5/11 Phasors: Passive RL Circuit Response

Today, we talked about sinusoids and phasors.
A sinusoid is a signal that has the form of the sine or cosine function.

where
Vm = the amplitude of the sinusoid
ω = the angular frequency in radians/s
ωt = the argument of the sinusoid

A sinusoid can be expressed in either sine of cosine form. With these identities, it is easy to express both as either sine or cosine with positive amplitudes.

By using these relationships, we can transform a sinusoid from sine form to cosine form or vice versa.
The picture below is the graphical approach.

To add A cos ωt and B sin ωt, we note that A is the magnitude of cos ωt while B is the magnitude of sin ωt.
The magnitude and argument of the resultant sinusoid in cosine form is readily obtained from the triangle.


Next, we talked about phasors, which are more convenient than sinusoids to work with sine and cosine functions.
A phasor is a complex number that represents the amplitude and phase of a sinusoid.


To deal with the complex numbers calculations, the following operations are important.

We did a little example of how to transform rectangular form to polar form.

Also, we did a example of the calculation of polar form.

Next, we introduced how to determine which one the leading direction and lagging direction are.

Then, we did a example of two sinusoids function addition.


The derivative v(t) is transformed to the phasor domain as jωV
The integral of v(t) is transformed to the phasor domain as V/jω
These are useful in finding the steady-state solution, which does not require knowing the initial values of the variable involved.
The picture below is the example of the transformation of the derivative of v(t) and the integral of v(t) to the phasor domain.

The table below summarizes the time-domain and phasor-domain representations of the circuit elements.

Then, we did lab.

Phasors: Passive RL Circuit Response
Pre-lab

The picture above is the amplitude gain and phase difference between the input voltage and the input current with different frequency.

The picture below is the basic set up for this lab.


Result:

With the cutoff frequency, the phase difference is (16.87)/(135)*360 = -44.9degree, and the gain is 13.28/668.3 = 0.01987


With ten times the cutoff frequency, the phase difference is (3.102)/(13.5)*360 = -82.72degree, and the gain is 2.155(mA)/223.2(mV) = 0.009655


With one tenth of cutoff frequency, the phase difference is (0.03)/1.45 = -7.448degree, and the gain is 0.001675/1.013 = 0.0016535


The picture above is the comparison of experimental and theoretical data.
From the data, we can see that we successfully finished the lab, especially on the phase difference.

Summary

We learned about sinusoids and phasors, which are different ways to deal with AC circuit. Also, we learned how to determine the phase difference and gain according to the graph. By comparing the data, we can see the experiment matched what we expected.