Advertisement

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

The charging of a capacitor is not instant as capacitors have i-v characteristics which depend on time and if a circuit contains both a resistor (R) and a capacitor (C) it will form an RC charging circuit with characteristics that change exponentially over time.

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 Constantτ ).

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

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:

kirchhoffs voltage law

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

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

rc time constant formula

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:

capacitor voltage
  • 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.

Tutorial Example No1

Calculate the RC time constant, τ of the following circuit.

rc charging circuit example
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.

335 Comments

Leave a Reply to roobee Cancel reply
Error! Please fill all fields.

  • Khalid Mehmood

    Good One

  • michelle

    thank you

  • Marcus Vinícius

    cool maaaaan

  • Jacky Joy

    Uhm, I just soldered up a 7 minute timer using an NE555 at 12V with an R/C of 10M and 33uf.
    I’m a bit puzzled because when I had the unit of the plug board the R/V for 7 miutes was 470 K and 33uf. That’s a factor of 20:1 variation. Needless to say neither circuit coincided with the formula on this page. I guess the input of the NE555 may add or subtract loads to make things unpredictable.
    Just looking at the equation on this page, however, – quote
    “The time constant τ is given as: T = R x C = 47k x 1000uF = 47 Secs
    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”
    (Somewhere 3 zeros seem to be dropped)
    Without doing any calcs 5V @ 47K & 1000uf looks to me more like 30 seconds.
    Any comments?

    • Wayne Storr

      The time constant, tau of a series RC circuit is given as: R (in ohms) multiplied by C (in farads). Thus T = R x C (as given)
      If R = 47 kilo-ohms and C = 1000 micro-farads, then one time constant (T) equals R x C = 47000 x 0.001 = 47 seconds (basic maths)
      As a capacitor is classed as fully charged at a time period of five time constsants, or 5T, if 1T = 47 seconds, then 5T = 5 x 47 = 235 seconds (again basic maths)
      Then the tutorial is correct as given.

  • Ivy

    If a capacitor with a capacitance of 3 farads is connected to a 5-volt battery, then each conducting plate would have change q=cv or q=(3 farads) ×(5 volt)=15 coulombs of change of each conducting plate

  • Ahmet kelem

    Hi from Turkiye
    I am 72 old year.But I love education and technology.
    Thanks and very thanks.

  • M.Devi sri prasad

    can i get the pdf off above RC circuit theory

  • George foot

    How did you get the answer (0.1191) to the first part of this? 5(1 – e-2.1277) = 5(1 – 0.1191)

    • Wayne Storr

      ex = Power of e. The exponential constant e = 2.718, thus e-2.1277 = 2.718-2.1277 = 0.1191 as given in the tutorial.

  • Niamat Sandhu

    The analytic explanation on grounds of graphical representation are very much sound and whole is narrated in a very much understandable way of possessing the core knowledge of RC series circuit components up to full depth of the theme of the time constant related to the RC series circuit copmonents………

  • luke

    Hi , how could you work out the capacitor value if you know the resistance and voltage?

    • Charles C

      Luke, here is the formula solved for “C”: C = -t / (R(ln(-Vc\Vs+1))). The transposing of the equation is straightforward up until reversing the “e^(-t\RC)” part. In order to reverse an exponent, we need to take a log using the base of the exponent as the base of the log. In this particular case, the base is Euler’s number which can be simplified to Natural Log (ln) as the base of a natural log is Euler’s number. However, if the base were any other number (we’ll use ‘a’ for example), you would simply take the natural log of the expression and divide it by the natural log of ‘a’.

    • Wayne Storr

      Transpose the formula for Vc

  • Yash dhavale

    Nice teaching

  • Bonongwe Venancio

    Well explain and easy to understand

  • Unknownic

    Had a doubt regarding how time t is related to potential difference as
    V (DIRECTLY PROPORTIONAL)e^xt
    (V as exponential of time)

  • Pakeeza Anjum

    Capacitance is maximum at
    1)t=0
    2)t=RC
    3)t=infinity

    • Wayne Storr

      Capacitance, given in farads (F), is the ability of a capacitor to store an electric charge on its plates and is therefore related to the physical properties of its design and size. That is, a capacitor always has the same value of capacitance no matter its application.

  • KANYECURE MICHAEL

    Good work seen

  • Peter A Lloyd

    it would be useful for the “RC Charging Table” to be extended out to for example 10tau. this would provide a quick reference for those assessing “step” input settling time to high accuracy ADC’s etc

    • Wayne Storr

      Not really. As stated in the tutorial, a capacitor is said to be fully charged at 5 time constants (5T, 5RC, or 5tau), then at 10T it will be in the same state as 5T

  • Pooja varatha

    Thanks

  • jonno

    My last comment did not copy a useful diagram relating to CR voltage versus time delay.
    Maybe the following link can be explored.
    Go to google.com and search

  • zohaib balouch

    nice work. well explained

  • Farhan

    Very well explained