Combination Of Capacitors - Parallel And Series

Combination Of Capacitors - Parallel And Series

Edited By Vishal kumar | Updated on Jul 02, 2025 06:02 PM IST

Capacitors are essential components in electronic circuits, playing a crucial role in storing and managing electrical energy. When capacitors are combined in different configurations—either in series or parallel—their collective behaviour changes, affecting the overall capacitance and performance of the circuit. Understanding these combinations is vital for designing efficient electronic systems, from everyday gadgets like smartphones and remote controls to more complex devices like computers and medical equipment. For instance, in camera flashes, capacitors are arranged to quickly release stored energy, providing the burst of light needed to capture a photo. This article explores how capacitors function in parallel and series arrangements, and how these configurations are applied in real-world scenarios.

This Story also Contains
  1. Combination of Capacitors
  2. Solved Examples Based on Combination of Capacitors - Parallel and Series
  3. Summary
Combination Of Capacitors - Parallel And Series
Combination Of Capacitors - Parallel And Series

Combination of Capacitors

A capacitor is a fundamental component in electronics, widely used for storing electrical energy. When multiple capacitors are combined in a circuit, they can be arranged either in series or parallel configurations. These combinations have a significant impact on the overall capacitance and behaviour of the circuit. In a series combination, the capacitors share the same charge, resulting in a decreased total capacitance, while in a parallel combination, the capacitances add up, allowing for more energy storage.

Capacitors can be combined in two ways.

1. Series

2. Parallel

Series Combination

If capacitors are connected in such a way that we can proceed from one point to another by only one path passing through all capacitors then all these capacitors are said to be in series.

Here three capacitors are connected in series and are connected across a battery of potential difference ‘V’.

Charge: 'q' given by battery deposits at the first plate of the first capacitor. Due to induction, it attracts '–q' on the opposite plate. The pairing +ve q charges are repelled to the first plate of the Second capacitor which in turn induces -q on the opposite plate. The same action is repeated to all the capacitors and in this way, all capacitors get q charge. As a result; the charge given by battery q, every capacitor gets charge q.

Potential difference: V is the sum of potentials across all capacitors. Therefore

$\begin{gathered}V=v_1+v_2+v_3 \\ v_1=\frac{q_1}{c_1}, v_2=\frac{q_2}{c_2}, v_3=\frac{q_3}{c_3}\end{gathered}$

Equivalence equation: The equivalent capacitance for the combination of capacitance in series can be calculated as $C_e=\frac{q}{V}$.

Or,

$\begin{aligned} 1 / \mathrm{C}_{\mathrm{e}} & =\mathrm{V} / \mathrm{q} \\ & =\left(\mathrm{v}_1+\mathrm{v}_2+\mathrm{v}_3\right) / \mathrm{q} \\ & =\mathrm{v}_1 / \mathrm{q}+\mathrm{v}_2 / \mathrm{q}+\mathrm{v}_3 / \mathrm{q} \\ 1 / \mathrm{C}_{\mathrm{e}} & =1 / \mathrm{C}_1+1 / \mathrm{C}_2+1 / \mathrm{C}_3\end{aligned}$

For 2 capacitor system $C=\frac{c_1 c_2}{c_1+c_2}$, and $\quad v_1=\frac{c_2}{c_1+c_2} \cdot V$

If n capacitors of capacitance 'c' are joined in series then equivalent capacitance $C_e=\frac{c}{n}$.

Parallel Combination

If capacitors are connected in such a way that there are many paths to go from one point to another. All these paths are parallel and the capacitance of each path is said to be connected in parallel.

Here three capacitors are connected in parallel and are connected across a battery of potential difference ‘V’.

The potential difference across each capacitor is equal and it is the same as P.D. across Battery. The charge given by the source is divided and each capacitor gets some charge. The total charge.

Therefore, each capacitor has a charge

$\mathrm{q}_1=\mathrm{C}_1 \mathrm{~V}_1, \mathrm{q}_2=\mathrm{C}_2 \mathrm{~V}_2, \mathrm{q}_3=\mathrm{C}_3 \mathrm{~V}_3$

Equivalent Capacitance: We know that when divided by v on both sides, $\frac{q}{v}=\frac{\mathrm{q}_1}{v}+\frac{\mathrm{q}_2}{v}+\frac{\mathrm{q}_3}{v}$.

Therefore the equivalence capacitance will be:

C = c1+c2+c3

The equivalent capacitance in parallel increases, and it is more than the largest in parallel. In parallel combination, V is the same, therefore $v=\frac{q_1}{c_1}=\frac{q_2}{c_2}=\frac{q_3}{c_3}$. In parallel combination $q \propto c$. The larger the capacitance larger is charge.

Charge distribution : $q_1=c_1 v, q_2=c_2 v, q_3=c_3 v$

In 2 capacitor systems charge on one capacitor $q=\frac{c_1}{c_1+c_2+\ldots} \cdot q$.

Capacitors in parallel give C = nc.

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Solved Examples Based on Combination of Capacitors - Parallel and Series

Example 1: Figure shows a network of capacitors where the numbers indicate capacitances in micro Farad. The value of capacitance C if the equivalent capacitance between points A and B is to be 1 µF is :

1) $\frac{31}{23} \mu F$
2) $\frac{32}{23} \mu F$
3) $\frac{33}{23} \mu F$
4) $\frac{34}{23} \mu F$

Solution:

Series Grouping

$\frac{1}{C_{e q}}=\frac{1}{C_1}+\frac{1}{C_2}+\cdots$

Parallel Grouping

$\begin{aligned} & C_{e q}=C_1+C_2+\cdots \\ & C_{A B}=1 \mu f=\frac{C \times \frac{32}{9}}{C+\frac{32}{9}}=\frac{\frac{32 C}{9}}{\frac{32}{9}+C} \\ & \frac{32 C}{9}=\frac{32}{9}+C \\ & \Rightarrow C=\frac{32}{23} \mu F\end{aligned}$

Hence, the answer is the option (2).

Example 2: A capacitance of 2 µF is required in an electrical circuit across a potential difference of 1.0 kV. A large number of 1 µF capacitors are available which can withstand a potential difference of not more than 300 V. The minimum number of capacitors required to achieve this is :

1) 2

2) 16

3) 24

4) 32

Solution:

Series Grouping

$\frac{1}{C_{\mathrm{eq}}}=\frac{1}{C_1}+\frac{1}{C_2}+\cdots$

wherein

So

we need 4 capacitors to withstand 1000V

4 capacitance in series gives $C_{e q}=\frac{1}{4}$

to get $2 \mu F$ we have to connect such 8 branches in parallel

$\therefore$ Total number of capacitance $=8 \times 4=32$

Hence, the answer is the option (4).

Example 3: A $1 \mu F$ capacitor and a $2 \mu F$ capacitor are connected in parallel across a 1200-volt line. The charged capacitors are then disconnected from the line and from each other. These two capacitors are now connected to each other in parallel with terminals of unlike signs together. The charges on the capacitors will now be

1) $1800 \mu \mathrm{C}$
2) $400 \mu C$. and. $800 \mu C$
3) $800 \mu$ C..and. $400 \mu C$.
4) $800 \mu$ C..and. $800 \mu C$

Solution:

Series Grouping

$\frac{1}{C_{e q}}=\frac{1}{C_1}+\frac{1}{C_2}+\cdots$

wherein

Initially charge on capacitors can be calculated as follows
$Q_1=1 \times 1200=1200 \mu^{\prime} \mathrm{C}$ and $Q_2=2 \times 12002400 \mu_C$
Finally when the battery is disconnected and unlike plates are connected together then common potential

$V^{\prime}=\frac{Q_2-Q_1}{C_1+C_2}=\frac{2400-1200}{1+2}=400 \mathrm{~V}$

Hence, the New charge is $1^* 400=400 \mu \mathrm{C}$
And the New charge on is $C_2$ is $2^* 400=800 \mu \mathrm{C}$

Hence, the answer is the option (2).

Example 4: The figure shows a charge (q) versus voltage (V) graph for a series and parallel combination of two given capacitors. The capacitances are:

1) $40 \mu F$ and $10 \mu F$
2) $60 \mu \mathrm{F}$ and $40 \mu \mathrm{F}$
3) $50 \mu \mathrm{F}$ and $30 \mu \mathrm{F}$
4) $20 \mu \mathrm{F}$ and $30 \mu \mathrm{F}$

Solution:

For parallel

$
C_{e q}=C_1+C_2
$

So, $q_1=10\left(C_1+C_2\right)=500 \mu \mathrm{C}$
$
=>C_1+C_2=50 \mu C \text {. }
$

For series

$
\begin{aligned}
& C_{e q}=\frac{C_1 C_2}{C_1+C_2} \\
& q_2=10 C_{e q}=80
\end{aligned}
$

So,
$
\begin{aligned}
& 10 \times \frac{C_1 C_2}{C_1+C_2}=80 \\
& 10 \times \frac{C_1 C_2}{50}=80 \\
& C_1 C_2=400 \\
& C_2=10 \mu F \\
& C_1=40 \mu F
\end{aligned}
$

Hence, the answer is the option (1).

Example 5: Two capacitors C1 and C2 are charged to 120 V and 200 V respectively. It is found that by connecting them together the potential of each one can be made zero. Then :

1)9C1 = 4C2

2)5C1 = 3C2

3)3C1 = 5C2

4)3C1 + 5C2 = 0

Solution:

Parallel Grouping

$C_{\text {eq }}=C_1+C_2+\cdots$

wherein

$\begin{aligned} & 120 C_1=200 C_2 \\ & 6 C_1=10 C_2 \\ & 3 C_1=5 C_2\end{aligned}$

Hence, the answer is the option (3).

Summary

Capacitors are either connected in parallel or in series, increasing the total capacitance by means of parallel connection, since it simply involves the addition of two capacitors by their capacitances. Meanwhile, a series total capacitance actually undergoes a decrease, as the reciprocal of total capacitance is just the sum of reciprocals of individual capacitances. These combinations make the circuit extremely flexible in design, allowing the overall capacitance and voltage ratings for a specific application to be readjusted.

Frequently Asked Questions (FAQs)

1. What is the fundamental difference between connecting capacitors in series and parallel?
In series connections, capacitors are connected end-to-end, sharing the same current but dividing the total voltage. In parallel connections, capacitors share the same voltage across them, but the total charge is divided among them. This results in different effects on the overall capacitance of the circuit.
2. What happens to the electric field between the plates of capacitors when connected in series?
When capacitors are connected in series, the electric field between the plates of each capacitor increases compared to when they are isolated. This is because the same charge is distributed across all capacitors, but each capacitor only gets a fraction of the total voltage, resulting in a stronger field for the same charge.
3. Why does adding capacitors in parallel increase the total capacitance?
Adding capacitors in parallel increases the total capacitance because it effectively increases the total plate area available for storing charge. More plate area means more charge can be stored at the same voltage, resulting in a higher capacitance.
4. What is the significance of the fact that charge is conserved in capacitor combinations?
The conservation of charge in capacitor combinations is crucial for understanding how charge is distributed and how voltage divides among capacitors. In series, it ensures that each capacitor has the same charge. In parallel, it means the total charge is the sum of charges on individual capacitors. This principle is fundamental for deriving the equations for equivalent capacitance.
5. Why can't you simply add the capacitances of series-connected capacitors like you do with parallel-connected ones?
You can't simply add the capacitances of series-connected capacitors because in series, you're effectively increasing the distance between the plates, which decreases capacitance. The reciprocal sum reflects this inverse relationship between distance and capacitance. In contrast, parallel connection increases plate area, which directly increases capacitance.
6. How does the equivalent capacitance change when capacitors are connected in parallel?
When capacitors are connected in parallel, the equivalent capacitance increases. It is the sum of individual capacitances: Ceq = C1 + C2 + C3 + ... This is because parallel connection increases the total plate area, allowing for more charge storage at the same voltage.
7. What happens to the equivalent capacitance when capacitors are connected in series?
When capacitors are connected in series, the equivalent capacitance decreases. It is calculated as: 1/Ceq = 1/C1 + 1/C2 + 1/C3 + ... This is because series connection effectively increases the distance between the plates, reducing the overall capacitance.
8. Why does the formula for series capacitors use reciprocals?
The formula for series capacitors uses reciprocals because capacitance is inversely proportional to the distance between plates. When capacitors are in series, it's like adding the distances between their plates, which translates to adding the reciprocals of their individual capacitances.
9. Can you explain why the voltage across each capacitor in a parallel combination is the same?
In a parallel combination, all capacitors are connected to the same two points in the circuit. Since voltage is the potential difference between two points, each capacitor experiences the same voltage. This is analogous to how the pressure is the same for all pipes connected to the same water tank.
10. How is charge distributed among capacitors in a series combination?
In a series combination, the charge on each capacitor is the same. This is because the charge that leaves one plate of a capacitor must be equal to the charge that enters the plate of the next capacitor in the series. The total charge in the circuit is conserved and distributed equally among all capacitors.
11. How does the energy stored in a system of capacitors change when you switch from series to parallel connection?
When switching from series to parallel connection, the energy stored in the system typically increases. This is because parallel connection increases the total capacitance, and the energy stored is proportional to the capacitance (E = ½CV²). However, the exact change depends on whether the voltage or charge is kept constant during the switch.
12. How does the concept of equivalent capacitance simplify circuit analysis?
Equivalent capacitance simplifies circuit analysis by allowing complex combinations of capacitors to be replaced by a single equivalent capacitor. This makes calculations easier, especially in more complicated circuits, as it reduces the number of components to consider while maintaining the same electrical behavior.
13. Can you have a combination of both series and parallel capacitors in the same circuit? How would you analyze such a circuit?
Yes, you can have both series and parallel combinations in the same circuit. To analyze such a circuit, you would use a step-by-step approach: first, simplify parallel sections into their equivalent capacitances, then simplify series sections, and repeat until you have a single equivalent capacitance for the entire circuit.
14. How does the presence of a dielectric material affect capacitor combinations?
The presence of a dielectric material increases the capacitance of individual capacitors by a factor equal to its dielectric constant. In combinations, this affects the equivalent capacitance proportionally. For parallel combinations, the increase is straightforward. For series combinations, the effect is more complex due to the reciprocal nature of the calculation.
15. How does the concept of equivalent capacitance relate to Kirchhoff's laws?
Equivalent capacitance is consistent with Kirchhoff's laws. For parallel capacitors, it satisfies Kirchhoff's voltage law (KVL) as all capacitors have the same voltage. For series capacitors, it satisfies Kirchhoff's current law (KCL) as the same current flows through all capacitors. The equivalent capacitance preserves these relationships while simplifying the circuit.
16. What happens to the time constant of an RC circuit when you replace a single capacitor with a combination of capacitors?
The time constant (τ = RC) changes based on the equivalent capacitance of the combination. If the combination results in a larger equivalent capacitance (as in parallel connection), the time constant increases, slowing the charging/discharging process. If the equivalent capacitance decreases (as in series connection), the time constant decreases, speeding up the process.
17. How does the concept of capacitor combinations apply to real-world applications like camera flashes?
In camera flashes, capacitor combinations are used to achieve the desired voltage and energy storage. Parallel combinations might be used to increase the total energy storage capacity, while series combinations could be employed to handle higher voltages. The specific combination allows designers to optimize the flash's power, recharge time, and safety.
18. Can you explain why the equivalent capacitance of two identical capacitors in series is half of one capacitor?
When two identical capacitors are connected in series, the equivalent capacitance is half because the effective distance between the plates doubles. Since capacitance is inversely proportional to the distance between plates, doubling the distance halves the capacitance. Mathematically, 1/Ceq = 1/C + 1/C = 2/C, so Ceq = C/2.
19. How does the distribution of electric potential differ between series and parallel capacitor combinations?
In a series combination, the electric potential (voltage) is divided among the capacitors, with each capacitor having a fraction of the total voltage. In a parallel combination, each capacitor experiences the full electric potential of the circuit. This difference in potential distribution is key to understanding how charge and energy are stored in these configurations.
20. What role does the concept of capacitor combinations play in the design of power supplies and voltage regulators?
In power supplies and voltage regulators, capacitor combinations are crucial for filtering, smoothing, and stabilizing voltages. Parallel combinations are often used to increase capacitance for better voltage smoothing and energy storage. Series combinations might be used to handle higher voltages or create voltage dividers. The ability to tailor capacitance and voltage handling is essential for efficient and stable power supply design.
21. How does the concept of capacitor combinations apply to the human body's electrical properties?
The human body can be modeled as a complex combination of capacitors and resistors. Different tissues and cell membranes act as capacitors in various series and parallel configurations. This understanding is crucial in biomedical applications, such as designing medical imaging techniques (like MRI) or understanding the effects of electric currents on the body.
22. What happens to the charge stored in a system of capacitors when you change from a parallel to a series configuration?
When changing from parallel to series configuration, if the voltage is kept constant, the total charge stored in the system decreases. This is because the equivalent capacitance decreases in series, and Q = CV. If instead the charge is kept constant, the voltage across the system would increase, as V = Q/C, and C has decreased.
23. How do capacitor combinations affect the breakdown voltage of a capacitor system?
In a series combination, the breakdown voltage of the system increases because the voltage is divided among the capacitors. This allows the system to handle higher voltages than a single capacitor. In parallel, the breakdown voltage remains the same as for a single capacitor, but the total energy that can be stored before breakdown increases.
24. Can you explain how capacitor combinations are used in touch screen technology?
In touch screen technology, a grid of capacitors is formed across the screen. When a finger touches the screen, it changes the capacitance of nearby grid points. This change is detected as a touch event. The screen effectively uses a complex combination of capacitors in both series and parallel to create a sensitive and accurate touch-detection system.
25. How does temperature affect the behavior of capacitor combinations?
Temperature can affect the capacitance of individual capacitors, typically causing it to change slightly. In combinations, these effects can compound. In parallel combinations, the changes add directly. In series combinations, the effect is more complex due to the reciprocal nature of the calculation. Temperature effects are important to consider in precision applications or extreme environments.
26. What is the significance of the fact that the equivalent capacitance of parallel capacitors is always greater than any individual capacitance?
This fact is significant because it means that parallel combinations always increase the total charge storage capacity of the system at a given voltage. It's a fundamental principle that allows engineers to design circuits with higher capacitance without needing to manufacture extremely large individual capacitors, which can be impractical or impossible.
27. How do capacitor combinations affect the resonant frequency of LC circuits?
Capacitor combinations change the total capacitance, which directly affects the resonant frequency of LC circuits. The resonant frequency is given by f = 1 / (2π√(LC)). Increasing capacitance (as in parallel combinations) lowers the resonant frequency, while decreasing capacitance (as in series combinations) raises it. This principle is used in tuning circuits in radio and other applications.
28. Can you explain how the concept of capacitor combinations is applied in the design of capacitive sensors?
Capacitive sensors often use combinations of capacitors to enhance sensitivity and reduce interference. For example, a differential capacitive sensor might use two capacitors in a series-parallel arrangement to measure small changes in capacitance caused by the parameter being sensed (like pressure or proximity). The combination allows for more precise measurements and better noise rejection.
29. How does the concept of capacitor combinations relate to the principle of superposition in electric fields?
The principle of superposition states that the total electric field at a point is the vector sum of the fields due to individual charges. Similarly, in capacitor combinations, the total effect (charge, voltage, or capacitance) is the sum or reciprocal sum of individual capacitor effects. This parallel between field superposition and capacitor combinations reflects the fundamental linearity of electrostatic systems.
30. What happens to the time required to charge a system of capacitors when you switch from series to parallel connection?
When switching from series to parallel connection, the time required to charge the system typically decreases. This is because parallel connection increases the total capacitance, allowing more charge to flow at the same voltage. However, the exact change depends on the charging circuit's characteristics, particularly its current-supplying capability.
31. How do capacitor combinations affect the phase relationship between voltage and current in AC circuits?
Capacitor combinations affect the overall reactance of the circuit, which determines the phase relationship between voltage and current. In purely capacitive circuits, current leads voltage by 90°. The specific combination (series or parallel) changes the magnitude of this effect but not the 90° phase difference. In more complex circuits, capacitor combinations can significantly alter the overall phase relationship.
32. Can you explain how the concept of capacitor combinations is used in the design of high-voltage capacitor banks?
High-voltage capacitor banks often use series combinations to distribute the total voltage across multiple capacitors, allowing the bank to handle much higher voltages than a single capacitor could. Parallel combinations within these series strings increase the total capacitance and energy storage. This combination approach allows for the design of capacitor banks that can store large amounts of energy at very high voltages safely.
33. How does the concept of capacitor combinations apply to the design of supercapacitors?
Supercapacitors often use complex combinations of microscopic capacitive elements to achieve their high capacitance. At the macro level, supercapacitors can be combined in series to increase voltage handling, or in parallel to increase total capacitance and energy storage. The principles of capacitor combinations are crucial in designing supercapacitor modules for various applications, from consumer electronics to electric vehicles.
34. What role do capacitor combinations play in the functioning of a Van de Graaff generator?
In a Van de Graaff generator, the large dome acts as one plate of a capacitor, with the ground as the other plate. Multiple small capacitors are effectively created in parallel as charge is continuously added to different parts of the dome. This parallel combination allows for the accumulation of a large amount of charge at a high voltage, which is the key to the generator's operation.
35. How does the concept of capacitor combinations relate to the principle of electrostatic shielding?
Electrostatic shielding often involves creating a Faraday cage, which can be thought of as a complex combination of capacitors. The conducting shield forms many small capacitors in parallel with the object being shielded. This parallel combination effectively redirects electric fields around the shielded object. Understanding capacitor combinations helps in designing effective electrostatic shields for various applications.
36. Can you explain how capacitor combinations are used in the design of electromagnetic pulse (EMP) protection systems?
EMP protection systems often use combinations of capacitors to absorb and redirect sudden surges of electromagnetic energy. Parallel combinations of capacitors can quickly absorb large amounts of charge, while series combinations help handle high voltages. The specific arrangement is designed to provide a low-impedance path for the EMP energy, protecting sensitive electronics.
37. How does the concept of capacitor combinations apply to the functioning of a Marx generator?
A Marx generator uses a clever combination of capacitors to generate very high voltage pulses. Capacitors are charged in parallel and then rapidly switched to a series configuration. This series combination allows the voltages of individual capacitors to add, producing a much higher output voltage than the charging voltage. Understanding capacitor combinations is crucial to the design and operation of these high-voltage pulse generators.
38. What is the significance of the fact that the equivalent capacitance of series capacitors is always less than the smallest individual capacitance?
This fact is significant because it means that series combinations always decrease the total capacitance of the system. This property is useful when a smaller capacitance is needed but smaller individual capacitors are not available or practical. It's also important in understanding voltage division in series capacitor chains and in designing circuits that need to handle high voltages with lower-rated components.
39. How do capacitor combinations affect the energy density of a capacitive energy storage system?
Capacitor combinations can significantly affect the energy density of a storage system. Parallel combinations increase total capacitance and energy storage capacity without increasing voltage, potentially increasing energy density if it allows for more efficient use of space. Series combinations can increase voltage handling, allowing for higher energy storage in some cases, but may decrease overall capacitance. The optimal combination depends on the specific requirements of the storage system.
40. Can you explain how the concept of capacitor combinations is applied in the design of RF matching networks?
In RF matching networks, combinations of capacitors (often with inductors) are used to match impedances between different parts of a circuit, maximizing power transfer and minimizing reflections. Series and parallel capacitor combinations allow for fine-tuning of the network's impedance characteristics. Understanding how these combinations affect the overall impedance is crucial for designing effective matching networks for various RF applications.
41. How does the concept of capacitor combinations relate to the principle of charge redistribution in switched-capacitor circuits?
Charge redistribution in switched-capacitor circuits relies heavily on the principles of capacitor combinations. As switches connect and disconnect capacitors in various series and parallel configurations, charge is redistributed according to the rules of capacitor combinations. This forms the basis for many analog-to-digital converters, filters, and other signal processing circuits. Understanding how charge distributes in different capacitor combinations is key to designing these circuits.
42. What role do capacitor combinations play in the design of voltage multiplier circuits?
Voltage multiplier circuits use a combination of capacitors and diodes to generate high DC voltages from lower AC or pulsed DC inputs. The capacitors are arranged in a way that effectively puts them in series for voltage addition

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