AC Voltage Waveform: Frequency, Time, RMS & Instantaneous Value

[cschear87]

Homework Statement

Given an AC Voltage that varies in time according to the equation:

v(t) = 100 sin(250t)

(i) What is the frequency of this waveform?

(ii) Calculate the time for one complete cycle of the voltage?

(iii) Calculate the instantaneous value of the voltage at time = 6ms.

(iv) Calculate the Root-Mean Square (RMS) value of the AC voltage.

(v) The first time after t=0ms that the instantaneous voltage is 50V.

Homework Equations

v(t) sin (250 x 6 x 10^-3)

The Attempt at a Solution

I'm trying to, obviously, get the third part of this. What I have so far is:
100sin (250pi x 6 x 10^-3)
=100sin (4.71)
In an example in my book it then goes on to multiply 100 x .951... where does the .951 come from and is this what I use for this equation or should I use a different number?
The equation in the book I have is (still 6ms):
v(t) = 100sin (100pi t) V
Their solution is:
100 sin (100pi x 6 x 10^-3)
=100sin (1.88)
=100 x .951
=95.1V
Not sure what I'm missing?

 

[LCKurtz]

It looks like they are using $100\pi$ instead of the $250\pi$. Probably a typo somewhere.

 

[CWatters]

The equation in the book I have is (still 6ms):
v(t) = 100sin (100pi t) V
Their solution is:
100 sin (100pi x 6 x 10^-3)
=100sin (1.88)
=100 x .951
=95.1V

sin(1.8849) = 0.951 if using radians.

In your problem (which is different to the one in the book) you use a different number eg..

= 100sin(4.71)

 

[cschear87]

sin(1.8849) = 0.951 if using radians.

In your problem (which is different to the one in the book) you use a different number eg..

= 100sin(4.71)

OH I see, just got confused. Thank you!

 

[rezeew]

The frequency of this waveform is 250 Hz, as indicated by the 250π in the equation. This means that the voltage completes 250 cycles in one second.

To calculate the time for one complete cycle, we can use the formula T = 1/f, where T is the time and f is the frequency. In this case, T = 1/250 = 0.004 seconds or 4 milliseconds.

To calculate the instantaneous value of the voltage at time = 6ms, we can simply plug in t=0.006 into the equation. v(0.006) = 100sin(250π x 0.006) = 99.951 V.

The RMS value of an AC voltage is calculated by taking the square root of the average of the squared instantaneous values over one complete cycle. In this case, the RMS value would be 100/√2 = 70.71 V.

To find the first time after t=0ms that the instantaneous voltage is 50V, we can set the equation equal to 50 and solve for t. 50 = 100sin(250πt) then t = 1/(250π) x arcsin(0.5) = 0.002 seconds or 2 milliseconds.