Worksheet 7: Time-varying RC circuits¶

This lab introduces time-varying RC circuits, where a resistor and capacitor interact to produce characteristic charging and discharging behaviors. These circuits play a fundamental role in electronics, particularly in signal processing, filtering, and timing applications. To analyze RC circuits, we will use function generators and oscilloscopes to observe voltage waveforms and understand how capacitors respond to varying input signals.

Capacitance and Time Constants¶

A capacitor stores electrical energy, and its ability to store charge is defined by its capacitance $C$ . In an RC circuit, the time required for a capacitor to charge or discharge is determined by the time constant $\tau$ :

$\tau = R C$

where $R$ is the resistance and $C$ is the capacitance. The voltage across the capacitor follows an exponential function during charging and discharging:

Charging:

$V_C (t) = V_{\text{max}} \left( 1 - e^{-t/\tau} \right)$

Discharging:

$V_C (t) = V_{\text{max}} e^{-t/\tau}$

where $V_{\text{max}}$ is the maximum voltage applied to the circuit. The time constant $\tau$ represents the time required for the capacitor to charge to 63% of the maximum voltage or discharge to 37% of the initial voltage.

Frequency Response and Filtering¶

RC circuits react differently depending on the frequency of the input signal: - At low frequencies, capacitors charge and discharge almost fully, producing large voltage variations. - At high frequencies, capacitors do not have enough time to charge fully, resulting in a lower voltage response.

This behavior makes RC circuits useful as filters, such as low-pass filters, which allow low-frequency signals to pass while attenuating high-frequency signals.

Overview of Measurements¶

This lab consists of two key measurements:

Measurement 1: Observing RC Charging and Discharging with a Square Wave¶

You will construct a series RC circuit and connect it to a square wave input from a function generator.
Using an oscilloscope, you will observe how the capacitor charges and discharges in response to the square wave input.
The experimental time constant $\tau_{\text{exp}}$ will be estimated from the charging curve and compared to the theoretical value $\tau_{\text{theory}} = RC$ .

Measurement 2: Exploring Sine Wave Response and Frequency Dependence¶

The circuit will now be driven by a sine wave input instead of a square wave.
By varying the input frequency, you will observe changes in the amplitude of the capacitor voltage using an oscilloscope.
The circuit's behavior as a low-pass filter will be examined by measuring how the capacitor voltage decreases at higher frequencies.

Objectives¶

By the end of this lab, you will be able to:
- Understand and measure charging and discharging behavior in RC circuits.
- Use oscilloscopes to visualize voltage waveforms over time.
- Analyze the time constant $\tau$ of an RC circuit experimentally.
- Investigate the frequency response of an RC circuit and its filtering properties.
- Develop skills in using function generators and oscilloscopes for circuit analysis.

Materials List¶

Breadboard and Connecting Wires
Resistors: 10 kΩ
Capacitors: 0.1 µF
Function Generator

Please reference Wave Function Generator manual for details on how to operate the wave function generator. Pages 3-8 contain the front panel overview. Pages 11-13 contain information on the user interface.

Oscilloscope

Please reference Oscilloscope manual for details on how to operate the oscilloscope. Pages 7 and 10-14 contain the front panel overview. Pages 15-17 contain information on the user interface.

Circuit Set-up¶

RC Circuit