Question

Suppose you want to find the difference of two sinusoidal currents, given as follows: i1(t)=I1 cos(ωt+ϕ1) and i2(t)=I2 cos(ωt+ϕ2). If you stay in the time domain, you will have to use trigonometric identities to perform the subtraction. But if you transform to the frequency domain, you can simply subtract the phasors I1 and I2 as complex numbers using your calculator. Make sure that all of your time-domain currents are represented using the cosine function before you apply the phasor transform. Your answer will be a phasor, so you will need to inverse phasor-transform it to get the answer in the time domain. This is an example of a problem that is easier to solve in the frequency domain than in the time domain. Use phasor techniques to find an expression for i(t) expressed as a single cosine function, where i(t)=[250cos(377t+30∘)−150sin(377t+140∘)] mA. Enter your expression using the cosine function. Round real numbers using two digits after the decimal point.

Answer 1

120.51·cos(377t+4.80°)

Explanation:

We can use the identity ...

sin(x) = cos(x -90°)

to transform the second waveform to ...

i₂(t) = 150cos(377t +50°)

Then ...

i(t) = i₁(t) -i₂(t) = 250cos(377t+30°) -150cos(377t+50°)

A suitable calculator finds the difference easily (see attached). It is approximately ...

i(t) = 120.51cos(377t+4.80°)

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The graph in the second attachment shows i(t) as calculated directly from the given sine/cosine functions (green) and using the result shown above (purple dotted). The two waveforms are identical.