**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}* (

*) is directly related to the inductive reactance*

**v**_{R}*(resistive reactance R):*

**X**_{L}\({v_L} = \frac{{{X_L}}}{Z}{v_{source}}\) 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 * X_{L}* increases with increasing frequency changes (

**), the voltage drop across the inductor will also increase with frequency. However, the resistive reactance is independent of frequency changes.**

*XL = 2πfL*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 >> 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°***and*

**V****in phase). At increasing frequencies, inductive reactance will drown out the resistive element (**

*I**) and the circuit will be primarily inductive, resulting in a phase angle approaching 90° (*

**X**_{L}>> R*leads*

**V****by 90°).**

*I***Procedure**

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

*V*

_{L}*V*

_{R}*I*

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 voltagefor**V**_{L}increments for the frequency range of**1k Hz**to**1k Hz**. For each frequency change, reduce the amplitude of the voltage source to maintain a*10 kHz*!**4V** - Turn off the voltage source and interchange the positions of
and**R**in the circuit.*L***Measure**for the same range of frequencies with*v*_{R}maintained at*V*_{S}.*4V***This is a very important step.**Failure to relocate the resistorcan result in a grounding situation where the inductive reactance is shorted out!*R* **Calculate**for each of the frequencies and organize a data table that includes**I = v**_{R}/R,*v*_{L},**v**_{R}and**I****KVL**.**Plot**the voltagesand**v**_{L}versus frequency on a single graph and**v**_{R}versus frequency separately. Label the curves and clearly indicate each plot point.*I*- 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
, does**f = 0 Hz**? Explain why or why not.*v*_{R}= V_{source} - At the point where
, does*v*_{L}= v_{R}? Should they be equal? Why? Is so, identify this point on the voltage plots.**X**_{L}= R - Is
**KVL**satisfied ()? Explain why or why not.**v**_{L}+ v_{R}= V_{S} - 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
versus frequency, labeling the curve and clearly indicate each plot point. How do the curves of*I*vs.*I*compare to*f*vs. frequency? Is the sensing resistor then a good measure of the current? Explain your reasoning.*v*_{R}

####
Exp#2: *Z* versus Frequency

*Z*

- Using the data from Exp#1 (
and*V*_{S}= 4V), calculate the experimental and theoretical total impedance (**I**and**Z**_{expt}) for each frequency using the following equations:**Z**_{thy}

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
? If*v*_{R}> v_{L}, would*f = 0 Hz*? Explain why or why not.*Z = R* - Predict and compare (from your plot) at which frequency does
? For frequencies less than this frequency is the circuit primarily resistive or inductive? How about for higher frequencies?*X*_{L}= R

####
Exp#3: **θ** versus Frequency

**θ**

- Using the inductive reactance
data from Exp#2,*X*_{L}**calculate**and**plot**the phase angle () as shown in the table below.*θ = tan*^{-1}(X_{L}/R) - 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.