Download Half-Life of an LR Circuit

Physics 261
Half-Life of an LR Circuit
A circuit made up of an inductor and a resistor in series with one another and a wave generator
will behave similarly to an RC circuit. Denoting the combined resistance of the inductor and the
resistor as R then a Kirchhoff-like potential loop gives
Vs − IR − L
dI
=0
dt
(1)
If Vs is a square wave, it will be at either of two levels, 0 or Vmax , and so will be one constant
value half the time and another constant value the other half, except during the very brief transition
intervals. We denote
V ≡ Vs − IR,
(2)
and then during a constant Vs intervals,
d(Vs − IR)
dI
dV
=
= −R .
dt
dt
dt
Thus,
dI
1 dV
=
dt
R dt
Substituting Equations 2 and 3 into Equation 1, we get
−
V +
L dV
=0
R dt
(3)
(4)
As with the RC system, this is a first-order differential equation with a solution of the form
x(t) = x0 e−t/τ
where τ is the decay constant.
Solve Equation 4 this way and determine the decay constant. Then following the derivation in
the RC decay lab, find an expression for the half-life.
Choosing a 2.2 kΩ or 10 kΩ resistor and an inductor, measure the resistance of the resistor and
inductor. The sum of these is R.
Measure the inductance L of the inductor with a Meterman 27XT multimeter. The uncertainty
of this measurement is ±(5% + 30 lsd). [The uncertainty is large.]
Calculate an expected value for the half-life.
Set up the LR circuit as described above.
Set the wave generator to output a square wave with Vp−p ≈ 10 V. Based on your expected
half-life, set the frequency to a value that allows for about 4 half-lives to reach maximum voltage
and 4 half-lives to decay.
Attach the oscilloscope across the resistor and measure the half-life.
Compare the measured half-life with the calculated half-life.
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