**Introduction**

Capacitors are components that store electricity and electrical energy (potential energy), and play an important role in circuits such as tuning, bypassing, coupling, and filtering. Capacitors are connected in parallel to increase capacity, and capacitors are connected in series to decrease capacity.

When the capacitor is connected in series in the circuit, it can prevent the sudden change of voltage and absorb the overvoltage in the peak state. The series resistance plays a damping role, and the resistance consumes the energy of the overvoltage, thereby suppressing the oscillation of the circuit. When the capacitor is connected in parallel, the parallel resistor can absorb the electric energy of the capacitor, prevent the discharge current of the capacitor from being too large, and avoid damaging the devices (such as thyristors) connected in parallel with it.

This is a very comprehensive article including the calculation formulas, circuits, and related common problems of series capacitors and parallel capacitors.

Capacitors in Series & Parallel - Electronics Basics

**Catalog**

**I What are the ****C****apacitors in ****S****eries and ****P****arallel?**

**1.1 Parallel ****C****onnection of ****C****apacitors**

We can describe the capacitors in parallel as a "water tank", but the water tank stores water, and the capacitor stores electric charges. If multiple capacitors are connected in parallel, they can naturally store more charge.

(1) The equivalent capacitance after parallel connection is equal to the sum of the capacitance of each capacitance;

(2) The voltage at both ends of each capacitor after parallel connection is equal;

- The withstand voltage after parallel connection is equal to the smallest capacitor voltage, and the equivalent capacitance is C1+C2, as shown in the figure below.

Figure1. Parallel Connection of Capacitors

**1.2 Series ****C****onnection of ****C****apacitors**

(1) The equivalent capacitance capacity after series connection is equal to the sum of the reciprocal of each capacitance;

(2) The capacitance of each capacitor after series connection is equal;

(3) The withstand voltage after series connection is equal to the sum of each capacitor voltage.

After the capacitor is connected in series, it is equivalent to increase the distance between the two poles. The more the number in series, the smaller the capacitance, but the higher the withstand voltage. In actual circuit design, we generally rarely use capacitors in series, but capacitors in parallel are often used. Sometimes the capacity of a single capacitor is not enough, and one more is added.

Figure2. Series Connection of Capacitors

## II Calculation Methods of Capacitance of a Series/Parallel Network

### 2.1 The Series and Parallel Combination

**(1) How to calculate the series capacitance of a capacitor?**

Suppose there are n capacitors connected in series. The series combination of these n capacitors is connected across a voltage source of V volts. Let us consider that the voltages across capacitors 1, 2, 3...n are V 1, V 2, V 3... Vn, respectively. The capacitances of capacitors 1, 2, 3 ... n are C 1, V 2, V 3 ... C n farad. Since all capacitors are connected in series, each of them will get the same charge, ie it is Q Coulomb. Now we know that the charge at both ends of the capacitor is only the product of the potential difference between the two ends of the capacitor and its capacitance value.

Since the series combination of these capacitors is connected across the source of the voltage V volts, replacing the series combination n of multiple capacitors.

If we consider a single equivalent capacitor of C,

Now we get from equations 1 and 2,

Therefore, when multiple capacitors are connected in series, the reciprocal of the equivalent capacitance of the system is given by the arithmetic sum of the reciprocal of their respective capacitances.

**(2) How to calculate the capacitance in parallel circuits?**

Suppose there are n capacitors connected in parallel. The parallel combination of these n capacitors is connected across the V volt voltage source. Since the capacitors are connected in parallel to the same voltage source, the charge of each capacitor is different and depends on their respective capacitance values. Let us consider that the charges of capacitors 1, 2, 3...n are Q 1, Q 2, Q 3,..., Q n coulombs, respectively. The capacitances of capacitors 1, 2, 3,..., n are C_1, C_2, C_3,... C_n coulombs respectively. It is now known that a charging capacitor is just the product of the voltage across the capacitor and its capacitance value. therefore,

Now instead of connecting multiple capacitors in parallel, if we connect a single equivalent capacitor with capacitance C across the voltage source, then the total charge at both ends of the equivalent capacitor,

Since all capacitors are connected in parallel

We can get from equations 1 and 2,

Therefore, when multiple capacitors are connected in parallel, the capacitance of the system is given by the arithmetic sum of their respective capacitances.

Figure3. (a) Three capacitors are connected in parallel. Each capacitor is connected directly to the battery.

(b) The charge on the equivalent capacitor is the sum of the charges on the individual capacitors.

**(3) Other related calculation formulas**

When the capacitor is connected in parallel, the area of the electrode is increased, and the capacitance is increased. The total capacity when connected in parallel is the sum of each capacity. When the capacitors are connected in series, the resistance value of the capacitor should be smaller than the insulation resistance of the capacitor in parallel to make the voltage distribution on each capacitor even, so as not to damage the capacitor due to uneven voltage distribution.

The series and parallel calculations of capacitors are just the opposite of the series and parallel calculations of resistors.

Voltage is the voltage during charging. The relationship between capacity and current, voltage is similar to power, and is related to load.

When voltage and capacity are quantitative, the smaller the load resistance, the larger the current and the shorter the time.

When the voltage and load are quantitative, the larger the capacity, the longer the current and the longer the time.

But in the actual discharge circuit, the general load is unchanged, the voltage of the capacitor is gradually reduced, and the current is gradually reduced.

(1) Electric capacity (uf) = current (mA)/15

Current limiting resistance (Ω)=310/maximum allowable surge current

Discharge resistance (KΩ)=500/capacitance (uf)

(2) Calculation method C=15×I

C is the capacitance of the capacitor, the unit is microfarad; the i device is the working current, the unit is ampere.

For example, if the resistance of a bulb is 0.6 amps, the capacitance should be 15×0.6=9 microfarads, and a 9 microfarad capacitor in series is sufficient.

(3) Empirical formula, 1uF output 50mA (if it is linear, a 10000F super capacitor can reach a surge current of 500 megaamps)

(4) The calculation of the half-wave rectification method should provide about 30mA current per uF capacitance, which is a reference on the 50Hz220V line in China.

The current is doubled in full-wave rectification, that is, 60mA current can be provided per uF.

Formula: R*C≥(3～5)*T/2, you need to know the frequency of the lowest signal in the ripple component (that is, the maximum T), and then determine the value of C.

● Capacitor capacity

Capacitor capacity indicates the size of electric energy that can be stored. The obstructive effect of capacitors on AC signals is called capacitive reactance. The capacitive reactance is related to the frequency and capacitance of the AC signal. The capacitive reactance XC=1/2πf c (f represents the frequency of the AC signal, and C represents the capacitance of the capacitor).

● The capacity unit and withstand voltage of the capacitor.

The basic unit of capacitance is F (farad), and other units include: millifarad (mF), microfarad (uF), nanofarad (nF), picofarad (pF). Since the capacity of the unit F is too large, we generally see units of μF, nF, and pF. Conversion relationship: 1F=1000000μF, 1μF=1000nF=1000000pF.

Each capacitor has its withstand voltage value, denoted by V. Generally, the nominal withstand voltage of the electrodeless capacitor is relatively high: 63V, 100V, 160V, 250V, 400V, 600V, 1000V, etc. The withstand voltage of polar capacitors is relatively low. Generally, the nominal withstand voltage values are: 4V, 6.3V, 10V, 16V, 25V, 35V, 50V, 63V, 80V, 100V, 220V, 400V, etc.

Power capacitor calculation: such as a three-phase capacitor bank with a nominal voltage of 690v and a capacity of 15kvar. Used in 600v circuit, delta connection, the actual effective capacity is: s=15kvar*600*600/(690*690)=11.34kvar. That is: the capacity and voltage are proportional to the square.

### 2.2 Voltage Division

Due to the large capacity of large capacitors, the volume is generally large, and they are usually made by multi-layer winding, which leads to a relatively large distributed inductance of large capacitors (also called equivalent series inductance, or ESL for short).

The impedance of the inductor to the high frequency signal is very large, so the high frequency performance of the large capacitor is not good. Some small-capacity capacitors are just the opposite. Because of their small capacity, the volume can be made small (shortening the lead wire reduces the ESL, because a piece of wire can also be regarded as an inductance), and flat capacitors are often used Structure, such a small capacity capacitor has a small ESL so that it has a good high frequency performance, but due to the small capacity, the impedance to low frequency signals is large.

So, if we want to pass the low frequency and high frequency signals well, we use a large capacitor and then a small capacitor.

The commonly used small capacitor is 0.1uF CBB capacitor is better (ceramic capacitor is also OK), when the frequency is higher, you can also connect smaller capacitors in parallel, such as a few pF, hundreds of pF. In digital circuits, a 0.1uF capacitor is generally connected to the ground in parallel to the power pin of each chip (this capacitor is called a decoupling capacitor, of course, it can also be understood as a power filter capacitor, the closer the chip is, the better), because The signal in these places is mainly high-frequency signal, and it is enough to use a smaller capacitor to filter.

The impedance of an ideal capacitor decreases as the frequency increases (R = 1/jwc), but an ideal capacitor does not exist. Due to the distributed inductance effect of the capacitor pins, the capacitor is no longer a simple capacitor in the high frequency range. , It should be regarded as a series high-frequency equivalent circuit of capacitance and inductance. When the frequency is higher than its resonance frequency, the impedance shows the characteristic of increasing with the increase of frequency, which is the inductance characteristic. At this time, the capacitance is like An inductance. On the contrary, inductors have the same characteristics.

Large capacitors in parallel with small capacitors are widely used in power supply filtering. The fundamental reason is the self-resonance characteristics of the capacitor. The combination of large and small capacitors can well suppress low-frequency to high-frequency power interference signals. Small capacitors filter high frequencies (high self-resonant frequency), and large capacitors filter low frequencies (low self-resonant frequency). The two complement each other.

● Series voltage divider ratio: V1 = C2/(C1 + C2)*V...the larger the capacitance, the smaller the voltage divided, which is the same under AC and DC conditions

● Parallel shunt ratio: I1 = C1/(C1 + C2)*I...The larger the capacitance, the larger the current that passes. Of course, this is under AC conditions.

Explanation: When two or more capacitors are connected in series, it is equivalent to lengthening the insulation distance, because only the two polar plates on the two sides work, and because the capacitance is inversely proportional to the distance, the distance increases and the capacitance decreases; two or two When the above capacitors are connected in parallel, the area equivalent to the plate increases, and because the capacitance is proportional to the area, the area increases and the capacitance increases.

● Capacitors in series: After the capacitors are connected in series, the capacity decreases and the withstand voltage increases. Formula: 1\C1+1\C2=1\C If two 50uf are connected in series, it becomes 25uf.

● Withstand voltage = add the withstand voltage values of two capacitors. If two 100V withstand voltages are connected in series, it becomes 200V.

● The formula for calculating the capacity of the series circuit of the capacitor C: 1/C=1/C+1/C2+1/C3+.+1/Cn

C is the total capacitance value of the capacitor series circuit, C1, C2, C3, Cn are the capacitance values of each capacitor in the capacitor parallel circuit, that is, the reciprocal of the total capacitance of the series circuit is equal to the sum of the reciprocal of the capacitance of each capacitor in the series circuit.

Figure4. Capacitors in Series and Parallel

### 2.3 How to Divide the Voltage When Capacitors are Connected in Series?

For example: 4V voltage source, two capacitors of 0.5F and 1F in series. If it is a DC voltage source, according to the characteristics of capacitor series voltage division introduced in middle school physics:

(1) The total voltage across the capacitor series circuit is equal to the sum of the divided voltages across the capacitors. That is, U= U1+ U2+ U3+…+Un.

(2) When capacitors are connected in series, the voltage distributed on each capacitor is inversely proportional to its capacitance. That is, Un = Q / Cn (because in the capacitor series circuit, the amount of charge carried on each capacitor is equal, so the larger the capacitor, the lower the voltage, and the smaller the capacitor, the higher the voltage. .)

Then the voltage source of 4V, the voltage on the two capacitors of 0.5F and 1F are 8/3V and 4/3V respectively 2. If it is an AC voltage source, from the impedance of the capacitor Xc=1/jωC, we can see |Xc| and C In inverse proportion, the same result can be obtained by using |Xc| as a resistor to calculate the voltage divider.

### 2.4 What is the Voltage Division Formula When Connecting 2 Capacitors in Series?

This is a theoretical calculation problem. It is necessary to assume that the withstand voltage value of the capacitor has no margin, that is, a capacitor of 200pF is breakdown when it exceeds 500V; a capacitor of 300pF is breakdown when it exceeds 900V.

After adding 1000V voltage, the 200pF capacitor will withstand 600V voltage. Regardless of the capacitor's withstand voltage margin, the 200pF capacitor will break down; at this time, 1000V will all be added to the 300pF capacitor, which exceeds its withstand voltage, so it will breakdown.

Calculation formula:

If there are M capacitors connected in series, the actual voltage value Un of any capacitor Cn is:

Un=U*C/Cn Among them: U is the total voltage; C is the total capacity of M capacitors in series.

For two capacitors in series, the formula evolves into:

Assuming that the total voltage is U, the voltages on C1 and C2 are U1 and U2 respectively, then

U1=C2*U/(C1+C2)

U2=C1*U/(C1+C2)

## III The Equivalent Method of Series or Parallel Connection of Capacitors with Different Rated Voltages and Capacities

The equivalent method of using capacitors with the same rated voltage in series or in parallel is relatively simple and commonly used.

Several capacitors with different rated voltages and different capacities are connected in series or in parallel, and the equivalent methods are different. Now give examples to illustrate. There are three capacitors C1: 220µF /10V C2: 100µF/25V C3: 10µF/100V

Calculate their parallel and series equivalent values respectively.

(1) Parallel equivalent method

1) Equivalent capacitance

C and = C1 + C2 + C3

= 220µF + 100µF + 10µF/

= 330µF

2) Equivalent withstand voltage

U parallel = U1 = 10V (take the minimum withstand voltage value U1)

(2) Series equivalent method

1) Equivalent capacitance

1/C string ==1/C1 + 1/C2 + 1/C3

= 1/220 + 1/100 + 1/10

= 252/2200

C string == 2200/252

≈ 8 (µF)

2) Equivalent withstand voltage

● Compare the Q value of each capacitor

Q1= C1 X U1 Q2=C2 X U2 Q3=C3 X U3

= 220 X 10 =100 X 25 =10 X 100

=2200 (C) =2500 (C) =1000 (C)

Q = Q3 =1000 (C) (take the minimum power value Q3)

● Find the actual allowable withstand voltage value of each capacitor

U1 (actual) = Q/C1 U2 (actual) = Q/C2 U3 (actual) = Q/C3

= 1000/220 = 1000/100 = 1000/10

≈4.5(V) = 10 (V) =100 (V)

3) U string = U1 (actual) + U2 (actual) + U3 (actual)

≈4.5 + 10 + 100

≈114.5(V)

Figure5. Equivalent Capacitance

## IV Comparison Table of Capacitors in Series and Parallel

### 4.1 Calculation Comparison of Capacitors in Series and Parallel

### 4.2 Correspondence Between Magnetic Circuit and Electric Circuit

### 4.3 Basic Physical Quantities of Magnetic Field and Magnetic Circuit

## V Frequently Asked Questions about Capacitors in Series and Parallel

(1) Do capacitors charge faster in parallel or series?

If two capacitors with the same capacity are connected in parallel or in series in the same circuit, the capacitor in series will charge faster, because the capacity of the capacitor is reduced by half after the capacitor is connected in series, and the charging time becomes shorter. The capacity of the capacitor after parallel connection is doubled, and the charging time will be longer for the same charging circuit.

(2) The electric charge of each capacitor in the series circuit is equal. Why is the electric charge of each capacitor equal to the electric charge of the equivalent capacitor?

Capacitor voltage: U=Q/C

Q=I*t

So U=(I*t)/C

When the capacitors connected in series are connected to the power supply, the capacitors start to charge. The current flowing through each capacitor is the same. As time goes by, the voltage of each capacitor increases. However, due to the different voltage rise rates of C, the sum of the voltage of each capacitor is equal to the power supply. When the voltage is applied, charging stops and the current is zero.

Analyze this process: the current flowing through each capacitor during the entire charging process is the same, and the elapsed time is the same, so the current of each capacitor is the same over time, so the amount of charge is the same and equal to the capacity of the capacitor.

Figure6. A Charging State of Three Capacitors in Parallel

(3) Are the filter capacitors in the power amplifier power supply connected in parallel?

The filter capacitor of the power amplifier power supply is set to eliminate some of the AC components contained in the rectification from AC to DC (the purpose is to improve the audio quality), so all capacitors with larger capacity are selected, generally using electrolysis above tens of microfarads Capacitor. The parallel connection of capacitors is the addition of the capacity of each capacitor, usually forming a standard type 1 filter circuit: "capacitor-resistor (or inductance)-capacitor". If the capacitors are connected in series, the capacity will decrease, it will only increase the cost and occupy more space, meaningless.

The power supply line filter capacitor of the amplifier circuit of the power amplifier is generally grouped in parallel. Depending on the design of the power supply, the single power supply circuit may also be directly connected in parallel, or divided into two groups. The two groups are separated by power inductors or resistors into two filter circuits to form a pie-type filter circuit; if it is a dual power supply circuit, , It is generally divided into two groups as for the two groups of power lines.

The easiest way to increase the filter capacitor of the power amplifier is to see the positive and negative poles and the rated withstand voltage. Connecting them in parallel can improve the stability of the DC voltage and improve the low-frequency characteristics of the amplifier, making the low frequency of the speaker sound more full and round.

Capacitors are generally used in parallel, and capacitors of different capacities filter noise at different frequencies. Large-capacity capacitors can only be realized by electrolytic capacitors. Electrolytic capacitors have positive and negative polarity and are very loud when connected reversely.

## VI Electrolytic Capacitors in Series

### 6.1 Function and Purpose of 2 Electrolytic Capacitors in Anti-phase Series

In some circuit designs, it is seen that two electrolytic capacitors are connected in series in reverse phase. The capacity of the two components should be equal and the withstand voltage is the same. In AC circuits, the leakage current can be reduced. Just use a non-polar capacitor to get a large-capacity non-polar capacitor. . Large-capacity non-polar capacitors are more expensive. The electrolytic capacitor has a large capacity and is cheap, but it has polarity, and the two are connected in reverse series. It is non-polar. It can only be used in very low voltage applications (up to 1-2V). The voltage is slightly higher. When the capacitor is used in the opposite direction, the leakage will be large. The accumulated effect will cause the electrolytic capacitor to heat up and eventually cause the capacitor to explode.

Electrolytic capacitors are used in DC circuits. So its series connection should be the negative pole of the first one and the positive pole of the second (just like dry batteries in series).

But in the circuit, there is indeed a case where the negative poles of two electrolytic capacitors are connected to the negative pole (inverted series), and the two positive poles are used. This is because it is used in an AC circuit (in a circuit where DC and AC coexist) , There is no guarantee that the potential of one pole is always higher than the other pole), so that when the capacitor is under reverse voltage, serious leakage current will be generated. At this time, non-polar capacitors should be used, but non-polar capacitors are expensive and expensive. The volume is large, so some people use two electrolytic capacitors to "reverse series". Its working state is that when there is alternating current, one of them is in the reverse state. Due to its serious leakage, the voltage drop across it is very small. Almost all of the voltage falls on the positive capacitor, and when the other half cycle of the alternating current, the state of the two capacitors will be exchanged, so these two capacitors are used as one, and the total capacitance is equal to any one of them. The total withstand voltage value is equal to 2 times of any capacitor.

### 6.2 Is the Electrolytic Capacitor in Series a Non-polarised Capacitor?

Of course, two electrolytic capacitors in parallel will not work. If two electrolytic capacitors are connected in series, it will still not work without applying proper bias voltage. Applying a bias voltage is quite complicated, especially when both ends of the capacitor (two in series) are not grounded (the bias voltage must be floating). Considering the complexity of applying the bias voltage, it is better not to use this method: connect the negative poles of the two capacitors, and connect the two capacitors in parallel with a high-current diode. The positive of the diode is connected to the negative of the capacitor, and the negative is connected to the positive of the capacitor. Parallel connection of course still has polarity. If reverse parallel connection, it is non-polar, but it is non-polar.

Reverse series connection is also not advisable. If you do a test, you will find that there must be a capacitor withstands the back pressure. If the voltage is large, it will blow up. Unless special measures are taken, the voltage is always applied to the capacitor with the positive voltage. on.

Two electrolytic capacitors of the same capacity can be connected in series, but a diode must be connected in anti-parallel to prevent reverse breakdown of the electrolytic capacitor. After adding a diode, it is okay if it is used for filtering, but it is definitely not good for blocking DC. Because the electrolytic capacitor is only charged and not discharged. Two identical electrolytic capacitors connected in reverse series can replace non-polar capacitors with the same capacity. The dielectric loss of the electrolytic capacitor is very large, and it must be connected to the AC circuit after the voltage is greatly reduced. Otherwise, it is either burned or fried.

## VII Quiz

● Question

A network of five capacitors of C is connected to a 100 V supply, as shown in below figure. Determine

(a) the equivalent capacitance of the network

(b) the charge on each capacitor.

● Solution

In the given network , the top three Capacitance is in series, So equivalent capacitance of top part

1C1=1C+1C+1C

C1=C3

Similarly, the lower two Capacitance is in series, So equivalent capacitance of lower part

1C2=1C+1C

C2=C2

Now both C1 and C2 are in parallel, so equivalent Capacitance of the Network

Ceq=C1+C2=C3+C2=5C3

Now Charge on top part will be

Q1=C1V=CV3

Now Charge on lower part will be

Q2=C2V=CV2

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