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5.4 Power in a Purely Capacitive Circuit

In this case the current that flows through the circuit leads the voltage π/2 radians (Fig. 5.4). Therefore, the instantaneous values of current and voltage are:

u = 2·U·sin ωt (5.17)

i = 2·I·sin(ωt + π/2) (5.18)

A circuit diagram featuring an AC voltage source labeled "Vm sin wt" connected to a capacitor labeled "C". To the right, a graph shows sinusoidal waveforms with axes labeled "u" and "i". The horizontal axis is marked with angles π/2, π, 3π/2, and 2π, representing the phase of the wave over time "wt".

Fig. 5.4 Purely capacitive circuit.

In this instance, in order to find the instantaneous power, we multiply the instantaneous voltage and current as we see (Eq. 5.19).

p(t) = u(t)·i(t) = 2·U·I·sin ωt·sin(ωt + π/2) = U·I·sin 2ωt (5.19)

Then, the active power has twice the frequency of voltage and current waves.

Recalling the trigonometric functions:

sin(π/2 + θ) = cos θ

sin 2x = 2 sin x cos x

cos 2x = cos²x - sin²x

tan 2x = 2 tan x / (1 - tan²x)

As previously, we get the power curve. Once more, the capacitor's continuous charge and discharge process generate positive and negative cycles of instantaneous power. Then, we can plot the instantaneous power values in a purely capacitive circuit (Fig. 5.5).

Graph depicting power consumption and power returned over time. The x-axis represents time, labeled as t, and the y-axis represents power. The graph shows sinusoidal curves with annotations indicating "Power consumed" and "Power returned." The average power is marked along the x-axis. Key points include π/2 and 3π/2 on the time axis.

Fig. 5.5 AC power waveforms in a purely capacitive circuit.

Question What happens if p > 0? If p > 0, then capacitor stores energy.

Question What happens if p < 0? If p < 0, capacitor releases energy.

What kind of power does a capacitor generate? A capacitor does not generate active power but reactive power.

What is the active power?

Question The active power is:

P = 1/T ∫₀ᵀ U·I·sin 2ωt = 1/T ∫₀ᵀ U·I·2·sin ωt·cos ωt = 2·U·I/T ∫₀ᵀ sin ωt·cos ωt = 0 (5.20)

Remembering the trigonometric functions of:

sin 2x = 2 sin x·cos x

cos 2x = cos²x - sin²x

tan 2x = 2 tan x / (1 - tan²x)

If we compare the instantaneous power curves of the coil and the capacitor, we see that during the coil's energy discharge (negative power cycle), the capacitor charges with the same energy (positive power cycle). Consequently, we say that the reactive power of the capacitor is negative with respect to that of the coil, and thus their effects cancel each other out.

In a capacitor we have:

Q = UC·I = XC·I² (5.21)

We measure it in reactive volt-amperes (VAr).

练习题

In a purely capacitive circuit, what is the phase relationship between the current and the voltage?

A. The current lags the voltage by

π /2

radians

B. The current leads the voltage by

π /2

radians

C. The current and voltage are in phase

D. The current leads the voltage by

π

radians

The instantaneous power in a purely capacitive circuit is given by $p (t) = U \cdot I \cdot sin (2 ω t)$ . What is the frequency of this power waveform compared to the voltage and current waveforms?

A. Half the frequency of voltage and current

B. The same frequency as voltage and current

C. Twice the frequency of voltage and current

D. Four times the frequency of voltage and current

How does the reactive power of a capacitor compare to that of a coil (inductor)?

A. They have the same sign and add together

B. The capacitor's reactive power is negative with respect to the coil's, so their effects cancel

C. The capacitor has no reactive power, only the coil does

D. Both have zero reactive power

Which of the following statements correctly describe the behavior of instantaneous power in a purely capacitive circuit? Select all that apply.

A. When

p > 0

, the capacitor stores energy

B. When

p < 0

, the capacitor releases energy

C. The capacitor generates active power

D. The instantaneous power has both positive and negative cycles

E. The active power averaged over one cycle equals zero

Which of the following statements are true when comparing power in resistors, coils, and capacitors in AC circuits?

A. A resistor always consumes positive power and converts it to heat

B. A coil has zero average active power over one cycle

C. A capacitor has zero average active power over one cycle

D. Both coils and capacitors can have negative instantaneous power

E. A resistor can have negative instantaneous power

In a purely capacitive circuit, the active power is zero because the integral of $sin (2 ω t)$ over one complete period equals zero.

A capacitor in an AC circuit generates active power that can perform useful work.

The current in a purely capacitive circuit leads the voltage by $π /2$ radians, while the current in a purely inductive circuit lags the voltage by $π /2$ radians.

The reactive power in a capacitor is calculated using the formula $Q = U_{C} \cdot I =$ ___ $\cdot I^{2}$ , and is measured in reactive volt-amperes (VAr).

In a purely capacitive circuit, the instantaneous power is given by $p (t) =$ ___ $\cdot sin (2 ω t)$ , which has twice the frequency of the voltage and current waveforms.

Explain why a capacitor does not generate active power in an AC circuit, even though instantaneous power is not always zero.

Using the instantaneous voltage $u = 2 \cdot U \cdot sin (ω t)$ and current $i = 2 \cdot I \cdot sin (ω t + π /2)$ for a purely capacitive circuit, derive the expression for instantaneous power and state its key characteristics.

In a purely resistive circuit, the instantaneous power is always positive or zero, while in purely inductive and capacitive circuits, the instantaneous power alternates between positive and negative values. What is the fundamental reason for this difference?

A. Resistors have higher resistance than inductors or capacitors

B. Resistors only dissipate energy as heat, while inductors and capacitors store and release energy

C. The frequency of power is different in each circuit type

D. The voltage and current have different amplitudes in each circuit type

Which of the following statements correctly describe the active power (P) and reactive power (Q) in purely inductive and capacitive circuits?

A. The active power in a purely inductive circuit is zero

B. The active power in a purely capacitive circuit is zero

C. Both inductors and capacitors generate reactive power measured in VAr

D. The reactive power of a capacitor is positive with respect to that of a coil

E. The instantaneous power in both inductive and capacitive circuits has twice the frequency of the voltage and current

In a purely inductive circuit, the current lags the voltage by π/2 radians, while in a purely capacitive circuit, the current leads the voltage by π/2 radians. This phase difference causes the reactive power of the capacitor to be negative with respect to that of the inductor, allowing their reactive effects to cancel each other when connected together.

The instantaneous power expression for a purely resistive circuit is $p (t) = U \cdot I (1 - cos 2 ω t)$ , which is always positive. In contrast, the instantaneous power expressions for purely inductive and capacitive circuits are $p (t) = - U \cdot I sin 2 ω t$ and $p (t) = U \cdot I sin 2 ω t$ respectively. The key difference is that resistors only ___ energy, while inductors and capacitors alternately store and release energy.

Explain why both purely inductive and purely capacitive circuits have zero active power, but a purely resistive circuit has non-zero active power. Use the concepts of energy storage and energy dissipation in your explanation.

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5.4 Power in a Purely Capacitive Circuit