8:49 PM

Introduction to Filters and Capacitive Reactance

Introduction to Filters and Capacitive Reactance:

Introduction:

Electrical filter is a circuit, designed to reject all unwanted frequency components of an electrical signal and allows only desired frequencies. In other words a filter is a circuit which allows only a certain band of frequencies.The main applications of the filters are at audio equalizers and in sensitive electronic devices whose input signals should be conditional.These filters are mainly categorized into 2 types. They are Active filters and passive filters.

Passive Filters:

Passive filters do not contain any amplifying elements just they are made up of Resistor, Capacitor and inductors (passive elements). These filters will not draw any additional power from the external battery supply. The Capacitor will allow the high frequency signals and inductor allows low frequency signals. Similarly inductor restricts the flow of high frequency signals and capacitor restricts the lower frequency signals. In these filters output signal amplitude is always less than the amplitude of the applied input signal.The gain of passive filters is always less than unity.This shows that the gain of the signals are cannot be improved by these passive filters. Due to this the characteristics of the Filters are affected by the load impedances. These filters can work at higher frequency ranges nearly at 500 MHz also.

Active Filters:

Active filters contain amplifying elements such as Op-Amps, Transistors and FET’s (active components)in addition to the passive elements (Resistors, Capacitors and Inductors). By using these filters we can overcome the drawbacks of Passive filters. Active filters will depend on external power supply because it will amplify the output signals. Without any inductor element these can achieve the resonant frequency that is the input impedance and output impedances are nullified by each other. In following year’s inductor less filter design is made. Because inductors dissipate some amount of power and generates stray magnetic fields. Not only have these problems, but also due to inductor the size of the active filter increases. So, due to these reasons the use of inductors in active filters is reduced.

Some of the advantages of Active filters

The combination of op-amps, resistors, capacitors, transistors and FETs gives an integrated circuit which in turn reduces the size and weight of the filter.
The gain of an Op-amp can be easily controlled in the closed loop form. Due to this reason the input signal is not restricted.
These are applicable in Butterworth filters, Chebyshev and Cauer filters.

The main drawback in active filters is the operational frequency range is less. In many applications the operational frequency range of active filters is maximized to 500 kHz only. The active filters must require D.C power supply. When compared with the passive filters these active filters are more sensitive. The outputs may disturb even due to the environmental changes also.

The filter is a sensitive circuit and in which the output components are only frequency terms. To analyze the filter circuit the frequency domain representation is the best one. This representation is as shown below.

The magnitude of the filter M is called as the gain of the filter. Magnitude is generally represented in dB as 20log (M).

One of the important characteristic of the filters is cut-off frequency. It is defined as the frequency which separates both pass band and stop band in frequency response. Pass band is the range of frequencies that are allowed by the filter without any attenuation. Stop band is defined as the band of frequencies that are not allowed by the filter.

Filters are classified based on the frequency of signals that they allowing through them. There are four types of filters they are Low Pass Filters, Band Pass Filters, High Pass Filters and Band Stop Filters. Due to the usage of high speed op-amps and approximate values of the components the characteristics of ideal and practical responses are nearly equal.

Low Pass Filter:

Low Pass Filters will pass the frequency signals less than cut-off frequency ‘fc’. Practically a small range of frequencies will pass even after the cut-off frequency range. The gain of the filter will depend up on the frequency. If the input signal frequency increases then, gain of the filter decreases. At the end of the transition band the gain becomes zero. This is as shown below.

Fig: Low Pass1

Where dotted line indicates the ideal filter characteristics and continuous line indicates practical filter characteristics.

Applications of low pass filters are in sound system that is in various types of loudspeakers. To block the harmonic emissions these low pass filters are used in radio transmitters. These are also used at DSL splitters in telephone subscriber lines.

High Pass Filter:

They will pass the frequencies after the cut off frequency ‘fc’. In practical case a negligible frequencies below the cut off range is allowed by the filter. This is as shown below.

Fig: high pass1

The combination of high pass filter with low pass filter forms Band pass filter. Applications of the high pass filters are at RF circuits and are also used in DSL splitters.

Band Pass Filter:

The name of the filter itself indicates that itallows only a certain band of frequencies and blocks all the remaining frequencies. The upper and lower limits of the band pass filter depend on the filter design. Practical and ideal characteristics of band pass filter are shown below.

Fig: band pass1

Applications of band pass filters are at transmitter and receiver circuits. These are mainly used to calculate the sensitivity of the receiver circuits and to optimize the signal to noise ratio.

Band Stop Filter

These are also called as Band rejection or band elimination filters. These filters stop only a particular band of frequencies and allow all other frequencies.The frequency limits of the filter depend on the filter design.The dotted line indicates the ideal case where as a continuous line indicates the practical case. It has two pass bands and one stop band.

Fig: band stop1

Applications of band stop filters are at instrument amplifiers.

Ideal filters frequency response:

Now let us see the ideal response of different filters. Here fL indicates the lower cut-off frequency and fH indicates the higher cut-off frequency.

Ideal characteristics of Low pass filter:

Fig: low pass2

This response shows that the low pass filter will allow the signals up to lower cutoff frequency and stops the frequencies higher than the lower cutoff frequency.

Ideal characteristics of High pass filter:

Fig: high pass2

This shows the high pass filter will allow the frequencies greater than the higher cut off frequency and stops the frequencies lesser than the high cut off frequency.

Ideal characteristics of Band pass filter:

Fig: band pass2

This response shows that the band pass filter will pass the frequencies between the lower cutoff region and higher cut-off region only. It stops the frequencies which are lesser than the lower cutoff frequency and also stops the frequencies greater than higher cut-off frequencies.

Ideal characteristics of Band stop filter:

Fig: band stop2

The above figure shows that the frequencies which are greater than the lower cut off frequency and the frequencies which are lower than the higher cut off frequency are not processed.

Capacitive Reactance:

When the Resistor is connected in series with the capacitor forms RC circuit. In RC circuit the capacitor will charge from the D.C supply voltage and when supply voltage is decreased, eventually the capacitor also discharges by reducing its storage charge. Not only at the time of the D.C supply even in the case of A.C supply also according to the supply voltage level the capacitor will charge and discharge continually.

But due to internal resistance there will be some attenuation in the flow of the current through the capacitor. This internal resistance is called as Capacitive Reactance. ‘X_C’Indicates the Capacitive Reactance and it is measured in Ohms same as that of the resistance.

When the frequency is varied in the capacitive circuit according to the amount of frequency change this capacitive reactance value also changes. The electrons flow from one plate to the other plate causes the current flow in the circuit. But due to the movement of electrons the frequency level varies. When the frequency through the capacitor increases the capacitive reactance value decreases and when frequency through the capacitor decreases the capacitive reactance value increases. Thus, by this we can say that the capacitive reactance is inversely proportional to the applied frequency level. This shows that the capacitor connected in the circuit is dependent on supply frequency. This phenomenon is called complex impedance.

Capacitive Reactance Formula

Xc = 1/(2π1c)

Where Xc = Capacitive Reactance

π = 3.142

f = Frequency in Hz

c = Capacitance in Farads (F).

Capacitive reactance Example:

Let us consider two frequencies to observe the capacitive reactance phenomenon. Let f1=1kHz and f2=10kHz and the capacitor c= 220nF.

At first frequency level

X_C= 1/2πf1c = 723.4Ω

At second Frequency level:

X_C= 1/2πf2c= 72.34Ω

This shows clearly that with the increase of frequency the reactance decreases.

Capacitive Reactance Vs Frequency:

From the above frequency verses capacitive reactance plot we can observe that when the frequency is zero the reactance value reaches to infinity this shows the phenomenon of the open circuit. When the value of the frequency increases exponentially the reactance value decreases. When frequency reaches to infinity the reactance value at nearly zero this gives us closed circuit behaviour.

Voltage Divider concept

We have already studied the voltage divider concept in resistors topic and we know that the voltage divider circuit is able to produce the output voltage which is a fraction of the input voltage.

VOUT = VIN x (R2 / (R1 + R2))

By replacing the resistor R2 with a capacitor C in the above circuit the voltage drop across the two components changes with the input frequency because reactance of the capacitor varies with the frequency.Now the output voltage across the capacitor depends on the input frequency. Using this concept we can construct passive low and high pass filters by replacing one of the resistors with a capacitor in a voltage divider circuit.

Capacitor behaviour in Low Pass Filter:

For the Low pass filter the resistor R2 is replaced by the capacitor C1. At normal frequency the circuit is as shown in the above figure. When the frequency is zero the reactance value is very high which is nearly equal to infinity. At this condition circuit acts as an open circuit. When frequency is very high the reactance value reaches to zero and the circuit acts as a closed circuit. Both of these behaviour are shown in the above figure.

Capacitor behaviour in High Pass filters:

For the High pass filter the resistor R1 is replaced by the capacitor C1. From the above figure it is clear that at normal frequency the circuit acts like a high pass filter circuit. Initially at the zero frequency value the circuit behaves like an open circuit. When frequency increases the reactance will decrease exponentially. At some point the frequency reaches to infinity level thus, it affects the reactance to reach the zero state. These circuit behaviors are shown in the above figures.

REFERENCE /// Electronics Hub

7:57 PM

Capacitance in AC Circuits

Capacitance in AC Circuits:

Introduction:

When DC supply voltage is applied to the capacitor,the capacitor is charged slowly and finally it reaches to fully charged position. At this point the charging voltage of a capacitor is equal to the supply voltage. Here the capacitor acts as an energy source as long as voltage is applied. Capacitors don’t allow current (i) through them after they fully charged.The current flowing through the circuit depends on the amount of charge in the plates of capacitors and also the current is directly proportional to the rate of change of voltage applied to the circuit. i.e. i = dQ/dt = C dV(t)/dt.

If AC supply voltage is applied to the capacitor circuit then the capacitor charges and discharges continuously depending on the rate of frequency of supply voltage. The capacitance of a capacitor in AC circuits depends on the frequency of supply voltage applied to it. In AC circuits the capacitors allow current when the supply voltage is continuously changing with respect to time.

AC Capacitor Circuit:

In the above circuit we observed that a capacitor is directly connected to the AC supply voltage. Here the capacitor continuously charges and discharges depending on the changes in supply voltage, because the AC supply voltage value is constantly increases and decreases. We all know that the current flowing through the circuit is directly proportional to the rate of change of voltage applied.

Here the charging current has its high value, if the supply voltage crosses its value from positive half cycle to the negative half cycle and vice versa. i.e. at 00 and 1800 in sine wave signal. The current through the capacitor has its minimum value when the supply voltage in sine wave crosses over at its maximum or minimum peak value (Vm). Hence we can say that the charging current flowing through the circuit is maximum or minimum depending on the supply voltage levels in sine wave.

AC Capacitor Phasor Diagram:

Figure 2. Phasor diagram of AC capacitor.

The phasor diagram of AC capacitor is shown in the above figure, here the voltage and currents are represented in sine wave forms. In the above figure we observed that at 00 the charging current is at its maximum value because the voltage is increasing slowly in positive direction. At 900 there is no current flow through the capacitor because at this point the supply voltage is at its maximum peak value.

At 1800 point the voltage slowly decreases to zero and current is at maximum value in the negative direction. Again the charging reaches to its maximum value at 3600, because at this point the supply voltage is at minimum value.

From the waveforms in the above figure we can see that, the current is leading the voltage by 900. Hence we can say that in an ideal capacitor circuit the AC voltage lags the current by 900.

Capacitive Reactance:

We know that the current flowing through the capacitor is directly proportional the rate of change of applied voltage but capacitors also offer some form of resistance against the current flow same as like resistors. This resistance of capacitors in AC circuits is called as capacitive reactance or commonly known as reactance. Capacitive reactance is the property of a capacitor which opposes the flow of current in AC circuits. It is represented with symbol Xc and measured in Ohms same as like resistance.

We need some extra energy over capacitive reactance to charge up a capacitor in the circuit. This value is inversely proportional to the capacitance value and the frequency of supply voltage.

Xc∝ 1/c and Xc∝ 1/f.

The equation for capacitive reactance and parameters which influences them are discussed in below.

Capacitive Reactance,

XC = 1/2πfC = 1/ωC

Here,

XC = Reactance of capacitor

f = frequency in HZ

C = Capacitance of a capacitor in Farads

ω (omega) = 2πf

From the above equation we understood that capacitive reactance is high where the frequency and capacitance values are at low and at this stage the capacitor acts as a perfect resistor. If the frequency of supply voltage is high then the reactance value of capacitor is low and also at this stage capacitor acts as a good conductor. From the above equation it is clear that the reactance is zero if the frequency is infinity and the reactance value is infinity where the frequency is at zero.

Capacitive Reactance against Frequency:

Figure 3.Relation between Reactance and Frequency.

The above figure shows the relation between the capacitive reactance, current and frequency of the supply voltage. Here we observed that if frequency is low then the reactance is high. The charging current increases with increase in frequency, because the rate of change of voltage increases with time.The reactance is at infinite value where the frequency is zero and vice versa.

AC Capacitance - Example No1:

Find the rms value of current flowing through the circuit having 3uF capacitor connected to 660V and 40Hz supply.

Capacitive Reactance,

XC = 1/2πfC

Here,

f = 40HZ

C = 3uF

Vrms = 660V

Now,

XC = 1/(2 × 3.14 × 40HZ × 3 × 10-6) = 1326Ω

Irms = Vrms/XC = 660V/1326Ω = 497mA

AC Capacitance - Example No2:

Find the rms value of current flowing through the circuit having 5uF capacitor connected to 880V and 50Hz supply.

Capacitive Reactance,

XC = 1/2πfC

Here,

f = 50HZ

C = 5uF

Vrms = 880V

Now,

XC = 1/(2 × 3.14 × 50HZ × 5 × 10-6) = 636Ω

Irms = Vrms/XC = 880V/636Ω = 1.38 A

From the above two examples practically we observed that the reactance of a capacitor depends on the frequency of the supply voltage and it is inversely proportional relation. In example 1 , the reactance is 1326Ω for the frequency of 40HZ but the reactance value decreases to 636Ω when the frequency increases to 50HZ which is shown in example 2.Hence it is clear that the reactance of a capacitor is inversely proportional to the frequency and capacitance.

REFERENCE // Electronics Hub

7:48 PM

Capacitive Voltage Divider

Capacitive Voltage Divider:

Introduction:

In a voltage divider circuit,the supply voltage or circuit voltage is distributed among all the components in the circuit equally,depending on the capacity of those components.

The construction of capacitive voltage divider circuit is same as like resistive voltage divider circuit. But like resistors, the capacitive voltage divider circuit is not affected by the changes in the frequency even though it uses reactive elements.

The capacitor is a passive component which stores electrical energy in the metal plates. A capacitor has two plates and these two are separated by non-conducting or insulating material, such as called as “dielectric”.

Here the positive charge is stored on one plate and negative charge is stored on another plate.

When DC current is applied to the capacitor, it charges fully. The dielectric material between the plates acts as insulator and also it opposes the current flow through the capacitor.

This opposition to supply current through the capacitor is called reactance (XC) of a capacitor. The capacitor reactance is also measured in ohms.

A fully charged capacitor acts as an energy source, because a capacitor stores energy and discharges it to the circuit components.

If an AC current is applied to the capacitor then the capacitor continuously charges and discharges the current through its plates.In this time the capacitor also has reactance which varies depends on supply frequency.

We know that the charge which is stored in the capacitor depends on the supply voltage and the capacitance of a capacitor.

In the same way the reactance also depends on some parameters, now we see the parameters which influence the reactance of a capacitor.

If a capacitor has smaller capacitance value , then the time required to charge a capacitor is less , i.e. smaller RC time constant is required. In the same way the RC time constant is high for larger capacitance value of capacitors.

From this we observed that , larger capacitance value capacitor has less reactance value where as smaller capacitance value of capacitor has larger reactance value. i.e. the reactance of a capacitor is inversely proportional to the capacitance value of the capacitor.

XC∝ 1/C

If the frequency of applied current is low then the charging time of capacitor increases,it indicates that reactance value is high. In the same way if the frequency of applied current is at high, then the reactance of the capacitor is low.

From this we can observe that the reactance of a capacitor is inversely proportional to the frequency.

Finally, we can say that, the reactance (XC) of any capacitor is inversely proportional to the frequency (f) and the capacitance value (C).

XC∝ 1/f

Capacitive Reactance Formula:

Already we know that the capacitive reactance is inversely proportional to the frequency and capacitance value of the capacitor. Thus formula for reactance is

XC = 1/2πfC

Here,

XC = Reactance of a capacitor in ohms (Ω)

f = Frequency in Hertz’s (HZ)

C = Capacitance of a capacitor in Farads (F)

π = Numeric constant (22/7 = 3.142)

Voltage Distribution in Series Capacitors:

If the capacitors are connected in series , the voltage distribution between the capacitors is calculated. Because the capacitors have different voltage values depending on the capacitance values in series connection.

The reactance of a capacitor which opposes the flow of current , depends on the value of capacitance and frequency of the applied current.

So now let us see how the reactance affects the capacitors , by calculating the frequency and capacitance values. Below circuit shows the capacitive voltage divider circuit in which 2 capacitors are connected in series.

Capacitive Voltage Divider:

The two capacitors which are connected in series have the capacitance values of 10uF and 22uF respectively. Here the circuit voltage is 10V,this voltage is distributed between both capacitors.

In the series connection all the capacitors have same charge (Q) on it but the supply voltage (VS) is not same for all capacitors.

The circuit voltage is shared by the capacitors depending on the capacitance values of the capacitors.i.e. in the ratio of V = Q/C.

From these values we have to calculate the reactance (XC) of each capacitor by using frequency and capacitance values of capacitors.

Capacitive Voltage Divider - Example No1:

Now we will calculate the voltage distribution to the capacitors 10uF and 22uF which are given in the above figure which have 10V supply voltage with 40HZ frequency.

Reactance of 10uF capacitor,

XC1 = 1/2πfC1 = 1/(2*3.142*40*10*10-6) = 400Ω

Reactance of 22uF capacitor,

XC\2 = 1/2πfC2 = 1/(2*3.142*40*22*10-6) = 180Ω

Total capacitive reactance of a circuit is,

XC= XC1+ XC2= 400Ω + 180Ω = 580Ω

CT= C1C2/(C1+C2) = (10*22*10-12)/(32*10-6) = 6.88uF

XCT = 1/2πfCT = 1/(2*3.142*40*6.88*10-6) = 580Ω

The current in the circuit is,

I = V/XC = 10V/580Ω = 17.2mA

Now, the voltage drop across each capacitor is,

VC1 = I*XC1 = 17.2mA*400Ω = 6.9V

VC2 = I*XC2 =17.2mA*180Ω = 3.1V

Capacitive Voltage Divider - Example No2:

Now we calculate the voltage drops across the capacitors 10uF and 22uF which are connected in series and they operate with 10V supply voltage of 4000HZ (4KHZ) frequency.

Reactance of 10uF capacitor,

XC1 = 1/2πfC1 = 1/(2*3.142*4000*10*10-6) = 4Ω

Reactance of 22uF capacitor,

XC\2 = 1/2πfC2 = 1/(2*3.142*4000*22*10-6) = 1.8Ω

Total capacitive reactance of a circuit is,

XC= XC1+ XC2 = 4Ω+1.8Ω = 5.8Ω

CT = C1C2/(C1+C2) = (10*22*10-12)/(32*10-6) = 6.88uF

XCT = 1/2πfCT = 1/(2*3.142*4000*6.88*10-6) = 5.8Ω

The current in the circuit is,

I = V/XCT = 10V/5.8Ω = 1.72A

Now, the voltage drop across each capacitor is,

VC1 = I*XC1 = 1.72A*4Ω = 6.9V

VC2 = I*XC2 = 1.72A*1.8Ω = 3.1V

From the above two examples we can conclude that the lower value capacitor (10uF) will charge to a higher voltage (6.9V), and the higher value capacitor (22uF) will charge itself to a lower voltage level(3.1V).

Finally the sum of two capacitor voltage drops values are equal to the supply voltage (i.e. 6.9V+3.1V=10V). These voltage values are same for all frequency values, because the voltage drop is independent of frequency.

The voltage drops for the two capacitors is same in both the examples where the frequency is different. The frequency is either 40HZ or 40KHZ the voltage drops across capacitors is same in both cases.

The current flowing through the circuit changes depending on the frequency. Current will increase with increasing the frequency, it is 17.2mA for 40HZ frequency but it is 1.72A for the frequency 4KHZ, i.e. the current will increase almost 100 times by increasing frequency 4HZ to 4KHZ.

Finally we can say that the current flowing through the circuit is directly proportional to the frequency (I α f).

Summary:

The opposition for the flow of current in the capacitor is known as reactance (XC) of a capacitor. This capacitive reactance is influenced by the parameters like capacitance value,frequency of supply voltage and also these values are inversely proportional to the reactance.
The AC voltage divider circuit will distribute the supply voltage to all the capacitors depending on their capacitance value.
These voltage drops for the capacitors are same for any frequency of supply voltage. i.e. the voltage drops across capacitors are independent on frequency.
But the current flowing is depending on frequency and also these two are directly proportional to each other.
But in DC voltage divider circuits, it is not an easy task to calculate the voltage drops across capacitors as it depends on reactance value, because the capacitors block DC current flow through it after fully charged.
The capacitive voltage divider circuits are used in large electronics applications.Mainly used in capacitive sensitive screens those change their output voltage when it is touched by a person finger.
And also used in transformers to increase voltage drop where generally the mains transformer contains low voltage drop chips and components.
Finally one thing to say is in voltage divider circuit the voltage drops across capacitors are same for all frequency values.

REFERENCE // Electronics Hub

7:38 PM

Capacitors in Series and Parallel

Capacitors in Series and Parallel:

Capacitor in Series:

Capacitors in series meanstwo or more capacitors are connected in a single linei.e positive plate of the one capacitor is connected to the negative plate of the next capacitor. All the capacitors in series have equal charge (Q) and equal charging current (iC).

Consider N- numbers of capacitors are connected in series, then

QT =Q1 = Q2 = Q3 = ———- = QN

IC = I1 = I2 = I3 = ——— = IN

Capacitors in a Series Connection:

The following circuits show the series connection of group of capacitors.

series connection of N-number of capacitors.

series connection of two capacitors.

In this circuit the charge (Q) stored in all capacitors is same because every capacitor has the charge which is flowing from the adjacent capacitor. The voltage drop in all capacitors is different from each other. But the total voltage drop applied between input and output lines of the circuit is equal to the sum of all the individual voltage drops of each capacitor. The equivalent capacitance of the circuit is Ceq = Q/V.

Thus,

VT = V1 + V2

Ceq = Q/V1 + Q/V2

1/Ceq = (V1+ V2)/Q

VT = Q/Ceq = Q/C1 + Q/C2

Series Capacitors Equation:

1/Ceq = 1/C1 + 1/C2 +……… + 1/CN

When the capacitors are in series connection the reciprocal of the equivalent capacitance is equal to the sum of the reciprocals of the individual capacitances of the capacitors in the circuit.

From the figure 2, the reciprocal of equivalent capacitance value of the circuit is equal to the sum of reciprocal capacitances values of two capacitors C1 and C2, the expression is given below.

1/Ceq = 1/C1 + 1/C2

Capacitors in Series - Example No1:

We assume that the values of two capacitors in the above figure2 are 0.4uF and 0.5uF respectively. Now we can calculate the equivalent capacitance for the two capacitors in series as,

1/Ceq = 1/C1 + 1/C2

1/Ceq = (C1 + C2)/ C1C2

Ceq = (C1C2) / (C1 + C2)

Ceq = (0.4uF*0.5uF) / (0.4uF+0.5uF)

Ceq = 0.22uF

We know that the equivalent capacitance of the capacitors in series connection is less than the value of the smallest capacitor in the circuit. In that way here also we get the equivalent value 0.22 uF is less than the smallest capacitance 0.4 uF in the series connection of the given two capacitors circuit.

Capacitors in Series - Example No2:

Calculate the equivalent capacitance and the individual voltage drops across the set of two capacitors in series have 0.1uF and 0.2uF respectively when connected to a 12V a.c. supply.

Equivalent capacitance,

1/Ceq = 1/C1 + 1/C2

Ceq = (C1C2) / (C1 + C2)

Ceq = (0.1uF*0.2uF) / (0.1uF+0.2uF)

Ceq = 0.066uF = 66nF

Voltage drops across the two given capacitors in series are,

V1 = (C2*VT)/ (C1+C2) = (0.2uF*12V)/ (0.1uF+0.2uF) = 8Volts

V2 = (C1*VT)/ (C1+C2) = (0.1uF*12V)/ (0.1uF+0.2uF) = 4Volts

From these results we observed that the equivalent capacitance 66nF is less than the smallest capacitance 0.1uF of the given two capacitors. The individual voltage drops across the given two capacitors are different. But the sum of individual voltage drops of both the capacitors is equal to the total voltage. i.e 8V+4V=12V.

Now we calculate the charge stored in individual capacitor,

Q1 = V1* C1 = 8V * 0.1uF = 0.8uC

Q2 = V2* C2 = 4V * 0.2uF = 0.8uC

Here we observed that equal charge 0.8uC is stored in both the capacitors C1 and C2 which are connected in series.

Capacitors in Series Summary:

The equivalent capacitance of the capacitors connected in series chain is, the reciprocal of the equivalent capacitance is equal to the sum of the reciprocals of all the individual capacitances of the capacitors connected in series. All the capacitors which are connected in series have the equal charge (Q). The charging current ( iC) is also same for all the individual capacitors which are connected in series like iC=i1= i2 etc.

The voltage drops for individual capacitors in series connection are different. But the sum of all the individual voltage drops is equal to the total voltage across the circuit. i.e.VT = V1+V2 etc . A large value of capacitance will result in a smaller voltage drop while a small capacitance value will result in a larger voltage drop.

Capacitors in Parallel Circuits:

Capacitors in parallel means two or more capacitors are connected in parallel way, i.e. both of their terminals are connected to each terminal of the other capacitor or capacitors respectively. All the capacitors which are connected in parallel have the same voltage and is equal to the VT applied between the input and output terminals of the circuit. Then, parallel capacitors have a ‘common voltage’ supply across them .i.e. VT = V1 = V2 etc.

The equivalent capacitance, Ceq of the circuit where the capacitors are connected in parallel is equal to the sum of all the individual capacitances of the capacitors added together. This is because the top plateof each capacitor in the circuit is connected to the top plate of adjacent capacitors. In the same way the bottom plate of each capacitor in the circuit is connected to the bottom plate of adjacent capacitors.

The following circuits show parallel connection between groups of capacitors. Figure3 has N-number of capacitors and figure 4 has two capacitors connected in parallel.

Parallel connection of two capacitors,

In above figure the total charge (Q) across the circuit is divided between the two capacitors, means the charge Q distributes itself between the capacitors connected in parallel. Because the voltage drop across individual capacitors is equal and also it is equal to the total voltage applied to the circuit. But the total charge Q is equal to the sum of all the individual capacitor charges connected in parallel. i.e. From the above figure the two different capacitors C1 and C2 have two different charges Q1 and Q2 respectively. Here Q=Q1+Q2

Now we see the equivalent capacitance of the capacitors C1 and C2 connected in parallel which shown in the above figure.

We know the formula,

Q=Ceq VT

Here, Q = Q1+Q2

And VT = V1 = V2

Ceq=Q/VT = (Q1+Q2)/VT = (Q1/VT) + (Q2/VT)

Parallel Capacitors Equation:

Ceq = C1+C2+C3+ ———— +CN

The equivalent capacitance of the capacitors which are connected in parallel is equal to the sum of the individual capacitances of the capacitors in the circuit.

From the figure 4, the equivalent capacitance (Ceq) value is equal to the sum of both the capacitance values of C1 and C2, the expression is shown below.

Ceq = C1+C2

Capacitors in Parallel - Example No1:

Consider the capacitance values of the two capacitors C1 =0.2uF and C2 =0.3uF which are shown in above figure 4. Now calculate the equivalent capacitance of the circuit.

We know that the Equivalent capacitance,

Ceq = C1 + C2

Ceq =0.2uF + 0.3uF

Ceq =0.5uF

One important point to remember about parallel connected capacitor circuits, the equivalent capacitance (Ceq) of any two or more capacitors connected together in parallel will always be greater than the value of the largest capacitor in the circuit as we are adding together values. So in our example above Ceq=0.5uF whereas the largest value capacitor in the circuit is only 0.3uF.

Capacitors in Parallel - Example No2:

Calculate the equivalent capacitance across the set of two capacitors in parallel shown in figure 4.

(a) One capacitor of 0.1uF connected in parallel with 0.2uF.

(b) One capacitor of 750 nF connected in parallel with 0.5 uF.

(a) Equivalent capacitance,

Ceq = C1 + C2

Ceq =0.1uF + 0.2uF

Ceq =0.3uF

(b) Equivalent capacitance,

Ceq = C1 + C2

Ceq =750nF + 0.5uF

Ceq =750nF + 500nF

Ceq = 1250 nF =1.25uF

In the above two cases we observed that the equivalent capacitance (Ceq) value of the two capacitors connected together in parallel is greater than the value of the largest capacitor in the circuit as we are adding together values. In section (a) the value of Ceq = 0.3uF whereas the largest value capacitor is only 0.2uF. In section (b) the value of Ceq= 1.25uF whereas the largest value capacitor in the circuit is 0.5uF only.

Capacitors in parallel Summary:

The equivalent capacitance of the group of capacitors which are connected in parallel is equal to the sum of capacitances of the individual capacitors. i.e Ceq = C1+C2. Because the voltage drop across the individual capacitor is equal to the total voltage applied between the input and output terminals of the circuit. i.e. VT = V1=V2. The charges stored in individual capacitors are different but the sum of all the individual capacitor charges is equal to the total charge flows in the circuit. i.e. Q = Q1+Q2.

The charging current flow in circuit is distributed to all the capacitors in the circuit. But the total charging current is equal to the sum of all the individual charging currents of the capacitors in the circuit. i.e. iC=i1+i2 etc. The equivalent capacitance value of the group of capacitors which are connected in parallel is always greater than the value of the largest capacitor in the circuit.

REFERENCE // Electronics Hub