PHYS112 : Labs

10 LRC Circuits

In addition to reading this assignment, you may also need to refer to Appendix A on uncertainties, Appendix B on linear regressions, and Appendix C on the oscilloscope.

Introduction

Alternating current (AC) circuits involve potential differences and currents which vary sinusoidally with time. Time dependent electromagnetic effects (for example, time-dependent charging and discharging of capacitors) which are transient in DC circuits are persistent in AC circuits. Here, you will solidify some of these concepts, such as frequency dependent reactance and impedance, phase differences between voltages across various electronic components, and resonance in series LRC circuits.

**Figure 22:** A series LRC circuit.
$\includegraphics{LRC-circuit.eps}$

A series LRC circuit is shown in Figure 22. The inductor and capacitor impede the flow of current in the circuit by temporarily storing energy in magnetic and electric fields instead of by dissipating energy. Together, the inductor, resistor, and capacitor present a net impedance to the flow of current given by

$\begin{displaymath} Z = \sqrt{R^2 + (X_L - X_C)^2} \end{displaymath}$

(30)

where

is the inductive reactance

$\begin{displaymath} X_L = 2 \pi f^\mathrm{dr} L \end{displaymath}$

(31)

and

is the capacitive reactance

$\begin{displaymath} X_C = \frac{1}{2 \pi f^\mathrm{dr} C} \end{displaymath}$

(32)

The inductive and capacitive reactances and hence the total impedance

are frequency-dependent. At low frequencies, the capacitor dominates, while at high frequency the inductor dominates. The amplitude

of the current flowing through the circuit is related to the amplitude $\mathcal{E}^\mathrm{max}$ of the applied emf by Ohm's law

$\begin{displaymath} I = \frac{\mathcal{E}^\mathrm{max}}{Z} \end{displaymath}$

(33)

The phase angle between the potential difference across the resistor $\Delta v_R$ and the applied emf $\mathcal{E}$ is given by

$\begin{displaymath} \phi = \tan^{-1} \left( \frac{X_L - X_C}{R} \right) \end{displaymath}$

(34)

it is related to the time difference $\Delta t = t_{\Delta v_R} - t_\mathcal{E}$ between the two signals, illustrated in panel (a) of Figure 23, as

$\begin{displaymath} \phi = 2 \pi f^\mathrm{dr} \Delta t = 2 \pi \frac{\Delta t}{T^\mathrm{dr}} \end{displaymath}$

(35)

where $T^\mathrm{dr}$ is the period of oscillation of the driving emf.

**Figure 23:** Oscilloscope displays of the potential differences across the resistor and AC power source in a series LRC circuit. Panels (a) and (b) show the potential vs. time and X-Y mode displays for the circuit at a non-resonant driving frequency. Panels (c) and (d) show the same displays at the resonant frequency of the circuit.
$\scalebox{0.5}{ \includegraphics{LRC-nr.eps} }$ $\scalebox{0.5}{ \includegraphics{LRC-resonance.eps} }$

The impedance is minimal, the current is maximal, and the phase angle $\phi$ is zero at the resonant frequency of the circuit - the driving frequency at which and are equal. Setting Eqs. 31 and 32 equal and solving for the frequency yields

$\begin{displaymath} f = \frac{1}{2 \pi} \frac{1}{\sqrt{LC}} \end{displaymath}$

(36)

Panels (c) and (d) of Figure 23 illustrate the potential differences across the resistor ( $\Delta v_R$ ) and the AC source ( $\mathcal{E}$ ) at resonance.

Experiments and Analysis

Measuring Inductance Again

Use the DMM to measure the resistances of your resistor and your inductor. (You should have something close to a $1000 \Omega$ resistor. If not ask your instructor for one.)
In Lab 7, you measured capacitances with a LabPro interface by measuring the half-time of the charging/discharging of RC circuits. This time, use the oscilloscope to make the same kind of measurement. Remember to set the generator to produce a square waveform.
Construct the circuit shown in Figure 22. Connect the leads from channel 1 of the oscilloscope across the function generator ( $\mathcal{E}$ ) and the leads from channel 2 across the resistor ( $\Delta v_R$ ).
Caution! The black wire of each oscilloscope lead is grounded. It is important to place them at the same location in the circuit as shown in the diagram. Otherwise, you will ground the circuit in two places, effectively removing some devices from the circuit.
Set the display mode of your scope to CHOP so that you can see the signals from both channels on the display.
Set the function generator to produce a sinusoidal wave form.
Adjust the driving frequency $f^\mathrm{dr}$ (the frequency of the function generator) and the TIME/DIV scale of the scope to find the driving frequency at which the two signals are aligned in time as shown in panel (c) of Figure 23.
Also investigate the X-Y mode of the oscilloscope - the extreme CCW setting of the TIME/DIV knob. At resonance, the oscilloscope trace looks like a straight line as shown in panel (d) of Figure 23. Off resonance, it looks like an ellipse as shown in panel (b) Figure 23. The X-Y mode display allows you to ``tune in'' the resonant frequency more precisely.
Once you have the function generator set to the resonant frequency of the circuit, switch back to the potential difference vs. time display and measure the frequency. Also record its uncertainty.
Use Eq. 36 and your measurements of the resonant frequency and capacitance to determine the inductance and its uncertainty.

The Frequency Response of an LRC Circuit

Set the frequency of the function generator to $f^\mathrm{dr} = 100 \mathrm{Hz}$ .
Use the oscilloscope to measure the amplitude of the potential difference across the resistor ( $\Delta V_R$ )
Make sure that both signals are centered vertically in the display by setting the signal coupling switch temporarily to GND and carefully centering the flat line on the screen.
Measure the time difference $\Delta t$ between $\Delta v_R$ and $\mathcal{E}$ along the central horizontal line on the display as illustrated in panel (c) of Figure 23. (If $\mathcal{E}$ peaks before $\Delta v_R$ , record a positive time interval, and if $\mathcal{E}$ peaks after $\Delta v_R$ , record a negative time interval.)
Repeat steps 2 and 4 at driving frequencies of approximately 1000 Hz, 10000 Hz, 100000 Hz, and the resonant frequency of your circuit. Also make measurements at two frequencies to either side of the the resonant frequency to better map the behavior of the circuit near resonance.
Put your $\Delta V_R$ , $\Delta t$ , and corresponding $f^\mathrm{dr}$ measurements into columns in a spreadsheet.
In another column, use Ohm's law ( $\Delta V_R = IR$ ) to convert your $\Delta V_R$ measurements into currents.
In another column, use Eq. 35 to convert your $\Delta t$ measurements into phase angles.
Use Excel to produce graphs of vs. $f^\mathrm{dr}$ and $\phi$ vs. $f^\mathrm{dr}$ for your circuit.
Once you have made your graphs, put the frequency axes of both graphs on a logarithmic scale. (Right click on the values on the horizontal axis and choose 'Format Axis.' Once you do this, choose the 'Scale' tab and check the box marked 'Logarithmic Scale.')

Before You Leave Lab

Discuss with your instructor preliminary answers to the questions below.

Hand In ...

... a printout of your spreadsheet, including the graphs, and answers to the following.

Give your results for the inductance and capacitance in your original circuit and the associated uncertainties. Show the work you did to get them.
Part of your $\phi$ vs. $f^\mathrm{dr}$ graph is negative, part is positive, and it crosses zero somewhere in the middle. Explain each of these features and their ordering.
Is the circuit producing the signals in panels (a) and (b) of Figure 23 being driven above or below its resonant frequency? Explain.