Brooklyn College 1 RC Circuits Purpose 1. To study transient response of an RC series circuit. 2. To measure and calculate the time constant, . Introduction When a battery is applied to a capacitor and a resistor in series (at the instant that we close the switch, S) as shown in fig. 1, the capacitor takes a small time to charge, after that it becomes fully charged. The state during the time in which the capacitor is charging is called the ‘transient state’. When the capacitor reaches its full charge, the state is then called the ‘steady state’. We are going to study the transient state. Initially the capacitor is uncharged (has no net charge). When the switch is closed, charges flow fast to start charging the capacitor plates. The initial fast motion of charges causes a large initial current of the capacitor, . As the charges accumulate on the capacitor, the capacitor acts as a battery that opposes the flow of charges to it, therefore the charge flow slows down, and the capacitor current, decreases. The increase of charge of the capacitor follows an exponential growth curve. The voltage across a capacitor, is always directly proportional to the charge on the plates of the capacitor, , where is called the capacitance of the capacitor. The units of is the Farad, . . Therefore, the voltage across a capacitor also follows a similar exponential grow curve. Since the capacitor current is decreasing it comes out (from a mathematical derivation) that it follows an exponential decay curve. If we apply Kirchhoff’s voltage rule, KVL to the loop of the circuit of figure 1, with the switch closed we get: where is the source voltage. Using eqns. 1, 2 & 3, it can be shown (see the text book) that the equations, as a function of time, , for the capacitor charge, and the capacitor voltage, during the transient charging are inverse exponential: where, is called the ‘time constant’ and , for the circuit of fig. 1, And the equation describing the current of the capacitor, during the transient charging is exponential decay: 0 Figure 2: The curve for the capacitor charge, during transient charging Figure 1: Series RC circuit for capacitor charging S Figure 3: The curve for capacitor current decay during transient charging 0
Transcript
Brooklyn College 1
RC Circuits
Purpose
1. To study transient response of an RC series circuit.
2. To measure and calculate the time constant, .
Introduction
When a battery is applied to a capacitor and a resistor in series (at the instant that we close the switch, S) as shown in
fig. 1, the capacitor takes a small time to charge, after that it becomes
fully charged. The state during the time in which the capacitor is
charging is called the ‘transient state’. When the capacitor reaches its
full charge, the state is then called the ‘steady state’. We are going to
study the transient state.
Initially the capacitor is uncharged (has no net charge). When the
switch is closed, charges flow fast to start charging the capacitor
plates. The initial fast motion of charges causes a large initial current
of the capacitor, . As the charges accumulate on the capacitor, the
capacitor acts as a battery that opposes the flow of charges to it,
therefore the charge flow slows down, and the capacitor current, decreases. The increase of charge of the capacitor
follows an exponential growth curve. The voltage across a capacitor, is
always directly proportional to the charge on the plates of the capacitor, ,
where is called the capacitance of the capacitor. The units of is the Farad,
.
. Therefore, the voltage across a capacitor also
follows a similar exponential grow curve. Since the capacitor current is
decreasing it comes out (from a mathematical derivation) that it follows an
exponential decay curve. If we apply Kirchhoff’s voltage rule, KVL to the loop
of the circuit of figure 1, with the switch closed we get:
where is the source voltage. Using eqns. 1, 2 & 3, it can be shown (see the text book) that the equations, as a function
of time, , for the capacitor charge, and the capacitor voltage, during the
transient charging are inverse exponential:
where, is called the ‘time constant’ and , for the circuit of fig. 1,
And the equation describing the current of the capacitor, during the transient charging is exponential decay:
0
Figure 2: The curve for the capacitor
charge, during transient charging
Figure 1: Series RC circuit
for capacitor charging
S
Figure 3: The curve for capacitor
current decay during transient charging
0
Brooklyn College 2
Where, is the initial current of the capacitor. Notice from eqn. 2 that for fig. 1, during transient charging (and also in
the steady state) is always a constant equal to .
When the capacitor is fully charged, then no further charges flow to the capacitor plates, the charge on the plates of the
capacitor becomes constant and the capacitor current, becomes zero. Then the capacitor is said to have reached the
steady state.
When the time during charging from zero charge, reaches , then from eqn. 4 we get is equal to . The
time constant, is a measure of the charging and discharging time of the capacitor. A large time constant, means
slow charging. Also we can find, using eqn. 7, that for the capacitor current, becomes equal to .
If the battery, is replaced by a wire, then the capacitor discharges through the resistor and the charge and voltage of
the capacitor follow an exponential decay curve. This is the transient
discharge state. The eqn. for the capacitor charge and voltage during
discharging are:
as shown in fig. 4. Since the discharge starts fast then slows, the current
during discharge also starts large then decreases. It follows an
exponential decay curve like the charging case but it has an opposite
direction in the circuit, as shown in fig. 4. The eqn. is:
We can find , & at using equations 8, 9 & 10, and we find
the result as shown in figures 4 & 5. The curve for the capacitor voltage,
is similar to the curve for the capacitor charge except for a
multiplication factor (see eqn. 1 above).
We can cause the charging and the discharging transients by applying a
square wave in place of the battery . We can adjust the square wave
so that the peak is equal to the value of (charging case) and the
lower value of the square wave is equal to zero volts (discharging
case) , as shown in fig. 6.
Figure 4: The curve for capacitor charge
decay during transient discharging
0
Figure 5: The curve for capacitor current
decay during transient discharging
0
0
Figure 6: A square wave voltage source to
cause the charging and discharging transients.
Brooklyn College 3
Running the experiment (The data sheet is on page 6)
Finding the measured time constant for an RC circuit charging and discharging transients
1) Open the simulator http://falstad.com/circuit/ click the Run/Stop to stop the default simulation. Clock circuits from
the top menu and select black circuit. This is to clear the default circuit.
2) Click ‘Draw’ in the top menu and move the mouse to Inputs and Sources and select ‘Add A/C Voltage Source (2-
terminal). Click somewhere at the left of the black space and drag up to place your AC voltage source (make sure the +
sign is on the top of the circle of the voltage source. Move the mouse over the source till its color turns light blue and
right click and select edit. Set the Max voltage to 0.5 (the simulator already knows all the units so here do not write v, it
already knows that the unit is volts). Click the drop down menu of the waveform and select square wave. Set the DC
offset to 0.5 (this will cause the upper peak to be at 1v and the lower at 0v). The DC offset is a constant value of voltage
that is added to all point of the voltage of the wave. Set the frequency to 12 Hz (do not write Hz, it already knows the
unit). Keep the rest of the values at their default: phase offset = 0 (meaning the wave will start at 0 time) and the Duty
cycle = 50 (meaning the square wave is symmetrical. Meaning it has equal time duration for the max voltage as for the
time duration for its minimum voltage, as in fig 6 above). Click ‘Apply’ then click ‘OK’.
3) Click Draw from the top menu and select ‘Add Resistor’. Click and drag to place it as shown in fig. 7 below. Edit it to
change its value in a similar way as you did for the voltage source (moving the mouse over it till its color is light blue
then right click and select edit). Set it to 180 (it already knows the unit is ohms). Click Draw in the top menu, move the
mouse to ‘Passive Components’ and select ‘Add Capacitor’. Click and drag to place it as shown in fig. 7 below. Change its
value to 33u. We will call this capacitor C1.The simulator accepts ‘u’ for μ. 33μF means 33 micro and the simulator
already knows that the unit is Farad so do not write the unit Farad, F. The milli, m is , the micro, μ is and the
nano, n is .
4) To complete the circuit connections, click Draw and select Add wire. And click and drag to connect the components
with wires as shown in fig. 8 above. You need to click the square terminal of the wire (that shows when you place the
mouse over the round terminal of the circuit element (say the voltage source)) and drag a segment of wire (in the case
of the top terminal of the voltage source, two segments, one small vertically up from the top terminal of the voltage
source(as shown in fig. 8) and the other small segment towards the right to the resistor’s left terminal). Make sure not to
run a wire through a circuit element, otherwise you short circuit that element. After completing the circuit, click ‘Draw’
in the top menu and select ‘Select/Drag Sel’. This is to stop giving you more wires.
5) Move the mouse to the square wave voltage source and right click to select ‘View in Scope’. This will give a graph
below to display the square wave voltage as a function of time. Click the gear icon on the graph and choose’ Properties’.
Adjust the ‘Scroll Speed’ to 5ms/div, then click OK (do not click set as default). Again, repeat this step for the capacitor
Figure 7 (Left): Building the circuit. Figure 8 (Right): Completing wire connections.
