OBJECTIVES:
1. To study the transient response in storing an electrical charge on a capacitor in a RC circuit.
2. Also to study the transient decay of an initial charge on a capacitor thru a resistor.
3. To understand the time constant in a RC circuit and how it can be changed.
BACKGROUND & THEORY:
A capacitor has the ability to store an electrical charge and energy. The voltage across the capacitor is related to the charge by the equation V = Q/C for steady state values, or expressed as an instantaneous value, dv = dq/C
By definition i = dq/dt or dq = idt. Therefore
The derivation of the transient responses of both the capacitor current and voltage in an RC circuit when a source voltage is suddenly applied to that circuit is shown below. Note that the time constant = tau = RC.
The step response of an RC circuit can be analyzed using the following
circuit.
Immediately after the switch closes, KVL requires that -
(1)
If we differentiate (1) with respect to t, we get
(2)
The other two terms drop out because they are constants. Now divide
thru by R -
(3)
Separate the variables and indicate integration of both sides.
(4)
Do the integration of (4) and show limits.
(5)
Put in the limits.
(6)
Take the antilog:
(7)
The voltage across the capacitor at t = 0 (Vo) will be zero
because there cannot be an instantaneous change in voltage across the capacitor.
Therefore, the initial current in the circuit will be as follows:
(8)
Substituting (8) into (7) and plotting Normalized current = i(t)/Io
,versus Normalized time = t/RC.
Note that the time constant (t = J = RC) occurs at 36.8% of Io
or 0.368 Vs/R.
We also know that
(9)
Substituting i from (7) and Io from (8) into (9),
(10)
Putting in limits and simplifying gives -
(11)
Plotting normalized voltage ( Vc /VS ) versus
normalized time ( t/RC )
Note that the time constant ( t = J
= RC) occurs at 0.632 Vs.
EQUIPMENT AND PARTS LIST:
Variable Voltage Supply
Breadboard
Digital Multimeter (DMM)
Fixed Resistors: 120 kW, 68 kW
Electrolytic Capacitor 330 µF
PROCEDURE:
1. Refer to BACKGROUND & THEORY for how to calculate time constant . Calculate for a series RC circuit having C = 330 µF and R = 120 kW to gain a perspective of how long the transients will take.
2. Construct the following circuit using the values of R and C given in item 1. You will use a jumper wire for the switch to connect the resistor either to the voltage source or to the reference node (ground). Be sure to connect the negative side of the electrolytic capacitor to ground. The capacitor will either have a ,+, on one end and a ,-, on the other, or it will have a thick arrow with a ,-, inside it pointing to the negative end of the capacitor.
3. Set VS at 15 volts. Leave the jumper wire in the discharge
position until the voltage across the capacitor stabilizes at 0 volts.
(Note that if the capacitor has been charged before, a wire temporarily
shorting out the capacitor will speed up this process).
4. Then put the jumper wire in the charge position and record Vc
every 5 seconds up to 30 seconds then every 10 seconds from 30 seconds
to 2 minutes. Then leave the switch in up position until the voltage Vc
stabilizes at the maximum value (when the third digit is no longer changing
over 30 second period) and record that value.
Note: One person will need to call off time and the other person read the meter and write down voltage. You may need to do this more than once until you establish a good procedure for taking data.
5. Next put the jumper wire in the discharge position and record the capacitor voltage Vc at the same time intervals as in item 4.
6. Repeat items 4 and 5 with R = 68 kW.
QUESTIONS, COMPARISONS & GRAPHS:
1. Calculate the Thévenin Equivalent of the circuit seen by the capacitor including the effect of the voltmeter for each circuit, using the measured values of the resistors and source voltage.
2. Calculate the time constant for each circuit using the results of the previous calculation and the measured capacitor value.
3. Plot a graph of the charging curve for each of the two resistance values, VC (y axis) versus time (x axis). Draw in a dotted line to the graph at a point 63.2% of Thévenin Equivalent value of VS. What is the time in seconds which corresponds to this voltage on each curve. Make the graph as large as possible to enable more accurate readings of the two experimental values of the time constant. How does this time compare with your calculations of the time constants? Why might this time be slightly different from your calculation?
4. Plot a graph of the discharge curve for each of the two resistance values, VC (y axis) versus time (x axis). Draw in a dotted line to the graph at a point 36.8% of the initial value of VC. What is the time in seconds which corresponds to this voltage on each curve? How does this time compare with your calculations of the time constants?
Your discharge graph should look something like this.
5. What happened to the response time and the time constant with the smaller resistance?
CONCLUSIONS:
Based on what you've learned.
This page last updated
02/03/2006