Phasors: Passive RL Circuit Response

Phasors: Passive RL Circuit Response

In this lab, we wire the series RL circuit seen in the picture above. The circuit is treated as a system where a voltage is input and the resulting current is output. The system amplifies a sinusoidal input by a gain of I/V where the output is another sinusoidal wave with a phase shift of different amplitude. The experiment calls for a 47 Ω resistor in series with a 1 mH inductor. The cutoff frequency is calculated as ωc = R/L = 47 Ω/1 mH = 47000 rad/s. The linear frequency can be easily calculated by dividing by 2ᴨ, f = ωc/2ᴨ = 7.48 kHz. We repeat the calculation for 10ωc and ωc/10 and obtain 74.80 kHz and 748.03 Hz, respectively.

Now the theoretical gain is calculated by dividing the magnitude of the current and voltage phasors, G = |I/V| = |1/(R+jωL)| = 1/sqrt(R^2+ω^2*L^2). This is done for the three angular frequencies ωc, 10ωc, and ωc/10.

Finally, the phase shift is calculated as φ = -arctan(ωL/L). This is also done for all three frequencies. We obtain the following data.

We now test the circuit experimentally using a resistor R = 47.3 Ω (measured) and an inductor L = 0.998 mH. The input voltage is observed along with the oscilloscope along with the calculated current and inductor voltage. The current channel is calculated as i(t) = [vi(t) - vl (t)]/ R. The orange curve (C1) is the input voltage, the blue curve (C2) is the inductor voltage, and the black curve (M1) is the calculated current. We measure current to voltage, peak to peak to find the phase angle by measuring the change in time divided by the period then converting to degrees, φ = (Δt/T)*360°. The gain is then calculated by dividing the current amplitude by the input voltage amplitude, G = I/V.

1 V input @ 7.48 kHz

1 V input @ 74.8 kHz

 1 V input @ 748 Hz

The experimental data is now compared to the calculated theoretical data. The percent error for gain and phase shift of the experiment is very acceptable. This means the experiment was accurate.