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capacitors in series

Capacitors in Series

Capacitors are said to be connected together “in series” when they are effectively "daisy chained" together in a single line. The charging current ( iC ) flowing through the capacitors is THE SAME for all capacitors as it only has one path to follow.

Then, Capacitors in Series all have the same current flowing through them as iT = i1 = i2 = i3 etc. Therefore each capacitor will store the same amount of electrical charge, Q on its plates regardless of its capacitance. This is because the charge stored by a plate of any one capacitor must have come from the plate of its adjacent capacitor. Therefore, capacitors connected together in series must have the same charge.

QT = Q1 = Q2 = Q3 ….etc

Consider the following circuit in which the three capacitors, C1, C2 and C3  are all connected together in a series branch across a supply voltage between points A and B.

Capacitors in a Series Connection

capacitors in series circuit

In the previous parallel circuit we saw that the total capacitance, CT of the circuit was equal to the sum of all the individual capacitors added together. In a series connected circuit however, the total or equivalent capacitance CT is calculated differently.

In the series circuit above the right hand plate of the first capacitor, C1 is connected to the left hand plate of the second capacitor, C2 whose right hand plate is connected to the left hand plate of the third capacitor, C3. Then this series connection means that in a DC connected circuit, capacitor C2 is effectively isolated from the circuit.

The result of this is that the effective plate area has decreased to the smallest individual capacitance connected in the series chain. Therefore the voltage drop across each capacitor will be different depending upon the values of the individual capacitance’s. Then by applying Kirchoff’s Voltage Law, ( KVL ) to the above circuit, we get:

current through capacitors in series

Since Q = CV and rearranging for V = Q/C, substituting Q/C for each capacitor voltage VC in the above KVL equation will give us:

circuit current

dividing each term through by Q gives

Series Capacitors Equation

capacitors in series equation

When adding together Capacitors in Series, the reciprocal ( 1/C ) of the individual capacitors are all added together ( just like resistors in parallel ) instead of the capacitance’s themselves. Then the total value for capacitors in series equals the reciprocal of the sum of the reciprocals of the individual capacitances.

Capacitors in Series Example No1

Taking the three capacitor values from the above example, we can calculate the total capacitance, CT for the three capacitors in series as:

total capacitance

One important point to remember about capacitors that are connected together in a series configuration, is that the total circuit capacitance ( CT ) of any number of capacitors connected together in series will always be LESS than the value of the smallest capacitor in the series and in our example above CT = 0.055uF with the value of the smallest capacitor in the series chain is only 0.1uF.

This reciprocal method of calculation can be used for calculating any number of individual capacitors connected together in a single series network. If however, there are only two capacitors in series, then a much simpler and quicker formula can be used and is given as:

total capacitance in series

If the two series connected capacitors are equal and of the same value, that is: C1 = C2, we can simplify the above equation further as follows to find the total capacitance of the series combination.

equal value capacitors

Then we can see that if and only if the two series connected capacitors are the same and equal, then the total capacitance, CT will be exactly equal to one half of the capacitance value, that is: C/2.

With series connected resistors, the sum of all the voltage drops across the series circuit will be equal to the applied voltage VS ( Kirchoff’s Voltage Law ) and this is also true about capacitors in series.

With series connected capacitors, the capacitive reactance of the capacitor acts as an impedance due to the frequency of the supply. This capacitive reactance produces a voltage drop across each capacitor, therefore the series connected capacitors act as a capacitive voltage divider network.

The result is that the voltage divider formula applied to resistors can also be used to find the individual voltages for two capacitors in series. Then:

voltage across series capacitors

Where: CX is the capacitance of the capacitor in question, VS is the supply voltage across the series chain and VCX is the voltage drop across the target capacitor.

Capacitors in Series Example No2

Find the overall capacitance and the individual rms voltage drops across the following sets of two capacitors in series when connected to a 12V a.c. supply.

  • a)  two capacitors each with a capacitance of 47nF
  • b)  one capacitor of 470nF connected in series to a capacitor of 1uF

a) Total Capacitance,

total series capacitance

Voltage drop across the two identical 47nF capacitors,

capacitor voltage drop

b) Total Capacitance,

series capacitance

Voltage drop across the two non-identical Capacitors: C1 = 470nF and C2 = 1uF.

Individual Series Voltages

Since kirchoff’s voltage law applies to this and every series connected circuit, the total sum of the individual voltage drops will be equal in value to the supply voltage, VS. Then 8.16 + 3.84 = 12V.

Note also that if the capacitor values are the same, 47nF in our example, the supply voltage is divided equally across each capacitor as shown, because each capacitor shares an equal amount of charge ( Q = C x V = 0.564uC ) and therefore has half ( or percentage fraction for more than two capacitors ) of the applied voltage, VS.

When the capacitor values are different, the larger value capacitor will charge itself to a lower voltage and the smaller value capacitor to a higher voltage, and in our example above this was 3.84 and 8.16 volts respectively, to maintain the same amount of charge on the plates of both capacitors as shown.

charge on capacitors in series

Note that the ratios of the voltage drops across the two capacitors connected in series will always remain the same regardless of the supply frequency as their reactance, XC will remain proportionally the same.

Then the two voltage drops of 8.16 volts and 3.84 volts above in our simple example will remain the same even if the supply frequency is increased from 100Hz to 100kHz.

Although the voltage drops across each capacitor will be different for different values of capacitance, the coulomb charge across the plates will be equal because the same amount of current flow exists throughout a series circuit as all the capacitors are being supplied with the same number or quantity of electrons.

In other words, if the charge across each capacitors plates is the same, as Q is constant, then as its capacitance decreases the voltage drop across the capacitors plates increases, because the charge is large with respect to the capacitance. Likewise, a larger capacitance will result in a smaller voltage drop across its plates because the charge is small with respect to the capacitance.

Capacitors in Series Summary

Then to summarise, the total or equivalent capacitance, CT of a circuit containing Capacitors in Series is the reciprocal of the sum of the reciprocals of all of the individual capacitance’s added together.

Also for capacitors connected in series, all the series connected capacitors will have the same charging current flowing through them as iT = i1 = i2 = i3 etc. Two or more capacitors in series will always have equal amounts of coulomb charge across their plates.

As the charge, ( Q ) is equal and constant, the voltage drop across the capacitor is determined by the value of the capacitor only as V = Q ÷ C. A small capacitance value will result in a larger voltage while a large value of capacitance will result in a smaller voltage drop.

49 Comments

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  • M
    Mayen M Deng

    what easy ways can i manage to understand capacitors especially the calculation section?

  • A
    Atiq

    How Q = CV? Need proof, please provide.

  • D
    Darachhanthou hem

    Hello!

  • D
    Darachhanthou hem

    How can I understand about Capacitor voltage drop?

  • D
    Den

    Or to put it another way, how can Qt=Q1=Q2=Q3? I see why Q1=Q2=Q3, because I is the same for all, and IT=Q. But why isn’t Qt=Q1+Q2+Q3??

    • Wayne Storr

      For capacitors connected in series, the quantity of charge is the same across each capacitor, that is Q1=Q2=Q3=Q.. etc. but the voltage across each capacitor is different. QT was a convenient notation to use but could also have been: QA, or QB, or QC, etc. The point is to show its the SAME charge and not individual charges as explained in the tutorial, so if its easier for you to understand then you can have: V1=Q1/C1 or V1=(Q1=Q2=Q3)/C1 or V1=QA/C1, etc, etc, they are all the same.

      Your idea of: Qt=Q1+Q2+Q3 is correct and valid for parallel connected capacitors as the voltage across each capacitor is the same.

      • D
        Den

        I just hadn’t thought about it in awhile. Makes sense now. Thanks. Do you have a tutorial, and can you point me to some place that rigorously explains how parallel resistors help balance series caps? I understand the cause of imbalance. Differing cap tol and changing leakage current. A lot of sites show how to approximate the resistor values, but none I can find, show mathematically how to arrive at those approximations, or how to derive the actual voltages and currents after the resistors are put in place. Thank you

  • D
    Den

    Where do you get the idea that V1=QT/C1 ?

  • S
    Shruthi

    In series circuit.How the circuit knows that how much voltage to be supplied to depending on capacitance value physically

  • S
    Shruthi

    If c2 has less capacitance than c1 or c3. then charge deposited on c2 plates may be less than c1 or c3. if so,I2 may differ from I1 and I3. Then how can we say that I2=I1=I3. When they are in series.
    Very lucky to have this website.
    Thank you sir

    • Wayne Storr

      For series connected capacitors, they have the same charge as Q = CV. Therefore the voltage drop across each capacitor is in proportion to its capacitance value and may not be the same voltage for each capacitor. Then the series current is Ic = Vc/Xc which will be the same for all capacitors.

  • L
    Luan

    How do you work out the charge across a series circuit?

  • D
    David

    Sorry to be dense but if “this series connection means that in a DC connected circuit, capacitor C2 is effectively isolated from the circuit” how does C2 effect the voltage or current in the circuit at all? I can’t see how current or potential could even reach C2. Excellent site BTW. Thanks for your help.

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