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RL circuit on the oscilloscope

Curricula: APC, Sek II

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Aim

Observe the rise and decay of the current in an RL circuit and measure the time constant \(L / R\).

The bench

AC voltage source — a square wave from 0 to 5 V (amplitude 2.5 V, DC offset 2.5 V), 10 Hz. A 1 H inductor (winding 10 Ω) and a 100 Ω resistor in series. Oscilloscope: CH1 — the source, CH2 — resistor (the voltage across it is proportional to the current), DUAL mode, 2 V/div, 5 ms/div, triggered on CH1 at 2.5 V.

Procedure

  1. Examine the CH2 waveform.
  2. Measure the time it takes the voltage across the resistor to reach 63% of its steady-state value.
  3. In the resistor's settings, change 100 Ω to 220 Ω and repeat.
Expected result

The current rises and decays exponentially. The steady-state voltage across the resistor is about 4.5 V (5 V · 100 / 110); 63% of it — about 2.9 V — is reached after \(\tau = L / R\) ≈ 9 ms, a little under two divisions. The half-period of 50 ms is more than five \(\tau\), so the current has time to settle; at a higher frequency it would not reach 4.5 V, and the 63% would have to be taken of a smaller value. With 220 Ω, \(\tau\) falls to about 4.4 ms.

Questions

  1. Why does the time constant decrease as the resistance increases?
  2. How is an RL circuit similar to an RC circuit, and how does it differ?
Answers
  1. At the first moment the whole source voltage is across the inductor, so the current starts rising at the rate \(U / L\) whatever the resistance. A larger resistance lowers the steady current \(U / R\), and at the same starting rate the current gets close to it sooner, hence \(\tau = L / R\). With 220 Ω the total resistance is about 230 Ω instead of 110 Ω, and \(\tau\) falls to about 4.4 ms.
  2. In both circuits the voltages and the current change exponentially with a time constant: 63% of the change after one \(\tau\), practically complete after five; the resistor waveform here has the same shape as the capacitor waveform in an RC circuit. They differ in what cannot jump and in the steady state: in RC the capacitor voltage is continuous and the current dies away to zero, in RL the inductor current is continuous and settles at \(U / R\). \(\tau = R \cdot C\) grows with the resistance, \(\tau = L / R\) falls.