Phasors: Passive RL Circuit Response Lab

Phasors: Passive RL Circuit Response Lab

Overview: 

In this lab, we measured the gain and phase responses of a passive RL circuit, and compared the measurements with expected values.

Design:
We designed the following circuit to execute the measurements:

R was set to 1.1 ohms

^^^^^ L was set to 1 microHenry ^^^^

Based on the fact that the input voltage and current are complex exponentials, we can 

Calculate the circuit's input/output relationship (the amplitude gain) using the equation:

I/V = Amplitude gain =  1/(R+j*omega*L)

Calculate the phase shift relationship between input and output using the equation:

phi-omega = -arctan((omega*L)/R)

And finally, calculate the cutoff frequency of the circuit using the equation: 

omega cutoff = R/L

We wanted to analyze the circuit at three different frequencies: cutoff frequency,  cutoff frequency/10, and cutoff frequency*10

Below are our calculations for the expected phase shifts and amplitude gains:

Calculated values:

Cutoff Frequency = 1.1*10^6 Hz

Phase Shift:
Cutoff Frequency = -45 degrees
Cutoff Frequency/10 = -5.71 degrees
Cutoff Frequency *10 = -84.29 degrees

Amplitude Gain:
Cutoff Frequency = 0.6482
Cutoff Frequency/10 = 0.9046
Cutoff Frequency*10 = 0.0905

These gains make sense, as a higher frequency will cause a much larger impedance response from the inductor, while a lower frequency will cause the inductor to act more like a short.

Construction and Execution:

 ^^^^ The resistor was measured to be approximately 1.35 Ohms ^^^^

This much of a variation unfortunately called for all new calculations of phase shifts and amplitude gains:

 ^^^^ Recalculated values, accommodating for the change in resistance ^^^^

Updated calculated values:

Cutoff Frequency = 1.1*10^6 Hz

Phase Shift:
Cutoff Frequency = -39.17 degrees
Cutoff Frequency/10 = -4.66 degrees
Cutoff Frequency *10 = -83.00 degrees

Amplitude Gain:
Cutoff Frequency = 0.5742
Cutoff Frequency/10 = 0.7383
Cutoff Frequency*10 = 0.0902

We then constructed the circuit as illustrated previously in the schematic:

Results:

Using a 1 volt amplitude sinusoidal signal, these were our results:

The screenshot above shows the oscilloscope output of our circuit. 

Analysis:

Unfortunately, due to limitations of our equipment, we were only able to record data at 15 KHz. This is 1/11.7 of our cutoff frequency, so the data is quite skewed. That being said, the oscilloscope graph is quite true to out expectations at that frequency. As you can see, the voltage across the inductor (blue) is TINY in comparison to the voltage across the resistor (notice the differences in scale). Overall, the general behavior of an inductor has been shown: 

An inductor in a circuit that is oscillating at below its cutoff frequency will only marginally affect the circuit, as its impedance will be minuscule in comparison to the impedance of the other resistance.

An inductor in a circuit that is oscillating at its cutoff frequency will significantly affect the circuit, as its impedance will be at least equal to that of the other resistance.

And an inductor in a circuit that is oscillating at above its cutoff frequency will drastically affect the circuit, as its impedance will be massive in comparison to the other resistance.