,875 MHz, inside MF, 300 kHz to 3 MHz. HF begins only at 3 MHz, so the manual's table lists 160 m as the MF amateur band.

UHF

F = 300 ÷ λ = 300 ÷ 0,23 = 1304 MHz, about 1,3 GHz, between 300 MHz and 3 GHz: UHF, which also holds the 70 cm band. SHF starts only at 3 GHz.

VHF

VHF, 30 to 300 MHz, has wavelengths from 300 ÷ 30 = 10 m down to 300 ÷ 300 = 1 m. HF, 3 to 30 MHz, runs from 100 m down to 10 m, the range below.

It counted as high when the names were chosen

When the name 'high frequency' was given, these were among the highest frequencies in use; higher ranges were named later, and 'high' was kept for historical reasons. It describes the frequency, not how high the waves travel.

HF, VHF and perhaps UHF

Most amateurs work on HF and VHF, and many on UHF as well, so these three ranges matter most. Leaving out HF, where most amateur bands lie, is wrong, and SHF is for the few.

Ultra

UHF is ultra-high frequency, 300 MHz to 3 GHz, the range above VHF, very high frequency. The table names its ranges low, medium, high, very, ultra, super, extremely and tremendously high.