Assuming a Purely Inductive Circuit, Current Will Lag the Voltage by 90°: Design & Testing

The 90-Degree Rule: Why Current Lags Voltage in Pure Inductance

When assuming a purely inductive circuit, current will lag the voltage by exactly 90 degrees (π/2 radians). This is not just a theoretical abstraction; it is a direct consequence of Lenz’s Law and the physics of magnetic field collapse. In an ideal inductor, the component resists changes in current by generating a back-electromotive force (back-EMF). When AC voltage is applied, the voltage reaches its peak precisely when the rate of change of current is at its maximum (which occurs when the current itself is crossing zero). Conversely, when the voltage crosses zero, the current is at its peak and its rate of change is zero.

To visualize this on the bench, we use a simple series topology. Consider a circuit with three primary nodes:

• Node A: AC Source High (Function Generator Output)
• Node B: Inductor Input Lead
• Node C: Inductor Output Lead / Ground Return

Think of the inductor as a heavy water wheel in a pipe. The water pressure (voltage) must be applied at its maximum to get the stationary wheel to start moving. By the time the wheel is spinning at its fastest (peak current), the pressure required to maintain that speed drops to zero. This mechanical inertia perfectly mirrors electrical inductance, resulting in the current waveform trailing the voltage waveform by one-quarter of a cycle.

Component Selection & Design Walkthrough

A 'purely' inductive circuit is a theoretical ideal. Every physical inductor possesses parasitic winding resistance (DCR) and parallel capacitance. To approximate a purely inductive circuit at a specific frequency, we must select an inductor where the inductive reactance ($X_L$) vastly dwarfs the DCR.

Let us design a 1 kHz test circuit. We will target an inductive reactance of roughly 60Ω to keep currents in the safe milliamp range for standard breadboard components.

Calculating the Target Inductance

The formula for inductive reactance is $X_L = 2\pi f L$. Rearranging for L:

$L = X_L / (2\pi f) = 60 / (2 \times 3.14159 \times 1000) \approx 9.55 \text{ mH}$.

We will round to a standard off-the-shelf value of 10 mH. A highly reliable, readily available through-hole part for this is the Bourns 78F103K-RC (10mH, 10% tolerance, 350mA max DC current). According to the Bourns 78F series datasheet, this part has a maximum DCR of 1.2Ω.

Verifying the 'Pure' Assumption

At 1 kHz, the reactance of our 10mH inductor is $X_L = 62.83\Omega$. The phase angle ($\theta$) of the real component is calculated as:

$\theta = \arctan(X_L / R_{DCR}) = \arctan(62.83 / 1.2) = 88.9^\circ$.

At 88.9 degrees, the current lags the voltage by nearly a perfect 90 degrees. The 1.2Ω DCR introduces only a 1.9% resistive error, making this an excellent physical approximation of a purely inductive circuit for bench testing.

Behavior Matrix: Parameter Shifts and Failure Extremes

Understanding how a circuit reacts when variables drift or components fail is critical for debugging. The table below maps the behavior of our 10mH / 1kHz topology under stress.

Failure Extremes: Open vs. Short

Short Circuit (Node B jumpered to Node C): The inductor is bypassed. The function generator sees a near-dead short (only the resistance of the jumper wire and sense resistor). Current spikes massively. Most modern signal generators will trigger their internal over-current protection and shut down the output stage, but older models may blow an internal fuse or damage the output op-amp.

Open Circuit (Inductor removed or lead broken): Impedance becomes infinite. Current drops to absolute zero. The full source voltage will appear across the open nodes (Node B to Ground). Because $V = L(di/dt)$, if the circuit is opened while current is flowing, the sudden collapse of the magnetic field ($di/dt$ approaches infinity) will generate a massive voltage spike, potentially arcing across breadboard contacts or destroying sensitive driving transistors.

Breadboard Testing: Step-by-Step Verification

To prove the 90-degree lag empirically, you cannot measure current directly with a standard oscilloscope; you must measure the voltage drop across a known sense resistor. We will use the Tektronix oscilloscope basics methodology for phase measurement.

Numbered Build and Measure Steps

1. Build the Topology: Insert the Bourns 10mH inductor into the breadboard. Connect Node A (Function Generator BNC center pin) to the inductor input. Connect the inductor output to one leg of the 10Ω sense resistor. Connect the other leg of the sense resistor to Ground (Node C).
2. Configure the Source: Set the function generator to a 1.000 kHz Sine wave, 2.0V peak-to-peak (Vpp), with 0V DC offset.
3. Probe Channel 1 (Voltage): Connect Oscilloscope CH1 probe across the inductor (Node A to Node B). *Note: Because the sense resistor is small, Node A to Ground is a close approximation of the inductor voltage, but measuring strictly across the inductor is more accurate.*
4. Probe Channel 2 (Current): Connect Oscilloscope CH2 probe across the 10Ω sense resistor (Node B to Ground). The voltage on CH2 is directly proportional to the current ($I = V_{CH2} / 10\Omega$).
5. Trigger and Scale: Set the scope to trigger on CH1 rising edge. Adjust the timebase to 250 µs/div to display exactly one full 1ms cycle across the 10-division screen.
6. Measure Phase: Use the scope's cursor function. Place Cursor 1 on the zero-crossing (rising) of CH1. Place Cursor 2 on the zero-crossing (rising) of CH2. The delta time ($\Delta t$) should read approximately 250 µs. Since one full cycle is 1000 µs (360°), a 250 µs delay equals exactly 90°.

Decision Path: Pure Inductive vs. RL and RC Topologies

Why choose a purely inductive topology over an RC (Resistor-Capacitor) or RL (Resistor-Inductor) configuration? The decision hinges on energy storage mechanisms, DC power loss, and phase shift requirements. Capacitors store energy in an electric field and block DC; inductors store energy in a magnetic field and pass DC. All About Circuits notes that inductive reactance increases with frequency, whereas capacitive reactance decreases.

The Final Verdict: When to Commit to Pure Inductance

If your goal is to create a phase-shift network where minimal real power is consumed (such as in a high-efficiency AC motor start winding or a resonant tank circuit), you must use a pure inductive or capacitive topology. Adding a series resistor (RL) intentionally burns power as heat to force the phase angle away from 90°.

Concrete Recommendation: For educational bench testing, signal filtering, or demonstrating the 90-degree lag without the confounding variables of heavy resistive loading, default to the Bourns 78F103K-RC (10mH) driven at 1 kHz. This specific pairing guarantees an 88.9° phase shift, keeps current safely under 20mA, and provides a clean, easily measurable waveform on any entry-level digital storage oscilloscope. Do not substitute an unshielded power inductor for this application, as the open magnetic field will induce noise into adjacent breadboard traces and skew your phase measurements.