Download RC Circuits tutorial

RC Circuits
1. Draw a circuit diagram showing an RC circuit with a switch (as
in figure 31. of your text).
Assume the capacitor is initially
uncharged.
a) Before the switch is closed
I=
Vc =
Vr =
b) What is the voltage drop across the switch?
At t= 0 s the switch is closed and current flows.
2. Write the loop equation for this circuit.
3. At the instant the switch is closed – you have the initial
conditions for the circuit. Use the loop equation and knowledge of
circuits to determine the initial value for:
I=
Vc =
Vr =
Qc =
4. Describe the changes in each of the initial values as current
flows in the circuit –
Qc
Vc
Vr
I
5. After enough time has passed, the circuit reaches a final state
(called steady state in engineering texts) where the values no
longer change. What is the steady state value of each of the circuit
parameters you have been evaluating?
Solving the loop equation –
6. Rewrite the loop equation from part 1 as a first order differential
equation. The general form will usually have all the terms with the
variable on one side.
You should be able to derive the solution to this equation by
separating the variables and integrating. It is necessary to correctly
identify the limits of integration. In this case the limits will go from
the initial value of Q (which is 0 C for this example) the any final
value, Q and the initial value of t (which is usually t = 0 s) to t.
8. Sketch a graph of Q vs t for this equation. Confirm that it
exhibits the expected change in Q as the capacitor charges
(explain)
9. Use the final solution for Q to find an expression for the current
I as a function of time. Explain what this shows about current over
time.
10. Sketch a graph of I vs.t.
The Time Constant
11. The product RC has the units of time (go ahead and check). It
is called the time constant. It is represented with the symbol . So
= RC. When t = this is one time constant.
a) Evaluate Q(t) and I(t) when t = .
b) How would doubling R or C change the time constant?
c) On your original sketch – add a line showing the Q vs t for a
circuit with R (or C) doubled compared to the original.
12. Discharging the capacitor-
Once the capacitor is charged, the
battery can be removed without affecting the capacitor charge.
When the loop is closed (figure 31.----) the capacitor discharges.
a) Sketch the circuit and write the loop equation
b) Solve this using integration
Problems – refer to the examples in Chapter 31.
1. An uncharged capacitor and resistor are connected to a battery
with EMF= 12 V, R =4Ω, C=6F. The switch is closed at time
t=0.
a) Find the time constant, the maximum charge on the capacitor
and the maximum current in the circuit.
b) Find the charge on the capacitor at t = 2
c) Find the voltage across the capacitor at t = 2(Vc = 10 V)
d) What is the voltage across the resistor at t = 2 ?
2. A capacitor of value C is discharging through a resistor, R. In
terms of time constant = RC, when will the charge on the
capacitor be one- half its initial value?
(ans t = 0.69 )
b) When will the energy be one-half its initial value? (ans t=0.35)