A sinusoidal current has a maximum amplitude of 20 A. The current passes through one complete cycle in 1 ms. The magnitude of the current at zero time
is 10 A.
a) What is the frequency of the current in hertz?
b) What is the frequency in radians per second?
c) Write the expression for $i\left(t\right)$ using the cosine function. Express $\phi$ in degrees.
d) What is the rms value of the current?

Question

A sinusoidal current has a maximum amplitude of 20 A. The current passes through one complete cycle in 1 ms. The magnitude of the current at zero time is 10 A.

a) What is the frequency of the current in hertz?

b) What is the frequency in radians per second?

c) Write the expression for [latex]i\left(t\right)[/latex] using the cosine function. Express [latex]\phi[/latex] in degrees.

d) What is the rms value of the current?

Accepted Answer

a) From the statement of the problem, T = 1 ms;

hence [latex]f=\frac{1}{T} = 1000 Hz[/latex].

b)[latex]\omega=2\pi f = 2000\pi \frac{rad}{s} [/latex].

c) We have [latex]i\left(t\right)=I_{m}\cos\left(\omega t+\phi\right)= 20 \cos\left(2000\pi t+\phi\right)[/latex], but [latex]i\left(0\right)=10A[/latex]. Therefore [latex]10=20\cos\phi[/latex],and [latex]\phi=60^{\circ }[/latex].
Thus the expression for [latex]i\left(t\right)[/latex] becomes

[latex]i\left(t\right)=20 \cos\left(2000\pi t+60^{\circ }\right)[/latex].

d) From the derivation of Eq. 9.6[latex]V_{rms}=\frac{V_{m}}{\sqrt{2}}[/latex], the rms value of a sinusoidal current is [latex]\frac{I_{m}}{\sqrt{2}}[/latex] .Therefore the rms value is [latex]\frac{20}{\sqrt{2}}[/latex] , or 14.14 A.

Question 9.1: A sinusoidal current has a maximum amplitude of 20 A. The cu...

Related Answered Questions