The output voltage of an AC generator is given by Δv= (1.20 ×10^2 V) sin(30 πt) The generator is

The output voltage of an AC generator is given by Δv= (1.20...

The output voltage of an AC generator is given by $\Delta v=$ $(1.20 \times 10^{2} \mathrm{V}) \sin (30 \pi t)$ The generator is connected across a 0.500 - H inductor. Find the (a) frequency of the generator, (b) rms voltage across the inductor, (c) inductive reactance, (d) rms current in the inductor, (e) maximum current in the inductor, and (f) average power delivered to the inductor. (g) Find an expression for the instantaneous current. (h) At what time after t 5 0 does the instantaneous current first reach 1.00 A? (Use the inverse sine function.)

Key Concepts

Phase Relationship in Inductors

Inductors inherently cause a phase shift between voltage and current in AC circuits, typically resulting in the current lagging the voltage by 90 degrees in an ideal inductor. This phase difference is fundamental in the understanding and analysis of AC circuit behavior, especially in power calculations.

Inverse Trigonometric Solutions in AC Equations

Applying inverse trigonometric functions is crucial when solving for time or phase angles from sinusoidal equations in AC circuits. This approach allows one to determine specific points in time where a certain instantaneous condition, such as a given current or voltage value, is met.

Instantaneous Versus Average Values

In AC circuits, distinguishing between instantaneous values which vary with time, and average values such as RMS and average power, is vital. While instantaneous expressions provide the detailed time-dependent behavior, RMS and average power values give concise measures of the net effect over a complete cycle.

AC Circuit Analysis with Inductive Components

The analysis of AC circuits containing inductors requires the application of Ohm’s law in the context of reactance rather than resistance. The relationship, along with phase differences between current and voltage, is used to compute parameters like RMS current and instantaneous current, as well as to understand the overall behavior of the circuit.

Inductive Reactance

Inductive reactance is the opposition that an inductor offers to the change of current in an AC circuit and is dependent on the angular frequency and inductance. It is defined by the relation XL = ?L and plays a central role in determining both phase shifts and current amplitudes in circuits containing inductors.

Root Mean Square (RMS) Values

The RMS value of an AC voltage or current represents the effective value that would produce the same heating effect as an equivalent DC value. It is calculated by taking the square root of the mean of the squares of the instantaneous values over one cycle, which is essential for power calculations and circuit analysis.

Frequency and Angular Frequency

Frequency refers to how often the waveform repeats per unit time, measured in hertz (Hz), while the angular frequency is a related measure expressed in radians per second. The relationship between the two is given by ? = 2?f, which is crucial when connecting time-domain function forms with inherent energy and reactance calculations.

Sinusoidal Waveform

This concept involves understanding that alternating currents (AC) and their corresponding voltages are typically represented by sinusoidal functions. The sinusoidal nature characterizes the periodic change over time along with key parameters such as amplitude, angular frequency, and phase angle which are fundamental in AC circuit analysis.

Transcript

00:01 So we first have the given output voltage, and we can say v .l, it's going to be equaling 1 .20 times 10 to the second volts.

00:12 Or better yet, we can simply say 120 volts multiplied by sine of 30 pi t, where the inductance is equalling half of 0 .5 .00 henry's.

00:32 And so we can say for part a by inspection, omega is equaling 30 pi radians per second.

00:43 And so we find that then the frequency f is equaling omega divided by 2 pi and 30 pi divided by 2 pi 15 .0 hertz.

00:57 So this would be your answer for part a.

01:00 For part b then we find that the maximum voltage is equaling 120 volts.

01:11 So we can say that the rms voltage, delta v sub lrms, is equaling 120 volts, that maximum voltage divided by radical 2.

01:26 And so this is equaling 84 .9 volts.

01:31 This would be our answer for part b.

01:33 For part c then we want to find the inductive reactants.

01:38 This would be equalling 2 pi multiplied by the frequency times l.

01:44 The inductance, this would be simply omega times l.

01:48 So this would be equaling 30 pi radiance per second multiplied by 0 .50 henry's.

02:01 And this is giving us 47 .1 ums.

02:07 So this would be our answer for part c.

02:09 For part d then, we can say that the rms current will be equalling the rms voltage divided by the inductive reactants.

02:25 And so this would be equaling 84 .9 volts divided by 47 .1 ums.

02:32 This is equaling 1 .80 amperes or amps...

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