RC Series Circuits

Transient RC Series Circuits

Charging a Capacitor:

• The voltage across the capacitor is not instantaneously equal to that of the voltage across the battery when the switch is closed. The voltage on the capacitor builds up as more and more charges flows onto the capacitor until the battery is no longer able to "push" any more charge onto the capacitor, at which point the capacitor becomes fully charged.
  
• The initial flow of charges from the battery to the capacitor means that there is a current flowing through the system until the capacitor is charged. This current flow decays exponentially from some initial value to zero.
  
• The exponential nature of the change (of charge or voltage) with time results from applying Kirchoff's voltage rule and the definition of current as the rate of change of charge with time. (See Next Section)
  
• Note that in any real system the wires have some resistance, so a more correct model includes some resistance in series with the capacitor before it is charged up (Again see Next Section on RC circuits).

 Charging and Discharging a Capacitor in a RC Circuit (1.1M)

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The exponential nature of charging or discharging a capacitor:
Mathematically, an exponential change occurs when the derivative of a quantity with time is proportional to the quantity itself. For example if,

Let us apply this to the discharge of a capacitor through a resistor when the switch is closed and the capacitor is initially charged. After the switch is closed Kirchoff's voltage rule applies at all times and gives the equation.

Using Ohm's Law V = IR, the definition Capacitance C= Q/V, and the definition of current I = dQ/dt, we can rewrite this last equation so that the derivative of the charge is proportional to the negative of the amount of charge on the capacitor.

Thus the charge on the capacitor decays exponentially with a time constant τ.

Instead of Q_o we have use Q_max since the capacitor starts off with its maximum charge and decays away exponentially.

Mathematically, discharging a capacitor takes an infinite amount of time.The time constant τ represent the time for the system to make significant change in charge, voltage, or current whenever a capacitor is charging or discharging. After a time equal to one time constant, t = τ, the charge on the capacitor has dropped to e-1 = 36.8% of its maximum value. After 5τ the charge has dropped to 0.7% of its maximum value.	t	e–t/τ
	τ	.368
	2τ	.135
	3τ	.050
	4τ	.018
	5τ	.007

Other quantities such as the current and the voltage drop across the resistor or the capacitor can be found using the definition of current I = dQ/dt, Ohms Law V = IR, and the definition of capacitance C= Q/V.

Thus the current also decays away from its initial value when the switch is first closed. Ohms law shows that this also true for the voltage across the resistor.

The mathematical results for charging a capacitor are similar but more complicated. The main difference is that the charge and voltage across the capacitor approach a maximum value exponentially. For example, the charge on the capacitor start at zero an goes to a maximum value,

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Transient RC Conditions:Charging a Single Capacitor in a Series with a Resistor

	Start t = 0	Some Time Later at t	Long Time Later t >> t
	CAPACITOR
Voltage	0		VC,max = QC,max / C
Charge	0		QC,max
	RESISTOR
Voltage	VR,max = IR,max R		0
Current	IR,max		0

Note that these equations can only be used in a complex RC circuit if the circuit can be reduced to a simple RC circuit.

When the switch is first closed all the voltage is across the resistor and the circuit look like a simple DC Ohms law circuit with a resistor and no capacitor. This condition gives the maximum value of the voltage across the resistor and the maximum value of the current.

As charge flows onto the capacitor, the current drops exponentially (derivation) from its maximum value when the switch is first closed, I_max = V_b / R and I(t) = I_max e^-t/τ.

As charge flows onto the capacitor, the voltage across the resistor also drops exponentially. This happens because the capacitor now has a voltage drop and the circuit conforms to Kirchoff voltage rule at any time, V_b = V_c(t) + V_R(t) - the voltage drop across the capacitor and the resistor must equal to the voltage of the battery.

After a long time (t >> τ ) the current stops flowing in the circuit as the capacitor becomes fully charged and its voltage equals to that of the battery if it is a simple RC circuit. The voltage drop across the resistor is zero since there is no current flowing and V_R = I R.

The long time condition give the maximum charge on the capacitor. Here the circuit looks like a simple capacitor circuit with no resistor, Q_max = CV_b.

Note once again, that if the circuit is a more complex arrangement of capacitors and resistors them the maximum values may not be the voltage of the battery but the maximum value of the voltage across that component.

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Transient RC Conditions:Discharging a Single Capacitor Across a Resistor

	Start t = 0	Some Time Later at some time t	Long Time Later t >> τ
	CAPACITOR
Voltage	VC,max = QC,max / C		0
Charge	QC = QC,max		0
	RESISTOR
Voltage	VR,max = IR,max R		0
Current	IR = IR,max		0

When the circuit is first closed. the capacitor acts as a source of charge which can flow through the resistor. The capacitor acts like a emf source with a short lifetime compared to a batttery.

In energy terms, the energy in the electric field stored in the capacitor is converted into the energy of current flow which is in turn dissipated as heat as the charge flows through the resistor.

This is the simplest transient RC circuit problem since all the quantities that vary with time decay away expotentially.