Alternating currents

• The mean current is zero, the mean power is half the peak power, and the r.m.s. current is the d.c. that heats the same: \(I_0/\sqrt{2}\), for a sine wave only.
• Mean current zero, mean power half the peak, r.m.s. = the d.c. that heats the same. One diode removes, four diodes invert, a parallel capacitor fills in.

• 21.1.1 I can understand — understand and use the terms period, frequency and peak value as applied to an alternating current or voltage
• 21.1.2 I can use — use equations of the form x = x₀ sin ωt representing a sinusoidally alternating current or voltage
• 21.1.3 I can recall — recall and use the fact that the mean power in a resistive load is half the maximum power for a sinusoidal alternating current
• 21.1.4 I can — distinguish between root-mean-square (r.m.s.) and peak values and recall and use I_r.m.s. = I₀/√2 and V_r.m.s. = V₀/√2 for a sinusoidal alternating current
• 21.2.1 I can — distinguish graphically between half-wave and full-wave rectification
• 21.2.2 I can explain — explain the use of a single diode for the half-wave rectification of an alternating current
• 21.2.3 I can explain — explain the use of four diodes (bridge rectifier) for the full-wave rectification of an alternating current
• 21.2.4 I can — analyse the effect of a single capacitor in smoothing, including the effect of the values of capacitance and the load resistance

Units, significant figures and working are part of the physics. Give a calculated answer to the same number of significant figures as the least precise data, or one more; keep full precision in the working and round only at the end; write the unit with every final answer. A fifth of the qualification is experimental: Papers 3 and 5 test AO3 only, and their questions may be set in contexts outside the syllabus content, so the practical work in this chapter is set out as method, recording, graphs and uncertainties rather than as theory.

• “The r.m.s. current is the average current.” Correct The mean current is zero; the r.m.s. current is not; \(I_0/\sqrt2\) is for sine waves only. Over a whole cycle the positive and negative halves cancel, so the mean of I is zero. The r.m.s. value is the square root of the mean of the square, the equivalent steady current for heating. For a square wave, or a rectified output, go back to that definition.
• “The mains is 230 V, so the p.d. never goes above 230 V.” Correct A quoted a.c. p.d. is an r.m.s. value unless it says otherwise. The peak is \(230\sqrt2 = 325\) V, and the peak-to-peak value is 650 V.
• “The trace is 6 divisions high at 2 V div−1, so the peak is 12 V.” Correct The full height of the trace is the peak-to-peak value, 12 V. The peak is half of it: 6 V.
• “\(I = 2.4\sin(120\pi \times 0.0020) = 0.032\) A.” Correct In \(x = x_0\sin\omega t\), \(\omega t\) is in radians. In radian mode the answer is 1.64 A. A value at a time well before T/4 that comes out tiny is the degree-mode warning sign.
• “The mean power in the resistor is \(I_0^2R\).” Correct \(I_0^2R\) is the maximum power, reached only at the instants of peak current. The mean is half of it: \(\tfrac12 I_0^2R = I_{\mathrm{r.m.s.}}^2R\).
• “A half-wave rectifier turns the negative half-cycles positive.” Correct One diode removes them: the output is zero for half of every cycle. Turning them over is what a bridge of four diodes does.
• “The smoothing capacitor goes in series with the load.” Correct It goes in parallel with the load. In series it would block the steady current it is meant to supply.
• “The r.m.s. value is the average current.” Repair The average of a sinusoidal current over a cycle is zero. The r.m.s. value is the steady current that would dissipate energy at the same rate in a resistor.
• “\(I_{\mathrm{r.m.s.}} = I_0/\sqrt2\) for every alternating current.” Repair Only for a sine wave. A square wave of ±5.0 V has an r.m.s. value of 5.0 V; a half-wave output of peak V0 has V0/2.
• “The mains is 230 V, so the p.d. never exceeds 230 V.” Repair 230 V is the r.m.s. value; the peak is 325 V.
• “The trace is 6 divisions high at 2 V div−1, so V0 = 12 V.” Repair That is the peak-to-peak value. V0 = 6 V.
• “\(I = 2.4\sin(120\pi \times 0.002) = 0.032\) A.” Repair ωt is in radians. In radian mode the answer is 1.64 A.
• “Mean power = \(I_0^2R\).” Repair That is the maximum power. The mean is half of it: \(\tfrac12 I_0^2R = I_{\mathrm{r.m.s.}}^2R\).
• “The power in the resistor alternates between positive and negative.” Repair \(P = I^2R\) is never negative. It varies between 0 and P0 at twice the supply frequency.
• “A half-wave rectifier turns the negative half-cycles positive.” Repair It removes them. Turning them over is full-wave rectification.
• “In a bridge rectifier all four diodes conduct together.” Repair Two conduct, in series, on each half-cycle; the other two are reverse-biased.
• “The smoothing capacitor goes in series with the load.” Repair It is in parallel with the load. In series it would block the steady current.
• “A larger load resistance gives more ripple.” Repair A larger R draws less current, so the capacitor discharges more slowly (larger RC) and the ripple is smaller.
• “Full-wave output has the same frequency as the supply.” Repair Its humps repeat every T/2, so the output (and its ripple) has frequency 2f.

• Read the command word before you decide how much to write. This syllabus has fifteen of them: calculate, comment, compare, define, describe, determine, explain, give, identify, justify, predict, show (that), sketch, state and suggest. Define wants a precise meaning — for a physical quantity, usually an equation in words with every quantity named. State and give want a fact and nothing more. Describe wants the points or the features. Explain wants the reasons and the relationships — a describe-level answer to an explain question is incomplete however well written it is. Show (that) gives you the result and asks for the structured evidence that leads to it, so every step must appear — and a final value worked to one more significant figure than the one printed makes it plain that you calculated it rather than copied it. Sketch wants a freehand graph with its key features — intercepts, asymptotes, the shape — correct, but no plotted scale.
• Sketching them. Draw the input first, then each output directly beneath it with the time axes lined up. Half-wave: copy the positive humps and draw the axis itself where the negative humps were. Full-wave: copy the positive humps and reflect each negative hump in the time axis. Mark V0 on both and label a gap (half-wave) or a hump spacing (full-wave) as T/2.
• Sketching the smoothed output. Follow the rising edge of the first hump up to the peak; then draw a gentle, almost straight falling line (the start of an exponential) until it meets the next rising edge; follow that edge up to the peak; repeat. Draw the rectified humps faintly underneath. A larger RC gives a flatter falling line that meets the next hump nearer its peak.
• Which value does the question give you? A mains p.d. or a meter reading is r.m.s.; a value read from an oscilloscope trace or an equation is a peak (or, if it is the full trace height, a peak-to-peak). Convert once, at the start, and write down which is which.
• Interleave with the chapters that use this one. Topic 21 is the last electricity topic, so interleave it backwards: when you revise Topic 19, re-answer “why does a larger RC reduce the ripple?”; when you revise Topic 17, compare \(x = x_0\sin\omega t\) for a displacement and for a current, and the twice-frequency energy graph with the twice-frequency power graph; when you revise Topic 20, say where the sinusoidal e.m.f. of a supply comes from. Recalling a topic inside a new context is worth more than another pass over this chapter on its own; at A Level, Paper 4 assumes the whole of the AS content, so nothing here is ever finished with.

• Cambridge International AS & A Level Physics 9702 has five components. Topic 21 is A Level content, so it is examined in Papers 4 and 5. A Level content: examined in Paper 4 (A Level structured, which also requires the AS content) and, as practical context, Paper 5. AS Level candidates take Papers 1, 2 and 3; A Level candidates take all five, either staged over two years (Papers 1–3 in year one, Papers 4 and 5 in year two) or together in one series. Examinations are available in the June and November series, and in March in India.
• Across both the AS Level and the A Level the assessment objectives are weighted AO1 40% (knowledge and understanding), AO2 40% (handling, applying and evaluating information) and AO3 20% (experimental skills and investigations). AS candidates are graded a–e; A Level candidates A*–E. The Data and formulas sheet is printed as page 2 of Papers 1 and 2 and as pages 2 and 3 of Paper 4: it gives the constants and a short list of formulas. Every other equation in this chapter is one the syllabus says you must recall, and this chapter says which is which.
• Structured questions may ask you to define the r.m.s. value, state the period, frequency or peak value of a given trace or equation, sketch the output of a half-wave or a bridge rectifier, with and without a smoothing capacitor, and explain how one diode, four diodes or a capacitor produce that output. There is no multiple-choice paper on A Level content.
• The Data and formulas sheet prints \(x = x_0\sin\omega t\) (“alternating current/voltage”) and the capacitor discharge equation \(x = x_0\mathrm{e}^{-t/RC}\). You recall \(f = 1/T\), \(\omega = 2\pi f\), “mean power = half the maximum power”, \(I_{\mathrm{r.m.s.}} = I_0/\sqrt2\), \(V_{\mathrm{r.m.s.}} = V_0/\sqrt2\) and the power equations. Typical calculations: peak from r.m.s. and back, mean and peak power, a value at a given time (radian mode), and the size of a ripple.
• The oscilloscope measures p.d., time and frequency: read the peak-to-peak height and the length of several cycles. A plan might vary the smoothing capacitance and measure the ripple on an oscilloscope, then plot the ripple against 1/C. In this measurement the largest uncertainty is reading the edges of a small, thick trace. Both are set out in the practical-skills section.
• Read the command word before you decide how much to write. This syllabus has fifteen of them: calculate, comment, compare, define, describe, determine, explain, give, identify, justify, predict, show (that), sketch, state and suggest. Define wants a precise meaning — for a physical quantity, usually an equation in words with every quantity named. State and give want a fact and nothing more. Describe wants the points or the features. Explain wants the reasons and the relationships — a describe-level answer to an explain question is incomplete however well written it is. Show (that) gives you the result and asks for the structured evidence that leads to it, so every step must appear — and a final value worked to one more significant figure than the one printed makes it plain that you calculated it rather than copied it. Sketch wants a freehand graph with its key features — intercepts, asymptotes, the shape — correct, but no plotted scale.