Lab 14: Frequency Response of the Series R-L Circuit
Objective
- Observe the effect of frequency on the impedance of a series R-L circuit.
- Plot the voltages and current of a series RL circuit versus frequency & interpret them.
- Predict and plot the phase angle of the total impedance versus frequency for a series R-L circuit and relate them to the Phasor voltages.
Equipments
Background
Series R-L
According to the voltage divider rule (VDR), in a series R-L circuit, the voltage vL (vR) is directly related to the inductive reactance XL (resistive reactance R):
\({v_L} = \frac{{{X_L}}}{Z}{v_{source}}\) and \({v_R} = \frac{R}{Z}{v_{source}}\)
Where the internal resistance of the inductor is ignored and the total impedance is: \(Z = \sqrt {{R^2} + X_L^2} = \sqrt {{R^2} + {{(2\pi fL)}^2}} \)
Since the inductive reactance XL increases with increasing frequency changes (XL = 2πfL), the voltage drop across the inductor will also increase with frequency. However, the resistive reactance is independent of frequency changes.
The phase angle associated with the impedance Z is also sensitive to the applied frequency: \(\theta = {\tan ^{ - 1}}\frac{{{X_L}}}{R}\)
At very low frequencies the inductive reactance will be small compared to the series resistive element (R >> XL) and the network will be primarily resistive in nature. The result is a phase angle associated with the impedance Z that approaches 0° (V and I in phase). At increasing frequencies, inductive reactance will drown out the resistive element (XL >> R) and the circuit will be primarily inductive, resulting in a phase angle approaching 90° (V leads I by 90°).
Procedure
Exp#1: Plotting VL, VR, and I versus Frequency
Construct the following circuit.
All voltage measurements are peak-to-peak voltages.
- Maintaining the voltage source at a VS = 4VP-P, measure the voltage VL for 1k Hz increments for the frequency range of 1k Hz to 10 kHz. For each frequency change, reduce the amplitude of the voltage source to maintain a 4V!
- Turn off the voltage source and interchange the positions of R and L in the circuit. Measure vR for the same range of frequencies with VS maintained at 4V.
This is a very important step. Failure to relocate the resistor R can result in a grounding situation where the inductive reactance is shorted out! - Calculate I = vR/R for each of the frequencies and organize a data table that includes vL, vR, I and KVL.
- Plot the voltages vL and vR versus frequency on a single graph and I versus frequency separately. Label the curves and clearly indicate each plot point.
- Answer the following questions about the plots using short concise sentences:
- As the frequency increases, describe what happens to the voltage across the inductor and resistor using short concise sentences.
- At f = 0 Hz, does vR = Vsource? Explain why or why not.
- At the point where vL = vR, does XL = R? Should they be equal? Why? Is so, identify this point on the voltage plots.
- Is KVL satisfied (vL + vR = VS)? Explain why or why not.
- At low frequencies the inductor approaches a low-impedance short-circuit equivalent and at high frequencies a high-impedance open-circuit equivalent. Does the data from your table (as well as the plots) verify this? Explain your reasoning.
- Plot current I versus frequency, labeling the curve and clearly indicate each plot point. How do the curves of I vs. f compare to vR vs. frequency? Is the sensing resistor then a good measure of the current? Explain your reasoning.
Exp#2: Z versus Frequency
- Using the data from Exp#1 (VS = 4V and I), calculate the experimental and theoretical total impedance (Zexpt and Zthy) for each frequency using the following equations:
Compare them using a percent difference. Organize your data into a table. - Plot Z, R and XL versus frequency on the same plot. Label the curve and clearly indicate each plot point.
- Answer the following questions about the voltage plots using short concise sentences:
- As the frequency increases, describe what happens to the resistive & inductive reactance and the total impedance.
- At low frequencies is vR > vL? If f = 0 Hz, would Z = R? Explain why or why not.
- Predict and compare (from your plot) at which frequency does XL = R? For frequencies less than this frequency is the circuit primarily resistive or inductive? How about for higher frequencies?
Exp#3: θ versus Frequency
- Using the inductive reactance XL data from Exp#2, calculate and plot the phase angle (θ = tan-1(XL/R)) as shown in the table below.
- Answer the following questions about the voltage plots using short concise sentences:
- At low frequency, does the phase angle suggest resistive or inductive behavior? Explain why. Draw a Phasor voltage diagram showing this.
- At high frequencies, does the phase angle suggest a resistive or inductive behavior? Explain why? Draw a Phasor voltage diagram showing this.