ELECTENG 101 Chap.9 AC Circuit Concepts and Power
AC Circuit Concepts and Power
Sinusoidal signals are described by amplitude, angular frequency and phase. RMS gives the DC-equivalent heating value, while capacitors and inductors link voltage and current through derivatives and energy storage. AC power separates apparent, real and reactive quantities, with power factor describing the real share.
Period and frequency describe repetition, while a phase difference compares the timing of equal-frequency waveforms. Capacitor voltage cannot jump without an impulse current, and inductor current cannot jump without an impulse voltage. Their derivative laws produce the familiar quarter-cycle phase relations in sinusoidal steady state.
Power calculations must use compatible RMS quantities: apparent power is the voltage-current product, real power includes the cosine of phase difference, and reactive power represents reversible energy exchange with storage elements.
What this chapter covers
- 01
Cosine signal parameters
- 02
Period and frequency
- 03
Lead and lag
- 04
Average and RMS values
- 05
Capacitor and inductor laws
- 06
Real and reactive power
- 07
Power factor
Worked example · free
Differentiate a capacitor waveform
- +2Differentiate voltage to obtain −1000 sin(100t).
- +2Multiply by 20 µF to obtain −0.02 sin(100t) A.
- +2Rewrite as 0.02 cos(100t+90°) A.
Key terms
- RMS value
- The varying quantity’s DC-equivalent value for resistive heating.
- Reactive power
- The oscillatory power exchange associated with energy-storage elements in AC operation.
- Power factor
- Real power divided by apparent power, equal to the cosine of phase difference for sinusoids.
AC Circuit Concepts and Power FAQ
What is the RMS value of a sine wave?
For a sinusoid, RMS magnitude is peak magnitude divided by the square root of two. State whether a supplied value is peak or RMS before using it. The formula depends on waveform shape, so it should not be transferred automatically to a square wave or an arbitrary signal. RMS is useful because it predicts equivalent resistive heating.
Does capacitor current lead voltage?
Under passive references in sinusoidal steady state, capacitor current leads its voltage by 90 degrees. Inductor current lags its voltage by 90 degrees. Derive the relation from i equals C dv/dt or v equals L di/dt when signs are uncertain. The statement also assumes the same positive frequency and a consistent cosine reference.
Why are watts and volt-amperes different?
Watts measure average real-power transfer, while volt-amperes measure apparent power. Their ratio is power factor, so the numbers coincide only at unity power factor. Reactive power is measured in vars and accounts for the quadrature share. These units are related through the power triangle but should not be substituted for one another.
How do I decide whether one sinusoid leads another?
Write both signals with the same trigonometric function, positive amplitude and frequency, then compare their phase constants. The waveform with the larger phase in that common form reaches the same feature earlier and leads. A sketch of one period guards against sign errors, especially when converting between sine and cosine forms.
What happens to capacitors and inductors at DC steady state?
After transients have settled, an ideal capacitor carries no current and behaves as an open circuit, while an ideal inductor has no voltage drop and behaves as a short circuit. These are steady-state limits, not switching rules. Capacitor voltage and inductor current continuity must be respected immediately after a change.
How can I check an AC power calculation?
Confirm that voltage and current are both RMS, compute apparent power first, and verify the magnitude of real power does not exceed it. The power factor must lie between zero and one in magnitude. Use the stated lead-lag convention to assign reactive behaviour and retain watts, vars and volt-amperes on separate results.
Exam move
Sketch one period and label peaks, zero crossings and relative phase before doing algebra. Convert all peak or RMS values consistently. For energy-storage elements, apply the derivative law before using remembered phase language. In power questions, calculate apparent power first, then use phase or power factor to divide it into real and reactive parts.
Build a waveform table containing amplitude, angular frequency, frequency, period and phase for several cosine expressions. Rewrite sine forms as cosine before comparing lead and lag. Differentiate one capacitor voltage and one inductor current symbolically, then identify both the magnitude factor and quarter-cycle shift. Create a power triangle for resistive, lagging and leading cases, attaching the correct unit to each side.
Finish with mixed peak and RMS data: convert everything to one convention before calculating, and reject any answer whose real-power magnitude exceeds apparent power. Add a transient-versus-steady-state comparison in which capacitor voltage and inductor current continuity set the initial conditions before sinusoidal rules apply.
Solve one fixed-real-power load at two power factors, recomputing current and conductor loss to expose the squared benefit. For each waveform comparison, give both the angular phase difference and its equivalent time shift, checking that the frequencies match. Practise explaining why the full-cycle average of a sinusoid may vanish while its RMS and heating effect remain nonzero.
Finally, annotate every result as peak, RMS, average, real, reactive or apparent so values with similar numerical sizes cannot be interchanged silently.
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