Explain why a
capacitor is considered fully charged at the end of 5 time constants.
Calculate the amount
of voltage on a capacitor after it has charged a given number of time constants.
Explain the meaning
of instantaneous capacitor voltage.
Describe each
of the terms in the equation for determining the instantaneous capacitor-charging voltage
in an RC circuit.
Calculate the
instantaneous capacitor charge voltage.
Describe each
of the terms in the equation for determining the instantaneous capacitor-charging current
in an RC circuit.
Calculate the
instantaneous capacitor charge current.
Describe the curves
for instantaneous resistor voltage and current in a capacitor-charging circuit.
Capacitor
Charge Voltage
Instantaneous
Capacitor Voltage, vc
While a capacitor is charging through a
resistance, the voltage across the capacitor at any given instant is: vc = Vs(1 - e-t/RC) where
vc = instantaneous voltage across the capacitor
in volts
Vs = source voltage in volts
e = constant 2.718
t = time in seconds
R = resistance value in ohms
C = capacitor value in farads
When
the capacitor is first connected to the DC source, the voltage across the capacitor
increases rapidly. The rate of increase then slows considerably as the capacitor voltage
reaches the source voltage level.
Since the time constant of a circuit is given by T
= RC, you can replace the RC terms in the equation for capacitor voltage
to get this version: vc = Vs(1 - e-t/T)
For this circuit:
Vs = 16 V
R = 220 kW
C = 1 mF
Calculate:
The time constant for the circuit.
The voltage across the capacitor after 100 ms of charge time.
Ans: T = 220 ms, vc = 5.84 V
Step 1
T = RC
T = 220 ms
Step 2
vc
= Vs(1 - e-t/T)
vc = 16(1 - e -100/220)
vc = 5.84 V
While a capacitor is charging through a
resistance, the capacitor current at any given instant is: ic
= (Vs/R) e-t/RC where
ic = instantaneous capacitor current in
amperes
Vs = source voltage in volts
e = constant 2.718, unitless
t = time in seconds
R = resistance value in ohms
C = capacitor value in farads
When the capacitor is first connected to the DC voltage source, there is a
rush of charge current that is limited only by the value of resistor R.
Since the time constant of
an RC circuit is given by T = RC, you can replace the RC terms
in the equation for capacitor current to get this version:
ic = (Vs/R)
e-t/T
For this circuit:
Vs = 16 V
R = 100 kW
C = 10 nF
Calculate:
The time constant for the circuit.
The voltage across the capacitor after 100 ms of charge time.
Ans: T = 1 ms, ic = 145 mA
Step1
T = RC
T = 1 ms
Step 2
ic
= (Vs/R) e-t/RC
ic = (16 V/100 kW)e
-100 ms/1 ms
ic = 145 mA
Since the resistor and capacitor are connected in series, the
resistor current is equal to the capacitor charging current.
So while a capacitor is charging through a resistance, the resistor
current at any given instant is: iR = (Vs/R) e-t/RC
where
iR = instantaneous resistor current in
amperes
Vs = source voltage in volts
e = constant 2.718
t = time in seconds
R = resistance value in ohms
C = capacitor value in farads
Resistor Voltage
While a capacitor is charging through a
resistance, the resistor voltage at any given instant is: vR
= Vs e-t/RC where
vR = instantaneous resistor voltage in
amperes
Vs = source voltage in volts
e = constant 2.718
t = time in seconds
R = resistance value in ohms
C = capacitor value in farads
You should take note of the following
relationships concerning the charge current and voltage in a series RC circuit.
At any given instant, the
capacitor current, resistor current, and source current are equal. This is a
property of series DC circuits. iC = iR = iS
At any given instant,
the sum of the capacitor voltage and resistor voltage equals the source voltage.
This is an expression of Kirchhoff's Voltage Law. Vs = vc
+ vR
