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These interconnections of inductors produce more complex networks whose overall inductance is a combination of the individual inductors. However, there are certain rules for connecting inductors in series or parallel and these are based on the fact that no mutual inductance or magnetic coupling exists between the individual inductors.

Inductors are said to be connected in "Series" when they are daisy chained together in a straight line, end to end. In the Resistors in Series tutorial we saw that the different values of the resistances connected together in series just "add" together and this is also true of inductance. Inductors in series are simply "added together" because the number of coil turns is effectively increased, with the total circuit inductance L T being equal to the sum of all the individual inductances added together.

Inductor in Series Circuit

The current, ( I ) that flows through the first inductor, has no other way to go but pass through the second inductor and the third and so on. Then, series inductors have a Common Current flowing through them, for example:

In the example above, the inductors , and are all connected together in series between points A and B. The sum of the individual voltage drops across each inductor can be found using Kirchoff's Voltage Law (KVL) where, and we know from the previous tutorials on inductance that the self-induced emf across an inductor is given as: V = L di/dt.

So by taking the values of the individual voltage drops across each inductor in our example above, the total inductance for the series combination is given as:

By dividing through the above equation by di/dt we can reduce it to give a final expression for calculating the total inductance of a circuit when connecting inductors together in series and this is given as:

Inductors in Series Equation

Then the total inductance of the series chain can be found by simply adding together the individual inductances of the inductors in series just like adding together resistors in series. However, the above equation only holds true when there is "NO" mutual inductance or magnetic coupling between two or more of the inductors, (they are magnetically isolated from each other).

One important point to remember about inductors in series circuits, the total inductance ( ) of any two or more inductors connected together in series will always be GREATER than the value of the largest inductor in the series chain.

Inductors in Series Example No1

Three inductors of 10mH, 40mH and 50mH are connected together in a series combination with no mutual inductance between them. Calculate the total inductance of the series combination.

Mutually Connected Inductors in Series

When inductors are connected together in series so that the magnetic field of one links with the other, the effect of mutual inductance either increases or decreases the total inductance depending upon the amount of magnetic coupling. The effect of this mutual inductance depends upon the distance apart of the coils and their orientation to each other.

Mutually connected series inductors can be classed as either "Aiding" or "Opposing" the total inductance. If the magnetic flux produced by the current flows through the coils in the same direction then the coils are said to be Cumulatively Coupled. If the current flows through the coils in opposite directions then the coils are said to be Differentially Coupled as shown below.

Cumulatively Coupled Series Inductors

While the current flowing between points A and D through the two cumulatively coupled coils is in the same direction, the equation above for the voltage drops across each of the coils needs to be modified to take into account the interaction between the two coils due to the effect of mutual inductance. The self inductance of each individual coil, and respectively will be the same as before but with the addition of M denoting the mutual inductance.

Then the total emf induced into the cumulatively coupled coils is given as:

By dividing through the above equation by di/dt we can reduce it to give a final expression for calculating the total inductance of a circuit when the inductors are cumulatively connected and this is given as:

If one of the coils is reversed so that the same current flows through each coil but in opposite directions, the mutual inductance, M that exists between the two coils will have a cancelling effect on each coil as shown below.

Differentially Coupled Series Inductors

The emf that is induced into coil 1 by the effect of the mutual inductance of coil two is in opposition to the self-induced emf in coil one as now the same current passes through each coil in opposite directions. To take account of this cancelling effect a minus sign is used with M when the magnetic field of the two coils are differentially connected giving us the final equation for calculating the total inductance of a circuit when the inductors are differentially connected as:

Then the final equation for inductively coupled inductors in series is given as:

Inductors in Series Example No2

Two inductors of 10mH respectively are connected together in a series combination so that their magnetic fields aid each other giving cumulative coupling. Their mutual inductance is given as 5mH. Calculate the total inductance of the series combination.

Inductors in Series Example No3

Two coils connected in series have a self-inductance of 20mH and 60mH respectively. The total inductance of the combination was found to be 100mH. Determine the amount of mutual inductance that exists between the two coils assuming that they are aiding each other.

Inductors in Series Summary

We now know that we can connect together inductors in series to produce a total inductance value, equal to the sum of the individual values, they add together, similar to connecting together resistors in series. However, when connecting together inductors in series they can be influenced by mutual inductance.

Mutually connected series inductors are classed as either "aiding" or "opposing" the total inductance depending whether the coils are cumulatively coupled (in the same direction) or differentially coupled (in opposite direction).

In the next tutorial about Inductors, we will see that the position of the coils when connecting together Inductors in Parallel also affects the total inductance, of the circuit.

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Inductors in Series and Series Inductor Circuits

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These interconnections of inductors produce more complex networks whose overall inductance is a combination of the individual inductors. However, there are certain rules for connecting inductors in series or parallel and these are based on the fact that no mutual inductance or magnetic coupling exists between the individual inductors.

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Series and Parallel Inductors | Inductors | Electronics Textbook

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Read about Series and Parallel Inductors (Inductors) in our free Electronics Textbook

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Learning Objectives

By the end of this section, you will be able to:

Derive expressions for total capacitance in series and in parallel.

Identify series and parallel parts in the combination of connection of capacitors.

Calculate the effective capacitance in series and parallel given individual capacitances.

Several capacitors may be connected together in a variety of applications. Multiple connections of capacitors act like a single equivalent capacitor. The total capacitance of this equivalent single capacitor depends both on the individual capacitors and how they are connected. There are two simple and common types of connections, called series and parallel, for which we can easily calculate the total capacitance. Certain more complicated connections can also be related to combinations of series and parallel.

Capacitance in Series

Figure 1a shows a series connection of three capacitors with a voltage applied. As for any capacitor, the capacitance of the combination is related to charge and voltage by [latex]C=\frac{Q}{V}\\[/latex].

Note in Figure 1 that opposite charges of magnitude Q flow to either side of the originally uncharged combination of capacitors when the voltage V is applied. Conservation of charge requires that equal-magnitude charges be created on the plates of the individual capacitors, since charge is only being separated in these originally neutral devices. The end result is that the combination resembles a single capacitor with an effective plate separation greater than that of the individual capacitors alone. (See Figure 1b.) Larger plate separation means smaller capacitance. It is a general feature of series connections of capacitors that the total capacitance is less than any of the individual capacitances.

We can find an expression for the total capacitance by considering the voltage across the individual capacitors shown in Figure 1. Solving [latex]C=\frac{Q}{V}\\[/latex] for V gives [latex]V=\frac{Q}{C}\\[/latex]. The voltages across the individual capacitors are thus [latex]V_1=\frac{Q}{C_1},V_2=\frac{Q}{C_2},\text{ and }V_3=\frac{Q}{C_3}\\[/latex].

The total voltage is the sum of the individual voltages:

Now, calling the total capacitance C S for series capacitance, consider that

[latex]V=\frac{Q}{C_{\text{S}}}=V_1+V_2+V_3\\[/latex].

[latex]\frac{Q}{C_{\text{S}}}=\frac{Q}{C_{1}}+\frac{Q}{C_{2}}+\frac{Q}{C_{3}}\\[/latex].

Canceling the Qs, we obtain the equation for the total capacitance in series C S to be

[latex]\frac{1}{C_{\text{S}}}=\frac{1}{C_{1}}+\frac{1}{C_{2}}+\frac{1}{C_{3}}+\dots,\\[/latex]

where "..." indicates that the expression is valid for any number of capacitors connected in series. An expression of this form always results in a total capacitance C S that is less than any of the individual capacitances C ₁, C ₂, ..., as Example 1 illustrates.

Total Capacitance in Series, C s

Total capacitance in series:

[latex]\frac{1}{C_{\text{S}}}=\frac{1}{C_{1}}+\frac{1}{C_{2}}+\frac{1}{C_{3}}+\dots\\[/latex]

Example 1. What Is the Series Capacitance?

Find the total capacitance for three capacitors connected in series, given their individual capacitances are 1.000, 5.000, and 8.000 µF.

Strategy

With the given information, the total capacitance can be found using the equation for capacitance in series.

Solution

Entering the given capacitances into the expression for [latex]\frac{1}{C_{\text{S}}}\\[/latex] gives [latex]\frac{1}{C_{\text{S}}}=\frac{1}{C_{1}}+\frac{1}{C_{2}}+\frac{1}{C_{3}}\\[/latex].

[latex]\frac{1}{C_{\text{S}}}=\frac{1}{1.000 \mu\text{F}}+\frac{1}{5.000 \mu\text{F}}+\frac{1}{8.000 \mu\text{F}}=\frac{1.325}{\mu\text{F}}\\[/latex]

Inverting to find C S yields [latex]C_{\text{S}}=\frac{1.325}{\mu\text{F}}=0.755\mu\text{F}\\[/latex].

Discussion

The total series capacitance C s is less than the smallest individual capacitance, as promised. In series connections of capacitors, the sum is less than the parts. In fact, it is less than any individual. Note that it is sometimes possible, and more convenient, to solve an equation like the above by finding the least common denominator, which in this case (showing only whole-number calculations) is 40. Thus,

[latex]\frac{1}{C_{\text{S}}}=\frac{40}{40 \mu\text{F}}+\frac{8}{40 \mu\text{F}}+\frac{5}{40 \mu\text{F}}=\frac{53}{40\mu\text{F}}\\[/latex]

so that

[latex]C_{\text{S}}=\frac{40\mu\text{F}}{53}=0.755\mu\text{F}\\[/latex]

Capacitors in Parallel

Figure 2a shows a parallel connection of three capacitors with a voltage applied. Here the total capacitance is easier to find than in the series case. To find the equivalent total capacitance C p, we first note that the voltage across each capacitor is V, the same as that of the source, since they are connected directly to it through a conductor. (Conductors are equipotentials, and so the voltage across the capacitors is the same as that across the voltage source.) Thus the capacitors have the same charges on them as they would have if connected individually to the voltage source. The total charge Q is the sum of the individual charges: Q = Q ₁ + Q ₂ + Q ₃.

Canceling V from the equation, we obtain the equation for the total capacitance in parallel

Total capacitance in parallel is simply the sum of the individual capacitances. (Again the " ... " indicates the expression is valid for any number of capacitors connected in parallel.) So, for example, if the capacitors in Example 1 were connected in parallel, their capacitance would be

C p = 1.000 µF + 5.000 µF + 8.000 µF = 14.000 µF.

The equivalent capacitor for a parallel connection has an effectively larger plate area and, thus, a larger capacitance, as illustrated in Figure 2b.

Total Capacitance in Parallel, C p

More complicated connections of capacitors can sometimes be combinations of series and parallel. (See Figure 3.) To find the total capacitance of such combinations, we identify series and parallel parts, compute their capacitances, and then find the total.

Example 2. A Mixture of Series and Parallel Capacitance

Find the total capacitance of the combination of capacitors shown in Figure 3. Assume the capacitances in Figure 3 are known to three decimal places ( C ₁ = 1.000 µF, C ₂ = 3.000 µF, and C ₃ = 8.000 µF), and round your answer to three decimal places.

Strategy

To find the total capacitance, we first identify which capacitors are in series and which are in parallel. Capacitors C ₁ and C ₂ are in series. Their combination, labeled C S in the figure, is in parallel with C ₃.

Solution

Since C ₁ and C ₂ are in series, their total capacitance is given by [latex]\frac{1}{C_{\text{S}}}=\frac{1}{C_{1}}+\frac{1}{C_{2}}+\frac{1}{C_{3}}\\[/latex]. Entering their values into the equation gives

[latex]\frac{1}{C_{\text{S}}}=\frac{1}{C_{1}}+\frac{1}{C_{2}}=\frac{1}{1.000\mu\text{F}}+\frac{1}{5.000\mu\text{F}}=\frac{1.200}{\mu\text{F}}\\[/latex].

Inverting gives C S = 0.833 µF.

This equivalent series capacitance is in parallel with the third capacitor; thus, the total is the sum

[latex]\begin{array}{lll}C_{\text{tot}}&=&C_{\text{S}}+C_{\text{S}}\\\text{ }&=&0.833\mu\text{F}+8.000\mu\text{F}\\\text{ }&=&8.833\mu\text{F}\end{array}\\[/latex]

Discussion

This technique of analyzing the combinations of capacitors piece by piece until a total is obtained can be applied to larger combinations of capacitors.

Section Summary

Total capacitance in series [latex]\frac{1}{C_{\text{S}}}=\frac{1}{C_{1}}+\frac{1}{C_{2}}+\frac{1}{C_{3}}+\dots\\[/latex]

If a circuit contains a combination of capacitors in series and parallel, identify series and parallel parts, compute their capacitances, and then find the total.

Conceptual Questions

If you wish to store a large amount of energy in a capacitor bank, would you connect capacitors in series or parallel? Explain.

Problems & Exercises

1. Find the total capacitance of the combination of capacitors in Figure 4.

2. Suppose you want a capacitor bank with a total capacitance of 0.750 F and you possess numerous 1.50 mF capacitors. What is the smallest number you could hook together to achieve your goal, and how would you connect them?

3. What total capacitances can you make by connecting a 5.00 µF and an 8.00 µF capacitor together?

4. Find the total capacitance of the combination of capacitors shown in Figure 5.

5. Find the total capacitance of the combination of capacitors shown in Figure 6.

6. Unreasonable Results.(a) An 8.00 µF capacitor is connected in parallel to another capacitor, producing a total capacitance of 5.00 µF. What is the capacitance of the second capacitor? (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?

Selected Solutions to Problems & Exercises

1. 0.293 μF

3. 3.08 µF in series combination, 13.0 µF in parallel combination

4. 2.79 µF

6. (a) -3.00 µF; (b) You cannot have a negative value of capacitance; (c) The assumption that the capacitors were hooked up in parallel, rather than in series, was incorrect. A parallel connection always produces a greater capacitance, while here a smaller capacitance was assumed. This could happen only if the capacitors are connected in series.

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By the end of this section, you will be able to: Derive expressions for total capacitance in series and in parallel. Identify series and parallel parts in the combination of connection of capacitors. Calculate the effective capacitance in series and parallel given individual capacitances. Several capacitors may be connected together in a variety of applications.

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ENERGY VS POWER

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Comparison chart

Energy

Power

Definition

Energy is the capacity to do work. Energy is power integrated over time.

Power is the rate at which work is done, or energy is transmitted.

Unit

joules = watt-seconds or joule = Newton-meter

watt = joules/second

Common symbol(s)

W

P

Example

I left a 60W light bulb on for 30 days, which raised my electric bill by 43.2 kWh (kilowatt-hours).

My car's battery can provide 500 amps at 12 volts, which equals 6kW of power.

Different Forms of Power and Energy

There are different forms of energy. These include kinetic, potential, thermal, gravitational, electromagnetic, sound, light and elastic. The form of energy is dependent on the frame of reference, and can be transformed into other forms. For instance potential energy is dependent on the position of the object, whereas kinetic energy is the energy required to accelerate an object to a particular speed, and so on.

Different forms of power could be electric power, which is the rate at which electrical energy is transferred by a circuit, human power, and optical power.

Transformation of Energy

Various devices can be used to convert one form of energy into another. For instance, a battery converts chemical to electric energy, chemical explosion converts chemical energy in to kinetic and thermal energy and so on. Power cannot be converted or transformed.

Measuring Energy vs. Power

Although it is not possible to directly measure energy, the work done can be defined and measured. The methods involves using a calorimeter, which measures the heat absorbed or released in chemical reactions or physical changes, thermometer, which measures temperature or bolometer that is employed to measure the intensity of radiation. Energy generated can be stored whereas power cannot.

Since power is energy per unit of time, in theory it can be calculated after measuring the energy used per second. When calculating the real power consumption of an electrical device, it is essential to measure the voltage applied and the current consumed, taking into account the power that is dissipated in the circuit.

Difference in Units

Energy is usually measured in Joules, the other units include ergs and calories. Power is measured in Watts, which is Joules per second, or ergs per second. Machine are usually described by its power rating, the higher the number the more powerful the machine.

Equation

Power is an energy per unit of time. As a rate of change of work done or the energy of a subsystem, power is:

where P is power, W is work and t is time.

References

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What's the difference between Energy and Power? In physics, energy is defined as the amount of work that can be performed by force, whereas power is defined as the rate at which work is performed. Different Forms of Power and Energy There are different forms of energy. These include kinetic, potential,...