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5.2 Power in a Purely Resistive Circuit

In AC circuits, we use the instantaneous values of voltage and current to find the instantaneous power. To better understand this concept, we plot curves based on the instantaneous power of receivers. As in DC circuits, we start by determining the current in the circuit. We will analyze the power in the resistive circuit (Fig. 5.1).

A simple circuit diagram showing an AC voltage source labeled "Vac" connected in series with a resistor labeled "R". The circuit forms a closed loop, with the voltage source and resistor connected by wires.

Fig. 5.1 Purely resistive circuit with AC voltage source.

The value of instant current is:

i=I0·senωt (5.3)

In addition, the instantaneous voltage function has the same frequency:

u=U0·senωt (5.4)

We can also express these equations as:

i=2·I·senωt (5.5)

u=2·U·senωt (5.6)

In order to find the power, we have to multiply voltage and current. Now we plot the voltage wave as well the power waveform in a resistive load (Fig. 5.2). When the voltage is zero, the power is also zero. Otherwise, the power waveform is always positive, because the resistor can only dissipate energy, not supply energy to the circuit. The resistor outputs power regardless of whether the voltage is positive or negative.

Fig. 5.2 AC power to its real.

We get the instantaneous power curve multiplying the sine function of current i=2·I·senωt by the sine function of voltage u=2·V·senωt. As we see, this curve is always positive.

Question What does this curve show about resistance? This graph illustrates that the resistor absorbs energy from the generator and converts it into heat.

Question What does the average value of this curve represent? The mean value of this power is the active power (P) or real power, and we measure it with a wattmeter.

In a resistor we know that power is:

P=R·I2W (5.7)

In this case, the instantaneous power equals the product of instantaneous current and voltage. Multiplying, we get

pt=ut·it=2·V·senωt·2·I·senωt=2·U·I·sen2ωt=U·I·1-cos2ωt (5.8)

Applying trigonometric identities, we know:

sen2u=1-cos2u2

P=1T∫0Tp·dt=U·I (5.9)

Then, we obtain the average power integrating the power over one cycle and dividing by the cycle time.

P=1T∫0Tp·dt=U·I·1-cos2wt·dt=UIT∫0Tdt-U·I·∫0Tcos2wtdt=UITt0T-UI2wsen2wt0T=U·IT·T=U·I (5.10)

Remember:

∫cos2xdx=12sin2xintegratingparts (5.11)

∫cos2xdx=12∫cosudu∫cosudu=sinuu=2x→dudx=2→dx=12du=2x→dudx=2∫cos2xdx=12sinu

Therefore, the most significant measure is the active power, which comes from the effective values of U and I.

练习题

In a purely resistive AC circuit, why is the instantaneous power waveform always positive or zero?

A. Because the voltage is always positive

B. Because the resistor can only dissipate energy, not supply energy to the circuit

C. Because the current leads the voltage by 90°

D. Because the power factor is always greater than 1

What is the average power (active power) in a purely resistive AC circuit with effective voltage $U$ and effective current $I$ ?

P = 2 U I

P = U \cdot I

P = \frac{U \cdot I}{2}

P = U^{2} \cdot I

The instantaneous power in a purely resistive AC circuit can be expressed as $p (t) = U \cdot I \cdot (1 - cos 2 ω t)$ . What does this expression reveal about the power frequency?

A. The power has the same frequency as the voltage and current

B. The power frequency is half the voltage and current frequency

C. The power frequency is double the voltage and current frequency

D. The power has no frequency component

Which of the following statements are true about power in a purely resistive AC circuit?

A. The power waveform is always positive or zero

B. The resistor converts absorbed energy into heat

C. The average power is called active power or real power

D. The resistor can supply energy back to the circuit during negative voltage cycles

E. Active power is measured with a wattmeter

In a purely resistive AC circuit, the instantaneous voltage function has the same frequency as the instantaneous current function.

In AC circuits, the instantaneous power is found by multiplying the instantaneous voltage and instantaneous current.

A resistor in an AC circuit can supply energy back to the generator when the voltage is negative.

The mean value of power in a resistive AC circuit is called the ___ power or real power, and it is measured with a wattmeter.

In a resistor, power can be calculated using the formula $P = R \cdot I^{2}$ , where $R$ is resistance and $I$ is the ___ value of current.

Explain why the power waveform in a purely resistive AC circuit is always positive or zero, and contrast this with the power behavior of coils and capacitors.

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5.2 Power in a Purely Resistive Circuit