(a) Use the results of part (a) of Exercise 31.21 to show...

(a) Use the results of part (a) of Exercise 31.21 to show that the average power delivered by the source in an $L-R-C$ series circuit is given by $P_{\mathrm{av}}=I_{\mathrm{rms}}^{2} R$ . (b) An $L-R-C$ series series circuit has $R=96.0 \Omega,$ and the amplitude of the voltage across the resistor is 36.0 $\mathrm{V} .$ What is the average power delivered by the source?

Key Concepts

Average Power in AC Circuits

In AC circuits, the average power delivered by a source is determined by taking the time average of the instantaneous power over a full cycle. For a circuit element, the instantaneous power is the product of the instantaneous voltage and current. In resistive components, this averaging yields a net nonzero value, whereas in reactive components (such as inductors and capacitors) the energy oscillates between the source and the reactive elements, resulting in zero net energy dissipation over a complete cycle. Thus, the average power is primarily associated with the work done by the resistive element.

Energy Dissipation in Resistive versus Reactive Components

Resistive components dissipate energy as heat, and their contribution to the average power is given by the RMS current squared multiplied by the resistance. In contrast, reactive components like inductors and capacitors store and release energy cyclically without net energy loss over a full cycle. This fundamental difference underlies the power equation for AC circuits, where only the resistor contributes to the net average power consumption.

AC Circuit Analysis

AC circuit analysis involves understanding the behavior of circuits when subjected to alternating currents and voltages. In such analyses, phasor techniques and frequency domain methods are typically employed, especially when dealing with circuits composed of resistors, inductors, and capacitors. The behavior of each component under AC excitation is characterized by its impedance, which dictates how it affects the amplitude and phase of currents and voltages in the circuit.

RMS Values

The root-mean-square (RMS) value of a periodic current or voltage is a measure that represents the equivalent direct current (DC) value that would deliver the same power to a load. For sinusoidal signals, the RMS value is obtained by taking the square root of the mean of the square of the function over one cycle. This value is crucial in AC power calculations as it accounts for both the amplitude and the time variation of the AC signal.

Transcript

00:01 Let's look at average power in a lrc series circuit.

00:05 So we can start out with this equation where we know that our average power is equal to the vrms, the rms voltage, times the rms current, times cosine of the phase angle.

00:20 Now, if we want, we can express this in a number of different ways.

00:25 So, for example, we can take advantage of the fact that cosine theta is also known as the power factor.

00:37 And this is equal to r over z.

00:41 So that means that we can rewrite this entire equation as power average equals vrms, irms times r over z.

00:54 Right? and now we can also take into account that, like, we know that, v rms equals irms times the impedance, which means that we can turn that around and say, hey, if we solve this for irms, we can say irms equals vrms over z.

01:19 So that means that we can take our vrms over z and we can replace vrms over z here.

01:27 So that means that our entire equation at first is going to turn into irms times r -i -m -s times r, which is irms -s -squared r.

01:45 So that gives us like a one form of our average power in terms of the rms current.

01:51 We can also get one in terms of the rms voltage if we look at the fact that vrms equals i -r -m -r -m -s current.

01:59 So that's our form of oms law.

02:04 So we can rewrite our average power as we can rewrite this.

02:10 Irms squared.

02:12 If we can solve this for irms again, we get vrms over r.

02:19 So instead of i squared rms, we can write this as v squared rms over r squared times r...

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