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Home / RC Networks / RC Charging Circuit

RC Charging Circuit

When a voltage source is applied to an RC circuit, the capacitor, C charges up through the resistance, R

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.

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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:

Where:

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

Time

Constant

RC Value

Percentage of Maximum

Voltage

Current

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.

The time constant, τ is found using the formula T = R x C in seconds.

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.

Next
RC Discharging Circuit

Read more Tutorials inRC Networks

1. RC Charging Circuit

2. RC Discharging Circuit

3. RC Waveforms

4. RC Integrator

5. RC Differentiator

276 Comments

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Jeff Darling

Great work.

I suspect you might see an increase in traffic to this page. I shared this page with some of my friends who will find this info useful.

Posted on October 24th 2020 | 3:41 pm

Reply

Nicola A Siciliano

It is possible to calculate the RC equivalent of the parallel of Series of RC)?

If i have several branches in parallel and each branch is made by R and C in series, like in this article. Assuming oll the R have the same value and all the C have the same value can i replace all the branches with only on RC equivalent?

(R+C)//(R+C)//(R+C)= ???

Regards,

Nico.

Posted on October 22nd 2020 | 8:45 pm

Reply

Wayne Storr

An interesting thought. Clearly with R1 = R2 = R3, and C1 = C2 = C3, the corner frequency would remain unchanged, and all you would have are parallel connected impedancies (R1 + Xc1) || (R2 + Xc2) || (R3 + Xc3) acting as a current divider. Then the parallel RC branches can be replaced with one single RC network at the chosen corner frequency.

Posted on October 23rd 2020 | 8:13 am

Reply

kenneth shanzekule

need more questions

Posted on October 13th 2020 | 7:05 am

Reply

Jackson Muyaka

Its been nice reading your logical writing. It’s really helpful.

Posted on October 13th 2020 | 5:44 am

Reply

Saara sally

What is the capacitor

Posted on October 01st 2020 | 3:56 am

Reply

Edward

device designed for package of charge

Posted on October 21st 2020 | 12:15 pm

Reply

Habib Nipon

In a RC series circuit, on the Resistor, current and voltage lies on Ohmic condition, but on the Capacitor, I leads the Vc by 90^o. So for KVL calculation, how can it be said that Ic =Ir?

Thank you

Posted on September 23rd 2020 | 11:24 am

Reply

johhn

You do not need to be involved with phase angles and capacitor reactive effects at this stage of the study. These can be seen in a later tutorial “AC Capacitance and Capacitive Reactance”. Don`t go there until you are confident with the basics.

A useful Rule of Thumb is; If components are in Series, establish that the current is the factor in common , then use Ohm`s Law and the Capacitor charging formula to establish the output voltage. (across the Capacitor ).

If the components are in Parallel then the Voltage is common and the Currents will differ. The real world resistance of the actual charging power source has to be included , otherwise a silly value for the capacitor charging surge will be seen.

Posted on September 24th 2020 | 11:46 am

Reply

Wayne Storr

For a series circuit, current is common to all components, thus Ic = Ir

Posted on September 23rd 2020 | 8:09 pm

Reply

Hans Laaby

What about constant current charging? Please show.

Posted on September 19th 2020 | 9:05 pm

Reply

johhn

Constant current charging of a capacitor means that the voltage across the capacitor , during the period of charging, is made to increase perfectly linearly e.g. as an arithmetical progression rather than the exponential seen with constant voltage charging via the series resistor as shown.

How to achieve this? You could have a high voltage battery and charge the capacitor through a high resistance . Capacitor voltage could be insignificant and current dependent on the battery volts /total circuit ohms. This will continue until the capacitor has charged to a fraction of the supply voltage.

Look at the charging graph. In the earliest stages of charging through a series resistor , this is almost a straight line.

Otherwise, use a special constant current source active circuit.

Posted on September 25th 2020 | 4:38 pm

Reply

Edward

What is meant by 63%

Posted on September 09th 2020 | 11:22 am

Reply

ALFONSO PEREZ

this the equation: Vc = Vs [1- eEXP(-1)] if you calculate the the square bracket value yields 0.63.

Posted on October 01st 2020 | 4:12 am

Reply

Akil

5. Assume a mobile phone charging unit is composed of a series RC network (R=5KΩ, C=7μF). If the charging unit is provided by 100V dc source along with a current limiting resistor of 15kΩ, then determine- [5 marks]

(i) When the phone will be fully charged?

(ii) For how long the phone will be able to hold the charge (you can convert the phone by a resistive load of 250 kΩ)?

Posted on September 02nd 2020 | 6:24 pm

Reply

ALFONSO PEREZ

does anybody know why tau is stated as R*C?

is it just because this equation simplifies the neperian power?

Vc = Vs [1 – e EXP(-t/R*C) becomes]

Vc = Vs [1- eEXP(-1)]

and why 4T is the value for transition?

thanks for your feedback

Posted on August 26th 2020 | 7:17 am

Reply

farah

Explain Time constant in an RC circuit. In an RC circuit during charging, If we doubled the resistance and halved the capacitance, the τ_c will remain the same or different?

Posted on July 28th 2020 | 6:57 pm

Reply

johhnf

The RC product i.e. the Time Constant is indeed the same.

Just bear in mind the surrounding electronic circuitry.

Maybe you are using very low consumption micropower circuits. These may well be unable to supply more than few micro amps of current. That capacitor needs a certain number of (micro) Coulombs to charge its plates. Thus it is justifiable to minimise the size of the capacitor and increase the charging series resistance in proportion.

The supply to the CR circuit will have its own Internal Resistance which has to be taken into account ( or designed away).

Modern batteries and lab power supplies have extremely low internal resistance.

Posted on July 28th 2020 | 7:30 pm

Reply

Ssd

Very clear this site is awesome

Posted on June 27th 2020 | 2:27 pm

Reply

Elton

I really liked the content, it is much better to study electronics knowing specifically what you want to be able to target!

Posted on June 19th 2020 | 8:53 am

Reply

Wahid nadeem

Description and explanation is good but understating for college students is too difficult

Posted on June 10th 2020 | 1:08 pm

Reply

CLeon

I am confused why there is a resistor. Can this circuit only has the capacitor and the battery?

Posted on May 23rd 2020 | 2:29 am

Reply

Moazam

Actually we are placing resistor here in order to measure noticeable change in capacitance with respect to time. According to the formula I=v/r if there would not be a resistor there would be infinite current in the circuit .this may lead to the charging of capacitor at 0 time .This sudden supply of infinite current may burn the capacitor. This is the reason which makes the presence of resistor a vital component of this circuit.

Posted on October 12th 2020 | 7:38 pm

Reply

Wayne Storr

If there was no resistance then it would not be an RC circuit

Posted on May 23rd 2020 | 6:28 pm

Reply

CLeon

the article has a lot of spelling mistakes, better polish up it firstly. an article like this is technically and fatal in case there are confusion.

Posted on May 23rd 2020 | 2:26 am

Reply

Wayne Storr

Having rechecked the tutorial, the British English given throughout is correct, however your English grammer regards the comment leaves a bit to be desired

Posted on May 23rd 2020 | 6:27 pm

Reply

Muhammad Saleem Hayat

I am totally satisfied with this site and it’s very helpful for learning any query about Electrical and electronics system

Posted on May 16th 2020 | 6:57 am

Reply

lucas

what does the e stand for

Posted on May 15th 2020 | 12:19 am

Reply

John

‘e’ stands for Euler’s number and it’s approximately equal to 2.7182818. Don’t know why that wasn’t covered here in this tutorial. Maybe in earlier chapters????

Posted on August 26th 2020 | 1:57 pm

Reply

drl

e = Exponent

ha haaa 🙂

Posted on July 01st 2020 | 1:44 am

Reply

odbal

The letter e denotes the exponential. Think of a process where the rate of flow is proportional to the amount remaining. For instance, a big water tank with a small hole in the bottom. A rapid flow at first dies away to a tiny trickle at the last. Compare this with the linear drop in water level produced by repeatedly removing the same small quantity of water.All kinds of DECAY processes, such as RC circuits, radioactivity, cooling of hot bodies… follow the same decay curve. Now look again at the curves and tables in the tutorial.https://betterexplained.com/articles/an-intuitive-guide-to-exponential-functions-e/might be a good place to go for more help.

Posted on June 11th 2020 | 5:07 pm

Reply

Darrell

https://en.wikipedia.org/wiki/E_(mathematical_constant)

Posted on May 28th 2020 | 8:12 pm

Reply

jonno

Think of a bank a/c with interest. Or a spreading virus. Or a decaying isotope etc etc.

Plotting the graph of any of these will give an identical shape of curve (s) depending on growth (upwards) and decay (downwards.

The fundamental shape is y=e^^x.

If you can put the bank interest rate or whatever into a statement with the magic e you have an instant graphical function making high end maths a lot easier.

Google exponential function.

Posted on May 17th 2020 | 12:04 pm

Reply

Plaid387hz

Euler number = 2.71828

Posted on May 15th 2020 | 5:54 am

Reply

More

Vineth

I like it thanku sir

Posted on May 08th 2020 | 8:27 am

Reply

jcc4987864

Lots of grammatical errors, ex : “the voltage across the value of the voltage across the capacitor”, commas are used in such a pitiful way…

Posted on April 02nd 2020 | 4:12 pm

Reply

View More

Read more Tutorials inRC Networks

1. RC Charging Circuit

2. RC Discharging Circuit

3. RC Waveforms

4. RC Integrator

5. RC Differentiator

Related Tutorials

RC Differentiator

The passive RC differentiator is a series connected RC network that produces an output signal which corresponds [...]

RC Integrator

The RC integrator is a series connected RC network that produces an output signal which corresponds to [...]

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