RC Circuit Transient Response

• An RC circuit's behavior, especially during charging and discharging, can be examined in this lab exercise, which also introduces the idea of a passive differentiator.
• Multisim will be used as a simulation tool.

Objectives

• $\tau$, the time constant, ought to be measured when charging and discharging RC circuits.
• Examine the transient response of RC circuits.
• Create and test a passive differentiator circuit.
• Use Multisim to simulate RC circuits.

Materials

• Resistors: $1 k\Omega$, $10 k\Omega$
• Capacitors: $0.1 \mu F$, $1 \mu F$
• Function Generator
• Oscilloscope
• Breadboard and connecting wires
• Multisim software

RC Circuit Transient Response

• When a DC voltage is applied to a series RC circuit, the capacitor charges over time with the formula: $V_C(t) = V_f + (V_i - V_f)e^{-t/RC}$

• $V_f$ refers to the final voltage across the capacitor.

• $V_i$ refers to the starting voltage across the capacitor.

• $t$ refers to the amount of time that has passed since the voltage was applied.

• $RC$ is the time constant, or $\tau$.

• The voltage reaches about 63.2% of its final value during charging or falls to 36.8% of its initial value during discharging after a period of time equal to the time constant $\tau$.

Differentiator

• A differentiator is a circuit that performs the mathematical operation of differentiation.

• A passive differentiator can be constructed using a resistor and capacitor with the output voltage of an ideal differentiator indicated by: $V_{out}(t) = RC \frac{dV_{in}(t)}{dt}$

• Differentiation can only be approximated by this circuit when the input signal's frequency is considerably lower than the circuit's "break frequency," $f_c$: $f_c = \frac{1}{2\pi RC}$

RC Circuit Step Response

Charging Phase

• Construct a series RC circuit with $R = 1 k\Omega$ and $C = 1 \mu F$.
• Use a square wave from the function generator, setting the frequency to 100 Hz and amplitude to 5V.
• Across the input signal, connect Channel 1. Across the capacitor, connect Channel 2.
• Measure how long it takes for the capacitor voltage to reach 63.2% of its maximum value while charging and record the waveform on the oscilloscope.
• Repeat steps with $R = 10 k\Omega$ and $C = 0.1 \mu F$.

Discharging Phase

• Observe the phase when the capacitor releases voltage as the square wave falls.
• Find how long it takes for the capacitor voltage to fall to 36.8% of its maximum.
• Repeat with different component values as in the charging phase.

Differentiator Circuit

• Construct the passive differentiator with $R = 10 k\Omega$ and $C = 0.1 \mu F$. Place the capacitor in series with the input signal, and connect the resistor between the capacitor and ground.
• Use a sine wave from the function generator, setting the frequency to 100 Hz and amplitude to 5V.
• Connect channel 1 to the input signal and channel 2 to the output across the resistor.
• Observe and document the output waveform.
• The frequency should be raised to $1 kHz$ and changes in output should be noted.

Multisim Simulation

• Use the same components and input signal, and simulate the RC circuit from section 3.1 in Multisim to measure the charging and discharging times and compare these results with experiments.
• Use the same components and input signal, and simulate the differentiator circuit from section 3.2 in Multisim to observe the output waveforms at different frequencies and compare these results with experiments.

Analysis

• Determine the time constant using known formulas.
• Compare these values with the measured time constants.
• Explain the differences between charging and discharging times.
• Discuss the performance of the differentiator at varying frequencies.
• Explain the discrepancy in observed data and the plausible causes.