Understanding Capacitors in Series

Previously, we discuss capacitors in parallel connection which we said is when both of its terminals are connected to each terminal of another capacitor. However, capacitors in series connection are when it is connected one after another and they have some current flowing through them. Today you’ll have an understanding of the definition, diagram, formula, working, and examples on how to calculate capacitors in series connection.

Capacitors in series connection

Capacitors are popularly known to work with other electrical components in a circuit. They are connected to other elements in either parallel or series. But we are looking at series capacitors here. Several capacitors are connected in series to make a functional block. Capacitors are said to be in series when they are daisy-chained together in a single line. For this system, the charging current (i_c) flowing through the capacitor is equal for all capacitors because it flows in one path.

Also, capacitors in series have the same current flowing through them, for instance, i_T = i₁ = i₂ = i_3, etc. Therefore, all the capacitors will store the same amount of electrical charge, Q on their plates regardless of its capacitance. This is due to the charge stored by a plate of each capacitor comes from the plate of its adjacent capacitor. So, capacitors connected in series must have the same charge.

Q_T = Q₁ = Q₂ = Q₃ …etc.

Take a look at the below circuit with three capacitors C₁, C₂, and C₃ are connected in series across a supply voltage between points A and B.

In the above series-connected circuit, the right-hand plate of the first capacitor, C₁ is connected to the left-hand plate of the second capacitor, C₂ which is also connected to the left-hand plate of the third capacitor, C₃. This means, in a DC-connected circuit, capacitor C₂ is effectively isolated from the circuit. This results in the decrease of the effective plate area to the smallest individual capacitance connected in the series chain. Thus, the voltage drop across each capacitor will be different depending upon the values of the individual capacitances.

Applying Kirchhoff’s Voltage Law, (KVL) to the above circuit, we get:

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

dividing each term through by Q gives Series Capacitors Equation

in a series capacitor, the reciprocal (1/C) of the individual capacitors are all added together just like resistors in a parallel circuit instead of the capacitance. The total value for capacitors in series equals the reciprocal of the sum of the reciprocals of the individual capacitances. Now let take a look at some examples of capacitors in series connections.

Example 1

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

You should know in connection like this, the total circuit capacitance (C_T) of any number of capacitors connected in series will always be less than the value of the smallest capacitor in the series. For instance, in the above example, C_T = 0.055μF with the value of the smallest capacitor in the series chain is 0.1μF which is greater than the C_T.

The method used is a reciprocal method, which can be used for calculating any number of individual capacitors connected in a single series network. However, if there are only two capacitors in series, then a much simpler and quicker formula can be employed:

Also, we can further simplify the above equation if the two series-connected capacitors are equal and of the same value, that is, C₁ = C₂. This helps us to get the total capacitance of the series combination.

Now we can say, if two series-connected capacitors are the same and equal, then the total capacitance, CT will be equal to one-half of the capacitance value, that is, C/2. With series-connected capacitors, the capacitive reactance of the capacitor acts as an impedance because of the frequency of the supply. This capacitive reactance produces a voltage drop across each capacitor, so the series-connected capacitor will act as a capacitive voltage divider network.

Example 2

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

• Two capacitors each with a capacitance of 47nF
• One capacitor of 470nF connected in series to a capacitor of 1μF

Total Equal Capacitance:

Voltage drops across the two identical 47nF capacitors,

Total Unequal Capacitance:

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

Watch the video below to learn more about capacitors in series connections:

Conclusion

The total or equivalent capacitance, CT of a circuit of a series capacitor is the reciprocal of the sum of the reciprocals of all of the individual capacitances added together. Also, you should know that all the series-connected capacitors contain the same charging current flowing through them. that is, two or more capacitors in series will always have equal amounts of coulomb charge across their plates.

That is all for this article, where the formula, calculation, diagram, and working of capacitors in series connections have been discussed. I hope you get a lot from the reading, if so, kindly share with other students. Thanks for reading, see you next time!