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Capacitive Reactance

Capacitive Reactance

Capacitive Reactance is the complex impedance of a capacitor who’s value changes with respect to the applied frequency

Capacitive Reactance is the complex impedance value of a capacitor which limits the flow of electric current through it. Capacitive reactance can be thought of as a variable resistance inside a capacitor being controlled by the applied frequency.

Unlike resistance which is not dependent on frequency, in an AC circuit reactance is affected by supply frequency and behaves in a similar manner to resistance, both being measured in Ohms. Reactance affects both inductors and capacitors with each having opposite effects in relation to the supply frequency. Inductive reactance (XL) rises with an increase in frequency, whereas capacitive reactance (XC) falls.

In the RC Network tutorial we saw that when a DC voltage is applied to a capacitor, the capacitor itself draws a charging current from the supply and charges up to a value equal to the applied voltage. Likewise, when the supply voltage is reduced the charge stored in the capacitor also reduces and the capacitor discharges.

But in an AC circuit in which the applied voltage signal is continually changing from a positive to a negative polarity at a rate determined by the frequency of the supply, as in the case of a sine wave voltage. For example, the capacitor is either being charged or discharged on a continuous basis at a rate determined by the supply frequency.

As the capacitor charges or discharges, a current flows through it which is restricted by the internal impedance of the capacitor. This internal impedance is commonly known as Capacitive Reactance and is given the symbol XC in Ohms.

Unlike resistance which has a fixed value, for example, 100Ω, 1kΩ, 10kΩ etc, (this is because resistance obeys Ohms Law), Capacitive Reactance varies with the applied frequency so any variation in supply frequency will have a big effect on the capacitor’s, “capacitive reactance” value.

As the frequency applied to the capacitor increases, its effect is to decrease its reactance (measured in ohms). Likewise as the frequency across the capacitor decreases its reactance value increases. This variation is called the capacitor’s complex impedance.

Complex impedance exists because the electrons in the form of an electrical charge on the capacitor plates, appear to pass from one plate to the other more rapidly with respect to the varying frequency.

As the frequency increases, the capacitor passes more charge across the plates in a given time resulting in a greater current flow through the capacitor appearing as if the internal impedance of the capacitor has decreased. Therefore, a capacitor connected to a circuit that changes over a given range of frequencies can be said to be “Frequency Dependant”.

Capacitive Reactance has the electrical symbol “XC” and has units measured in Ohms the same as resistance, ( R ). It is calculated using the following formula:

Capacitive Reactance

capacitive circuit capacitive reactance
 
Capacitive Reactance Formula
 
  • Where:
  •    Xc = Capacitive Reactance in Ohms, (Ω)
  •    π (pi) = 3.142 (decimal) or as 22÷7 (fraction)
  •    ƒ = Frequency in Hertz, (Hz)
  •    C = Capacitance in Farads, (F)

Capacitive Reactance Example No1

Calculate the capacitive reactance value of a 220nF capacitor at a frequency of 1kHz and again at a frequency of 20kHz.

 At a frequency of 1kHz:

capacitive reactance equation

 Again at a frequency of 20kHz:

reactance formula

 where: ƒ = frequency in Hertz and C = capacitance in Farads

Therefore, it can be seen from above that as the frequency applied across the 220nF capacitor increases, from 1kHz to 20kHz, its reactance value, XC decreases, from approx 723Ω to just 36Ω and this is always true as capacitive reactance, XC is inversely proportional to frequency with the current passed by the capacitor for a given voltage being proportional to the frequency.

For any given value of capacitance, the reactance of a capacitor, XC expressed in ohms can be plotted against the frequency as shown below.

Capacitive Reactance against Frequency

capacitive reactance against frequency

 

By re-arranging the reactance formula above, we can also find at what frequency a capacitor will have a particular capacitive reactance ( XC ) value.

Capacitive Reactance Example No2

At which frequency would a 2.2uF Capacitor have a reactance value of 200Ωs?

frequency formula

 

Or we can find the value of the capacitor in Farads by knowing the applied frequency and its reactance value at that frequency.

Capacitive Reactance Example No3

What will be the value of a capacitor in farads when it has a capacitive reactance of 200Ω and is connected to a 50Hz supply.

capacitance formula

 

We can see from the above examples that a capacitor when connected to a variable frequency supply, acts a bit like a frequency controlled variable resistance as its reactance (X) is “inversely proportional to frequency”. At very low frequencies, such as 1Hz our 220nF capacitor has a high capacitive reactance value of approx 723.3KΩ (giving the effect of an open circuit).

At very high frequencies such as 1Mhz the capacitor has a low capacitive reactance value of just 0.72Ω (giving the effect of a short circuit). So at zero frequency or steady state DC our 220nF capacitor has infinite reactance looking more like an “open-circuit” between the plates and blocking any flow of current through it.

Voltage Divider Revision

We remember from our tutorial about Resistors in Series that different voltages can appear across each resistor depending upon the value of the resistance and that a voltage divider circuit has the ability to divide its supply voltage by the ratio of R2/(R1+R2).

Therefore, when R1 = R2 the output voltage will be half the value of the input voltage. Likewise, any value of R2 greater or less than R1 will result in a proportional change to the output voltage. Consider the circuit below.

Voltage Divider Network

voltage divider network

 

We now know that a capacitor’s reactance, Xc (its complex impedance) value changes with respect to the applied frequency. If we now changed resistor R2 above for a capacitor, the voltage drop across the two components would change as the frequency changed because the reactance of the capacitor affects its impedance.

The impedance of resistor R1 does not change with changes in supply frequency as fixed value resistors are unaffected by changes in frequency. Then the voltage dropped across resistor R1 and therefore the output voltage is determined by the capacitive reactance of the capacitor at a given frequency.

This then results in a frequency-dependent RC voltage divider circuit. With this idea in mind, passive Low Pass Filters and High Pass Filters can be constructed by replacing one of the voltage divider resistors with a suitable capacitor as shown.

Low Pass Filter

low pass filter

High Pass Filter

high pass filter

 

The property of Capacitive Reactance, makes the capacitor ideal for use in AC filter circuits or in DC power supply smoothing circuits to reduce the effects of any unwanted Ripple Voltage as the capacitor applies an short circuit signal path to any unwanted frequency signals on the output terminals.

Capacitive Reactance Summary

So, we can summarize the behaviour of a capacitor in a variable frequency circuit as being a sort of frequency controlled resistor that has a high capacitive reactance value (open circuit condition) at very low frequencies and low capacitive reactance value (short circuit condition) at very high frequencies as shown in the graph above.

effect of frequency on capacitance

 

It is important to remember these two conditions and in our next tutorial about the Passive Low Pass Filter, we will look at the use of Capacitive Reactance to block any unwanted high frequency signals while allowing only low frequency signals to pass.

147 Comments

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  • ibrahim bangura

    there is a funny wrong statement on this note saying:
    “capacitive reactance is directly proportional to frequency” (This statement is wrong)!!!
    CAPACITIVE REACTANCE IS INVERSELY PROPORTIONAL TO FREQUENCY (correct)?
    example:
    1. high frequency cause low capacitive reactance.
    2. low frequency cause high capacitive reactance.
    ( This INVERSE proportionality)!
    I hope this mistake is rectified sooner.
    thanks

    • Wayne Storr

      We do not state anywhere in the tutorial that: “capacitive reactance is directly proportional to frequency”, then we can not correct what we have not written

      • Rares

        Actually you did say that:

        At Capacitive Reactance Example No3

        You said: We can see from the above examples that a capacitor when connected to a variable frequency supply, acts a bit like a “frequency controlled variable resistor” as its reactance (X) is directly proportional to frequency.

        We all know that reactance is inverse proportional to frequency.

        But this is a minor mistake.

  • md-Rakib hasan

    Calculate the reactance of 6 uF capacitor at 60 Hz.

  • Vinyasi

    How do I learn more about a certain type of wave which may best be described as being that of pulses of capacitive reactance?…

  • Charlie

    Great review.

  • chandra

    Excellent!!!!!

    Can u please sen the entire pdf to my mail id…please..

  • Dom

    How do I find the value of capacitor in a RC circuit, without the capacitor or capacitive reactance given in the question. The values I have in the question is the frequency (50 Hz a.c), current (0.4139 A) and voltages across resistor and capacitor.

  • Vinyasi

    Many many times during simulation I have seen the frequency go up in Eric Dollard’s analog computer in longitudinal magneto dielectric modality in which there is a minimum of two capacitors and two inductors in a circular ring and this ring is daisy-chained such that the inductors are in parallel and the capacitors are in serious with a minimum of two modules connected together in daisychain fashion and preferentially a maximum of about three. So, two or three is the optimal quantity of modules daisy-chained together. So, your article would imply that an inductor will increase the frequency of capacitive reactance coming from a capacitor and that increase in frequency would then be passed back to the capacitor in a positive feedback condition that spirals to infinity oblivion. The question is yes we know this doesn’t happen necessarily in an LRC circuit. And my question has always been how is it that a minimum of two capacitors and two inductors will do this (if the frequency is within a certain window and the alternating or square wave input is also kept very low and for a very short duration not to supply energy for the circuit but merely to provide a stimulus – a starting point – to initiate this escalation) and not a single capacitor in combination with a single inductor? Anybody have any ideas?…

  • ali

    So how changing resistance and capacitance affects the oscillator frequency when the resistor or capacitor is increasing and decreasing?

  • Abubakar

    What is formula of Avogadro’s number.

    • mouselb

      Avogadro’s constant is the number of elemental or molecular particles comprising one mol of that element or molecule. One mol of carbon·12 = 12grams ~= 6.02 x 10^23 atoms of carbon-12. One mol of O2·16 = 32grams ~= 6.02 x 10^23 diatomic atoms of Oxygen·16

  • ravichandra

    when cap reactance is decreased with increase in frequency why wee need to use another cap of 0.01micro farad cap(by pass cap) to remove the high freq signal where it can done by 100uf cap

    • David

      Because a real cap has some series resistance as if a resistor was in series with the ideal cap. This series resistance is different for different types of caps. For big electrolytic power supply caps, it’s fairly large so it makes it act like a filter and the cap can’t remove high frequency noise. The small cap in parallel has a much lower series resistance – and capacitance and can therefore filter out the high frequency noise.

    • mouselb

      Ideally a larger capacitance value (in an integrator/low-pass circuit) should be a more effective shunt at increased frequency. It’s because of the non-ideal properties of physically larger capacitors that smaller capacitors (even a few in spread-out sizes) is (are) used (in parallel) to bypass/shunt larger ones. The physically longer leads (and internal construction) of a large capacitor and circuit pathway trace lengths have ‘some’ small non-ideal inductance properties which can leave ‘noise’ at higher frequencies.

  • Jeremiah

    Notes to some extent area good but there is no straight defining terms. My opinion is that you could try to define the key terms.
    Also if there some physics practical questions and answers in PDF form you could share with me.

  • Steven Yam

    when capacitor 150mF Inductor 200H, find XC.

  • Ashokbhai

    Thanks…

  • fish

    how about a graph of Xc against 1/f

  • Nikko

    Hi, I have a question. If a DC source (frequency equal to zero) was used instead of AC, what should be the ideal resistor and capacitor voltages in an series RC circuit? Is their a mathematical way of getting these values?

  • Nanda wahyu

    Calculate the value of capacitor wich will take a current of 25A from a 230V 50Hz supply

  • loganathan p

    really clear and made easy. very useful informations

  • Morlo Moral

    It help a lot to me the formulas given also the example and I want to learn more thank you

  • Xhiwah

    Thanks,
    Can i use the idea of capacitive reactance to set a frequency a radio transimiter should transimite

  • charlston

    how do I calculate the Capacitance if the following is given:
    Voltage=220V
    frequency =100Hz
    Current =320mA