Lab 7. CAPACITIVE REACTANCE

7.1. Guiding Questions

How are are a capacitor’s current and voltage related in an ac circuit? How is a capacitor’s behavior affected by ac frequency? How can a capacitor’s capacitance be determined experimentally?

7.2. Equipment

Two capacitors, assorted resistors, variable-frequency ac source (signal generator), electric multimeter, wires, oscilloscope (optional if the signal generator is credible)

7.3. Background

In an RC circuit with a sinusoidal voltage source, the resistor and capacitor voltages are 90° (1/4 cycle) out of phase with each other. Consequently, their peak and rms voltages are not directly additive: instead, they add like the legs of a right triangle.

Vrms2 = VRrms

2 + VCrms2

The phase angle φ can be used to characterize the voltages by sin𝜙𝜙 = , cos𝜙𝜙 = ,

and tan𝜙𝜙 = .

Generalizing Ohm’s law, we can express the peak or rms voltage of the capacitor to its current (or vice versa) as

VC = ICXC,

where XC, the capacitave reactance, is given by XC = .

Since there are no branches in an RC circuit, all components pass the same current: I = IR = IC

𝐼𝐼 =𝑉𝑉𝑅𝑅 =

𝑉𝑉𝑋𝑋

tan𝜙𝜙 =𝑉𝑉𝑉𝑉 =

𝑋𝑋𝑅𝑅 =

1𝜔𝜔𝜔𝜔𝜔𝜔

φ

Vrms VCrms

VRrms

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If we know VCrms, VRrms, and R, we can find 𝑋𝑋 = 𝑅𝑅 . If we know ω, then we can

determine the capacitance 𝐶𝐶 = = .

7.4. Activities

In this lab, you will create RC circuits and measure voltages with the aim of finding capacitances.

1. First capacitor

Choose one capacitor and construct the circuit shown. Investigate the circuit at a range of source frequencies, ideally incrementing by a factor of 10. If you have a reliable function generator, it will tell you the frequency. If your function generator is suspect, you will need to measure the period with the oscilloscope. At each frequency, find a resistor that gives VRrms ≈ VCrms. Measure the resistance of the resistor with the ohmmeter independently, disconnected from all other circuit elements. Record the measured values of f or T, R, Vrms, VCrms, and VRrms. If you have an oscilloscope, you can use it to measure Vmax. Be sure to include the units with your measurements.

Description of capacitor:

T or f R Vrms VRrms VCrms tan φ φ XC C

Use the proper formulas to calculate the remaining columns.

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2. Second Capacitor Repeat the above procedure for the second capacitor. Description of capacitor:

T or f R Vrms VRrms VCrms tan φ φ XC C

3. Capacitors in series

Connect the two capacitors in series. Carry out the measurements as before, treating the pair of capacitors as a single unit.

T or f R Vrms VRrms VCrms tan φ φ XC C

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4. Capacitiors in parallel

Connect the two capacitors in parallel (but still in series with the resistor). Carry out the measurements as before, treating the pair of capacitors as a single unit.

T or f R Vrms VRrms VCrms tan φ φ XC C

7.5. Lab Report

This doesn’t have to be long, and it doesn’t have to be fancy. What I want to see are:

The data tables (this can be these original pages) Your best estimate of C for each circuit An evaluation of how well the capacitors in series conform to the series capacitance

formula An evaluation of how well the capacitors in parallel correspond to the parallel

capacitance formula

Write it up neatly and legibly.