# AC Circuits with Square Wave Sources and Inductors

Introduction
In this lesson, we will explore the behavior of an inductor in a circuit with an alternating current (AC) square wave source. We will use a simulation to understand how inductance, frequency, and waveform shape affect the current and voltage in the circuit.

Circuit Components and Properties

• Power Source: Our AC source is set to a peak voltage of 4 volts, with a frequency of 4.5 kHz. The waveform is square, which means the voltage switches instantly between its high and low states without a smooth transition as in sinusoidal waveforms.
• Inductor: The inductor in our circuit has an inductance of 5 millihenries (mH) with a tolerance of ±10%. This tolerance indicates the range within which the actual inductance might vary. The trapezoidal approximation is checked, suggesting that the simulation will approximate the behavior of the inductor using a trapezoidal rule, which is a numerical integration method to estimate the inductor’s response.

Learning Objectives

1. Understand the effect of inductance on the current in an AC circuit with a square wave.
2. Observe the phase relationship between voltage and current in an inductive circuit.
3. Learn about the significance of frequency in determining reactance and the circuit’s overall behavior.

Theoretical Background

• When an inductor is connected to a square wave source, the inductor resists changes in current due to its inherent property known as inductance.
• The instantaneous switching of a square wave causes abrupt changes in current, which an inductor opposes, leading to interesting waveforms.
• The frequency of the source influences the inductive reactance (X_L = 2πfL), which affects how much the inductor will oppose the current change.
• The phase offset between voltage and current is typically 90 degrees in a purely inductive circuit when driven by a sinusoidal source, but this can be different with square waves due to their harmonic content.

Simulation Analysis

• Using the provided simulation software, students can construct the circuit with the given values and run the simulation to observe the current through the inductor and the voltage across it.

• Students should pay attention to how the current waveform is shaped. Unlike with sinusoidal sources, the current waveform in response to a square voltage may have sharp rises and falls, but will not instantaneously change due to the inductor’s property of resisting changes in current.

• The effect of the inductor’s reactance will be more pronounced at higher frequencies, as reactance increases with frequency. At 4.5 kHz, we expect a significant reactance, which will limit the rate of change of current through the inductor.

• By observing the voltage across the inductor, students will notice that it spikes whenever the current changes. This is due to the inductor’s property ( V = L \frac{di}{dt} ), where ( V ) is the voltage across the inductor, ( L ) is the inductance, and ( \frac{di}{dt} ) is the rate of change of current.

• The trapezoidal approximation checkbox in the simulation suggests that it uses a numerical method to estimate the inductor’s behavior over time, which can be useful in understanding the approximate real-world behavior of the inductor in response to the square wave input.

Conclusion
Through this lesson, students will gain an understanding of the dynamic behavior of inductors in AC circuits with non-sinusoidal sources. The square wave input allows the exploration of concepts such as reactance, instantaneous power, and the effects of rapid changes in current, providing a comprehensive view of AC circuit behavior beyond the scope of traditional sinusoidal sources. Students are encouraged to alter the frequency and inductance values to observe and analyze the changes in the circuit’s response, reinforcing the theoretical concepts with practical simulation experience.