Trigonometry alternating currents

Question

The voltage of an alternating current flowing in an electrical circuit is given by V = Vo sin (ωt + a) where:

V is the voltage
Vo , ω and α are constants
T is the time in seconds since measurements of the voltage were begun

The maximum voltage in an electric circuit is 312 volts. The time between maximums is 0.02 seconds. The initial voltage measured was 200 volts.

a) Find the value of the constants Vo, ω and α and hence find an equation for V. (All values to 3 significant figures)

b) What will the voltage be after 1.2 seconds? (All times to 3 significant figures)

c) What are the first 3 times that the voltage will be 240 volts? (All times to 4 significant figures)

Answer 1

a) The AC source is a sine wave centered around the x-axis (as it is not shifted up or down) so the magnitudes of maximum and minimum voltage would be the same = 312 V.

 

Vo = 312 V.

 

The time between maximum two consecutive maximums is the time period T = 0.02 seconds.

Time period T is the reciprocal of frequency f.

 

f = 1/T = 1/0.02s = 50 Hz.

 

Relation between frequency and angular frequency ω: ω = 2πf = 2 * 3.14 * 50 = 314.159 rad/s

 

We can plug in the initial voltage V =200 and time t = 0 in the equation V = Vo sin (ωt + a) to get a.

 

200 = 312*sin(ω*0 + a) = 312*sin(a)

 

sin(a) = 200/312   -> a = sin^-1(200/312) = 39.868^o = 6.654 rad.

 

 

 

b) Plug in the values you found in a) and time, t in the original equation:  V = Vo sin (ωt + a) 

 

c) Plug in the values you found in a) and voltage, V in the original equation and solve for t.

 

V/Vo = sin(ωt + a)

 

lets suppose x = sin^-1(V/Vo) in degrees

 

(ωt + a) = x/2π is now in rad.

 

solve 

(ωt + a) = x/2π,

(ωt + a) = (π - x/2π) and 

(ωt + a) = x/2π + 2π 

 

to get the three times where V is 240 V.