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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)
  •    Ï€ (pi) = 3.142 (decimal) or as 22÷7 (fraction)
  •    Æ’ = 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: Æ’ = 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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  • Kiran

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  • Zin Aung

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  • Seth

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  • Alex P

    For the statement above that states “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”, did they mean the voltage across resistor R2?

    • Wayne Storr

      No. As explained in the tutorial, if fixed resistor R2 given in the previous circuit is replaced with a capacitor the reactance of the capacitor can be controlled by the frequency of the supply. Therefore the voltage dropped across R1 will also change. Then the tutorial is correct as given.

  • dilan

    Explain the effect on current when increasing the frequency of the supply voltage in a series RC

    • Wayne Storr

      XC reduces to approximately zero as frequency increases, R remains constant, then current I simply becomes V/R and not V/Z

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    What is the capacitive reactance of a 0.001-F capacitor at 60 Hz?

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  • stuart sjalund

    From what i see concerning capacitance is its non linearity of discharge. Capacitance follows the time constant of RC and 5 tc’s are required to obtain full charge of, 99.999%+, never 1.000. Discharge can never be to 0.000. From what I learned decades ago, is it safe to say that discharging the power stored in a capacitor is never linear? What is your point of view?
    Energy storage is entirely in the atom and depends on the cloud of atoms relative to a terminal, usually +. The atom cloud is around the + terminal as electrons are negative. Where am I wrong?

    • Wayne Storr

      Please read our tutorial about RC Discharging Circuit. A capacitor stores electric charge on its plates. One plate of a capacitor having a charge of +q coulombs, and the other plate has an equal but opposite charge of -q coulombs. The amount of electric charge is defined as: Q = CV

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    please check all notes generally on this learning website to make sure every information is put right! this is a very good tutorial site made easy to for students to learn, but be mindful of common mistakes not to mislead students. I love the setup of this website beautifully put together for learners, and the schematic diagrams are super lovely and plain to understand.
    please keep up the good work and rectify all mistakes in general.

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    Ibrahim
    ( London).

    • Wayne Storr

      Thank you for this. Being human we relay on experts like yourself to identify, in a polite and professional manner, any typing errors in our tutorials.