**Lab 14: Frequency Response of the Series R-L Circuit**

**Objectives**

- 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 **v**_{L} (**v**_{R}) is directly related to the inductive reactance **X**_{L} (resistive reactance R):

and

Where the internal resistance of the inductor is ignored and the total impedance is:

Since the inductive reactance **X**_{L} 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:

At very low frequencies the inductive reactance will be small compared to the series resistive element (**R >> X**_{L}) 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 (**X**_{L} >> R) and the circuit will be primarily inductive, resulting in a phase angle approaching 90° (**V** leads *I* by 90°).

**Procedure**

## Exp#1: Plotting *V*_{L}, *V*_{R}, and *I* versus Frequency

Construct the following circuit.

All voltage measurements are peak-to-peak voltages.

- Maintaining the voltage source at a
*V*_{S} = 4V_{P-P}, **measure** the voltage **V**_{L} 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 ***v*_{R} for the same range of frequencies with *V*_{S} 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 = v**_{R}/R for each of the frequencies and organize a data table that includes *v*_{L}, **v**_{R}, **I** and **KVL**.
**Plot** the voltages **v**_{L} and **v**_{R} 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 *v*_{R} = V_{source}? Explain why or why not.
- At the point where
*v*_{L} = v_{R}, does **X**_{L} = R? Should they be equal? Why? Is so, identify this point on the voltage plots.
- Is
**KVL** satisfied (**v**_{L} + v_{R} = V_{S})? 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 *v*_{R} 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 (
*V*_{S} = 4V and **I**), calculate the experimental and theoretical total impedance (**Z**_{expt} and **Z**_{thy}) 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
*v*_{R} > v_{L}? If *f = 0 Hz*, would *Z = R*? Explain why or why not.
- Predict and compare (from your plot) at which frequency does
*X*_{L} = 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
*X*_{L} data from Exp#2, **calculate** and **plot** the phase angle (*θ = tan*^{-1}(X_{L}/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.

**Questions**