The Amazing Inductor-Part 3 | LR Time Constants

📖Foundation

 

Colossians 3:23–24 (KJV)

23 “And whatsoever ye do, do it heartily, as to the Lord, and not unto men;”

24 “Knowing that of the Lord ye shall receive the reward of the inheritance: for ye serve the Lord Christ.”

What Is an LR Time Constant?

In Part 3 of The Amazing Inductor, we’re going to explore the LR time constant, usually represented by the Greek letter τ (tau).

One of the most important things to understand about an inductor is that current through an inductor cannot change instantaneously.

When voltage is first applied to an LR circuit, the current doesn’t immediately jump to its final value. Instead, the current rises exponentially as the magnetic field around the inductor builds.

The LR time constant tells us how quickly that current changes.

The LR Time Constant Formula

The time constant of a series LR circuit is:

τ = L / R

Where:

τ = Time constant in seconds
L = Inductance in henries
R = Total series resistance in ohms

An important detail is that we must include all resistance in the current path, including the internal DC resistance of the inductors themselves.

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Our Experimental LR Circuit

For this experiment, I connected eight 100 mH inductors in series.

That gives us:

L = 8 × 100 mH

L = 800 mH

L = 0.800 H

The measured DC resistance of all eight inductors is approximately:

Rinductor = 1.5 kΩ

Our external resistor measures:

Rexternal = 321 Ω

Therefore, our total series resistance is:

Rtotal = 1500 Ω + 321 Ω

Rtotal = 1821 Ω

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Calculating Tau

Now we can calculate the LR time constant:

τ = L / Rtotal

τ = 0.800 / 1821

τ = 0.000439 seconds

Therefore:

τ ≈ 0.439 ms

So one time constant for our circuit is approximately:

1τ = 0.439 milliseconds

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What Happens at Each Time Constant?

When voltage is applied, the current begins at zero and rises exponentially toward its final steady-state value.

At approximately:

1τ = 63.2%

2τ = 86.5%

3τ = 95.0%

4τ = 98.2%

5τ = 99.3%

By about 5τ, we normally consider the current essentially at its final steady-state value.

For our circuit:

5τ = 5 × 0.439 ms

5τ ≈ 2.195 ms

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Calculating the Final Current

For our calculations, we’re looking at a 3.8 V input step.

Once the current reaches steady state, the ideal inductive effect has essentially disappeared because the current is no longer changing.

The final current is determined by the total series resistance:

Ifinal = V / Rtotal

Ifinal = 3.8 V / 1821 Ω

Ifinal ≈ 0.002087 A

Therefore:

Ifinal ≈ 2.087 mA

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Why Measure Voltage Across the Resistor?

In our experiment, the oscilloscope is connected across the 321 Ω resistor, not directly across the inductors.

Why?

Because the resistor gives us an easy way to observe the current.

Ohm’s Law tells us:

VR = I × R

Since the resistor is a fixed 321 Ω, its voltage is directly proportional to the current flowing through it.

And because the resistor and inductors are connected in series, the same current flows through both.

So when we watch the voltage across the resistor rise on the oscilloscope, we’re effectively watching the inductor current rise.

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Final Voltage Across the 321 Ω Resistor

At steady state:

VR(final) = Ifinal × 321 Ω

VR(final) = 0.002087 A × 321 Ω

VR(final) ≈ 0.670 V

So our oscilloscope waveform should eventually approach approximately:

VR(final) ≈ 0.670 V

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Our Calculated LR Current Rise

Here are the theoretical values for our circuit:

And that’s exactly the type of exponential rise we’re looking for on the oscilloscope.

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The LR Current Rise Equation

The current through the circuit follows:

I(t) = Ifinal(1 − e^(-t/τ))

The voltage across our resistor follows the same exponential shape:

VR(t) = VR(final)(1 − e^(-t/τ))

This is why measuring across the resistor gives us such a useful picture of how the current through the inductor changes with time.

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Safety Warning

Inductors store energy in a magnetic field.

When current through an inductor is suddenly interrupted, the collapsing magnetic field can produce a voltage spike, or back EMF, that may be considerably higher than the applied voltage.

This voltage can potentially damage components or test equipment.

Always verify your circuit connections, voltage levels, and oscilloscope grounding before applying power. Appropriate protection should be used when experimenting with larger inductors or higher voltages.

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Foundation First

The goal isn’t simply to memorize:

τ = L / R

We want to understand what the inductor is actually doing.

By calculating the expected response and then viewing that response on a real oscilloscope, we can connect the mathematics to what’s physically happening inside the circuit.

That’s what Foundation First is all about.

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