5. Application of Ordinary Differential Equations: Series RL Circuit
RL circuit diagram
The RL circuit shown has a resistor and an inductor connected in series. A constant voltage V is applied when the switch is closed.
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Time constant
Two-mesh circuits
RL circuit examples
Two-mesh circuits
The (variable) voltage across the resistor is given by:
The (variable) voltage across the inductor is given by:
Kirchhoff's voltage law says that the directed sum of the voltages around a circuit must be zero. This results in the following differential equation:
Once the switch is closed, the current in the circuit is not constant. Instead, it will build up from zero to some steady state.
Solving the DE for a Series RL Circuit
The solution of this differential equation is:
[We did the same problem but with particular values back in section 2. Separation of Variables]
Here is the graph of this equation:
The plot shows the transition period during which the current adjusts from its initial value of zero to the final value V/R, which is the steady state.
The Time Constant
The time constant (TC), known as τ, of the function
is the time at which R/L is unity. Thus for the RL transient, the time constant τ = L/R seconds.
NOTE: τ is the Greek letter "tau" and is not the same as the time variable t.
At 1 τ
At this time the current is 63.2% of its final value.
Similarly at 2 τ,
1 - e-2 = 1 - 0.135 = 0.865
The current is 86.5% of its final value.
After 5 τ the transient is generally regarded as terminated. For convenience, the time constant τ is the unit used to plot the current of the equation
That is, since τ = L/R, we think of it as:
Let's now look at some examples of RL circuits.
EXAMPLE 1
An RL circuit has an emf of 5 V, a resistance of 50 W, an inductance of 1 H, and no initial current.
Find the current in the circuit at any time t. Distinguish between the transient and steady-state current.
EXAMPLE 2
A series RL circuit with R = 50 W and L = 10 H has a constant voltage V = 100 V applied at t = 0 by the closing of a switch.
Find
(a) the equation for i (you may use the formula rather than DE),
(b) the current at t = 0.5 s
(c) the expressions for VR and VL
(d) the time at which VR = VL
Two-mesh Circuits
The next two examples are "two-mesh" types where the differential equations become more sophisticated. We will use Scientific Notebook to do the grunt work once we have set up the correct equations.
EXAMPLE 3
In the two-mesh network shown below, the switch is closed at t = 0 and the voltage source is given by V = 150 sin 1000t V. Find the mesh currents i1 and i2 as given in the diagram.
EXAMPLE 4
The switch is closed at t = 0 in the two-mesh network shown below. The voltage source is given by V = 30 sin 100t V. Find the mesh currents i1 and i2 as given in the diagram.
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