Alternating Current (ac) Circuits The current I, in amperes,...

Alternating Current (ac) Circuits The current $I$, in amperes, flowing through an ac (alternating current) circuit at time $t$, in seconds, is
$$
I(t)=220 \sin (60 \pi t) \quad t \geq 0
$$
What is the period? What is the amplitude? Graph this function over two periods.

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Key Concepts

Graphing Sinusoidal Functions

Graphing sinusoidal functions involves plotting the function over the desired time interval, indicating key features such as amplitude, period, and phase shift (if any). This visualization helps in understanding the behavior of AC circuits, as it clearly shows the repeating nature and oscillations of the current or voltage over time.

Angular Frequency

Angular frequency is the coefficient of the time variable within the argument of a trigonometric function. It defines how rapidly the function oscillates over time and is instrumental in calculating the period using the relationship period = 2? divided by the angular frequency.

Amplitude

Amplitude refers to the maximum absolute value that a periodic function reaches from its midline. In the context of trigonometric functions and AC circuits, it represents the peak value of current or voltage and is crucial in determining the strength or magnitude of the signal.

Period

The period is the length of time required for one complete cycle of a periodic function. It is the reciprocal of the frequency and indicates how often the waveform repeats itself. In sinusoidal functions, the period is computed using the angular frequency, which is found inside the sine or cosine function.

Transcript

00:02 Okay, so we're going to graph this function, i of t equals 220 sine 60 pi t, and t is greater than equal to zero, over two periods.

00:14 Okay, so let's do that.

00:18 First, we're going to find this period.

00:22 Okay, how do we find the period? we're going to put the argument of this function, which is 60 pi t, between the typical sign period.

00:38 Ranges, which is 0, the 2 pi.

00:42 The reason we're doing that is because we want to isolate that t.

00:47 So we're going to divide by 1 over 60 pi.

00:51 And we'll get 0 is less than or equal to t, is less than or equal to 1 over 30.

00:59 That's pretty crazy, which means this period is 1 over 30.

01:06 Okay.

01:07 One, two, three, four, over 30.

01:11 Two, three, four, two over thirty, which is one -fifteenth, but we can leave it that way.

01:18 Okay.

01:19 What is the amplitude of this function? the amplitude of this function is going to be 220.

01:30 The amplitude always refers to the constant in front of the trigonometric function.

01:36 In this case, it's the 220 in front of that.

01:39 Sign.

01:42 So 220...

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