Worksheet 8: Time-Varying RL Circuits and High-Pass Filters¶

In this lab, you will investigate the behavior of RL circuits (resistor-inductor circuits) in response to both time-varying and frequency-varying inputs. These circuits are foundational in electronics for understanding inductance, transient response, and filtering behavior, especially in the context of signal processing. By applying square and sine wave inputs, you will explore how inductors behave in dynamic conditions and how RL circuits can function as high-pass filters.

Inductors and Time Constants¶

An inductor resists changes in current. When a voltage is applied across an inductor in series with a resistor, the current increases gradually according to the following exponential relationship:

Current rise: $$ I(t) = \frac{V_0}{R} \left( 1 - e^{-t/\tau} \right) $$
Inductor voltage drop: $$ V_L(t) = V_0 \cdot e^{-t/\tau} $$

The RL time constant $\tau$ is given by: $$ \tau = \frac{L}{R} $$

It represents the characteristic time for the current to reach approximately 63% of its final value or to decay to 37% after the input is removed. This behavior is analogous to RC circuits but with current instead of voltage as the primary variable of interest.

High-Pass Filter Behavior¶

In the frequency domain, RL circuits act as high-pass filters. This means they allow high-frequency signals to pass through more effectively while attenuating low-frequency signals. This frequency-selective behavior is described by the voltage transfer function:

$|H(f)| = \frac{V_{\text{out}}(f)}{V_{\text{in}}(f)} = \frac{L \cdot 2\pi f}{\sqrt{R^2 + (L \cdot 2\pi f)^2}}$

The cutoff frequency $f_c$ of the RL high-pass filter is:

$f_c = \frac{R}{2\pi L}$

At frequencies above $f_c$ , the resistor voltage increases, demonstrating the filter's ability to "pass" high-frequency content.

Overview of Measurements¶

This lab consists of two main measurements:

Measurement 1: Observing RL Transient Response with a Square Wave¶

You will construct a series RL circuit and apply a square wave using a function generator.
The voltage across the inductor will be observed using an oscilloscope, revealing the characteristic exponential rise and fall associated with current transients.
The experimental time constant $\tau_{\text{exp}}$ will be estimated and compared to the theoretical value using $\tau = L/R$ .
You will learn about sensing resistors, parasitic effects, and good measurement practices when using oscilloscopes.

Measurement 2: Exploring Sine Wave Response and High-Pass Filter Behavior¶

The same RL circuit will now be driven by a sine wave, and the frequency will be varied across a broad range (e.g., 10 Hz to 100 kHz).
You will observe how the voltage across the resistor changes with increasing frequency.
This will illustrate the high-pass filter behavior of the circuit, as low-frequency voltages are attenuated and high-frequency voltages are passed more effectively.
Data will be collected and visualized through oscilloscope screenshots and a frequency response table.

Objectives¶

By the end of this lab, you will be able to: - Understand the transient behavior of RL circuits in response to step inputs.
- Analyze the time constant $\tau = L/R$ and estimate it experimentally from oscilloscope data.
- Construct and evaluate an RL high-pass filter, analyzing its frequency response.
- Use function generators to produce time-varying signals and oscilloscopes to measure and analyze them.
- Compare theoretical and experimental results and identify causes of deviation (e.g., parasitic resistance or capacitance).

Materials List¶

Function Generator
Oscilloscope (preferably dual-channel)
Resistors: 1 kΩ
Inductors: 6.8 mH
Pasco RCL board

Circuit Set-up¶

RL Circuit