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5.3 Power in Circuits with Coil

Now we analyze the power in circuits with a single coil. We know that current delays to the voltage by 90 degrees, or π/2 radians (Fig. 5.3). Consequently, the instantaneous values of current and voltage are:

u=2·U·senωt (5.12)

i=2·I·senωt-π/2 (5.13)

A line graph depicting two sinusoidal waves on an X-Y axis. The horizontal axis is labeled "ωt" and the vertical axis is unlabeled. The waves intersect at regular intervals, with key points marked at π and 2π on the horizontal axis. The waves are labeled "u" and "i," showing their periodic nature.

Fig. 5.3 Power curve in a purely inductive circuit.

Similarly, only by multiplying the voltage at any given instant and the current provides the power at that instant. Then, we plot the instantaneous power curve in this circuit (Fig. 5.3).

Now, let us clarify what this means. As the power is:

pt=ut·it=2·U·senωt·2·I·senωt-π2=2·U·I·senωt·senωt-π/2=-2·U·I·senωt·cosωt=-U·I·sen2ωt (5.14)

Recalling the trigonometric functions:

sen2x=2senxcosxcos2x=cos2x-sen2xtg2x=2tgx1-tg2x

If we look at the results, we see that energy has twice frequency that of the original.(Remember: w=2·π·f).

Question What is the behavior if P > 0?

If p > 0, then the inductance L acts as a receiver (absorbs and stores energy).

Question What is the behavior when P < 0? If p < 0, inductance L acts as a generator (it returns the energy to the circuit). Only by analyzing the graphical representation we see that the power is positive during the quarter cycle in which the coil is charged (the coil receives energy from the generator and acts as a receiver). In the next quarter cycle, the coil releases its stored energy back to the generator, making the instantaneous power negative. Because the average value of the power during a cycle is zero, it is accurate to state that a pure inductive circuit generates no active power.

Question What is the average circuit power in a pure inductive circuit? Due to the fact that the mean value of the power is null, a pure inductive circuit does not generate real power. But, is there power during the coil's operation if it does not consume any actual energy for its operation? Although the coil itself does not consume energy during the operation, the constant charging and discharging process results in current flowing through the wires. Power's constant fluctuation through the conductors is known as reactive power (Q).

P = -1/T ∫₀ᵀ UI sin 2ωt = 0 (5.15)

In a coil we have that:

Q = UL·I = XL·I² (5.16)

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

练习题

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

A. Current leads voltage by 90 degrees

B. Current lags voltage by 90 degrees

C. Current and voltage are in phase

D. Current leads voltage by 180 degrees

If the voltage source in a purely inductive circuit has frequency $f$ , what is the frequency of the instantaneous power?

f /2

f

2 f

4 f

What is the correct formula for reactive power $Q$ in a coil?

Q = U_{L} \cdot I^{2}

Q = X_{L} \cdot I

Q = U_{L} \cdot I = X_{L} \cdot I^{2}

Q = X_{L} \cdot I^{2} /2

Which of the following statements correctly describe the behavior of instantaneous power in a purely inductive circuit?

A. When

p > 0

, the inductance absorbs and stores energy

B. When

p < 0

, the inductance returns energy to the circuit

C. The coil charges during one quarter cycle and discharges during the next

D. The average power over one complete cycle is positive

In a purely inductive circuit, the average value of power over one complete cycle equals zero.

In a purely inductive circuit, the instantaneous power is always positive, just like in a purely resistive circuit.

Reactive power is measured in watts (W), the same unit used for active power.

In a coil circuit, the current lags the voltage by ___ degrees, which equals $π /2$ radians.

The unit used to measure reactive power is the reactive volt-ampere, abbreviated as ___.

Explain why a pure inductive circuit has zero average active power, yet reactive power still exists during operation.

Compare the power behavior in a purely resistive circuit versus a purely inductive circuit. Explain why their average power values differ.

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

A. Resistors have higher resistance than inductors

B. Resistors only dissipate energy, while inductors store and return energy to the circuit

C. Inductors have higher power ratings than resistors

D. The current frequency is different in resistive and inductive circuits

Both resistors and pure inductors connected to AC sources have zero average power consumption over a complete cycle.

Which of the following statements correctly describe the differences between power in a purely resistive circuit and power in a purely inductive circuit?

A. In a resistive circuit, voltage and current are in phase; in an inductive circuit, current lags voltage by 90°

B. In a resistive circuit, the power frequency equals the source frequency; in an inductive circuit, the power frequency is twice the source frequency

C. Both circuits produce the same average active power

D. In a resistive circuit, power is measured in watts; in an inductive circuit, reactive power is measured in VAr

E. A resistor converts electrical energy to heat; an inductor stores energy in a magnetic field

The instantaneous power in a purely resistive AC circuit is given by $p (t) = U \cdot I \cdot (1 - cos (2 ω t))$ , while the instantaneous power in a purely inductive circuit is given by $p (t) = - U \cdot I \cdot sin (2 ω t)$ . In both cases, the power oscillates at ___ the frequency of the source voltage.

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5.3 Power in Circuits with Coil