All Electrical or Electronic circuits or systems suffer from some form of “time-delay” between its input and output terminals when either a signal or voltage, continuous, ( DC ) or alternating ( AC ), is applied to it.
This delay is generally known as the circuits time delay or Time Constant which represents the time response of the circuit when an input step voltage or signal is applied. The resultant time constant of any electronic circuit or system will mainly depend upon the reactive components either capacitive or inductive connected to it. Time constant has units of, Tau – τ
When an increasing DC voltage is applied to a discharged Capacitor, the capacitor draws what is called a “charging current” and “charges up”. When this voltage is reduced, the capacitor begins to discharge in the opposite direction. Because capacitors can store electrical energy they act in many ways like small batteries, storing or releasing the energy on their plates as required.
The electrical charge stored on the plates of the capacitor is given as: Q = CV. This charging (storage) and discharging (release) of a capacitors energy is never instant but takes a certain amount of time to occur with the time taken for the capacitor to charge or discharge to within a certain percentage of its maximum supply value being known as its Time Constant ( τ ).
If a resistor is connected in series with the capacitor forming an RC circuit, the capacitor will charge up gradually through the resistor until the voltage across it reaches that of the supply voltage. The time required for the capacitor to be fully charge is equivalent to about 5 time constants or 5T. Thus, the transient response or a series RC circuit is equivalent to 5 time constants.
This transient response time T, is measured in terms of τ = R x C, in seconds, where R is the value of the resistor in ohms and C is the value of the capacitor in Farads. This then forms the basis of an RC charging circuit were 5T can also be thought of as “5 x RC”.
RC Charging Circuit
The figure below shows a capacitor, ( C ) in series with a resistor, ( R ) forming a RC Charging Circuit connected across a DC battery supply ( Vs ) via a mechanical switch. at time zero, when the switch is first closed, the capacitor gradually charges up through the resistor until the voltage across it reaches the supply voltage of the battery. The manner in which the capacitor charges up is shown below.
RC Charging Circuit
Let us assume above, that the capacitor, C is fully “discharged” and the switch (S) is fully open. These are the initial conditions of the circuit, then t = 0, i = 0 and q = 0. When the switch is closed the time begins at t = 0 and current begins to flow into the capacitor via the resistor.
Since the initial voltage across the capacitor is zero, ( Vc = 0 ) at t = 0 the capacitor appears to be a short circuit to the external circuit and the maximum current flows through the circuit restricted only by the resistor R. Then by using Kirchhoff’s voltage law (KVL), the voltage drops around the circuit are given as:
The current now flowing around the circuit is called the Charging Current and is found by using Ohms law as: i = Vs/R.
RC Charging Circuit Curves
The capacitor (C), charges up at a rate shown by the graph. The rise in the RC charging curve is much steeper at the beginning because the charging rate is fastest at the start of charge but soon tapers off exponentially as the capacitor takes on additional charge at a slower rate.
As the capacitor charges up, the potential difference across its plates begins to increase with the actual time taken for the charge on the capacitor to reach 63% of its maximum possible fully charged voltage, in our curve 0.63Vs, being known as one full Time Constant, ( T ).
This 0.63Vs voltage point is given the abbreviation of 1T, (one time constant).
The capacitor continues charging up and the voltage difference between Vs and Vc reduces, so too does the circuit current, i. Then at its final condition greater than five time constants ( 5T ) when the capacitor is said to be fully charged, t = ∞, i = 0, q = Q = CV. At infinity the charging current finally diminishes to zero and the capacitor acts like an open circuit with the supply voltage value entirely across the capacitor as Vc = Vs.
So mathematically we can say that the time required for a capacitor to charge up to one time constant, ( 1T ) is given as:
RC Time Constant, Tau
This RC time constant only specifies a rate of charge where, R is in Ω and C in Farads.
Since voltage V is related to charge on a capacitor given by the equation, Vc = Q/C, the voltage across the capacitor ( Vc ) at any instant in time during the charging period is given as:
- Vc is the voltage across the capacitor
- Vs is the supply voltage
- e is an irrational number presented by Euler as: 2.7182
- t is the elapsed time since the application of the supply voltage
- RC is the time constant of the RC charging circuit
After a period equivalent to 4 time constants, ( 4T ) the capacitor in this RC charging circuit is said to be virtually fully charged as the voltage developed across the capacitors plates has now reached 98% of its maximum value, 0.98Vs. The time period taken for the capacitor to reach this 4T point is known as the Transient Period.
After a time of 5T the capacitor is now said to be fully charged with the voltage across the capacitor, ( Vc ) being aproximately equal to the supply voltage, ( Vs ). As the capacitor is therefore fully charged, no more charging current flows in the circuit so IC = 0. The time period after this 5T time period is commonly known as the Steady State Period.
Then we can show in the following table the percentage voltage and current values for the capacitor in a RC charging circuit for a given time constant.
RC Charging Table
|RC Value||Percentage of Maximum|
|0.5 time constant||0.5T = 0.5RC||39.3%||60.7%|
|0.7 time constant||0.7T = 0.7RC||50.3%||49.7%|
|1.0 time constant||1T = 1RC||63.2%||36.8%|
|2.0 time constants||2T = 2RC||86.5%||13.5%|
|3.0 time constants||3T = 3RC||95.0%||5.0%|
|4.0 time constants||4T = 4RC||98.2%||1.8%|
|5.0 time constants||5T = 5RC||99.3%||0.7%|
Notice that the charging curve for a RC charging circuit is exponential and not linear. This means that in reality the capacitor never reaches 100% fully charged. So for all practical purposes, after five time constants (5T) it reaches 99.3% charge, so at this point the capacitor is considered to be fully charged.
As the voltage across the capacitor Vc changes with time, and is therefore a different value at each time constant up to 5T, we can calculate the value of capacitor voltage, Vc at any given point, for example.
RC Charging Circuit Example No1
Calculate the RC time constant, τ of the following circuit.
Therefore the time constant τ is given as: T = R x C = 47k x 1000uF = 47 Secs
a) What will be the value of the voltage across the capacitors plates at exactly 0.7 time constants?
At 0.7 time constants ( 0.7T ) Vc = 0.5Vs. Therefore, Vc = 0.5 x 5V = 2.5V
b) What value will be the voltage across the capacitor at 1 time constant?
At 1 time constant ( 1T ) Vc = 0.63Vs. Therefore, Vc = 0.63 x 5V = 3.15V
c) How long will it take to “fully charge” the capacitor from the supply?
We have learnt that the capacitor will be fully charged after 5 time constants, (5T).
1 time constant ( 1T ) = 47 seconds, (from above). Therefore, 5T = 5 x 47 = 235 secs
d) The voltage across the Capacitor after 100 seconds?
The voltage formula is given as Vc = V(1 – e(-t/RC)) so this becomes: Vc = 5(1 – e(-100/47))
Where: V = 5 volts, t = 100 seconds, and RC = 47 seconds from above.
Therefore, Vc = 5(1 – e(-100/47)) = 5(1 – e-2.1277) = 5(1 – 0.1191) = 4.4 volts
We have seen here that the charge on a capacitor is given by the expression: Q = CV, where C is its fixed capacitance value, and V is the applied voltage. We have also learnt that when a voltage is firstly applied to the plates of the capacitor it charges up at a rate determined by its RC time constant, τ and will be considered fully charged after five time constsants, or 5T.
In the next tutorial we will examine the current-voltage relationship of a discharging capacitor and look at the discharging curves associated with it when the capacitors plates are effectively shorted together.