Capacitive Voltage Divider






Capacitive Voltage Divider
Voltage divider circuits may be constructed from reactive components just as
easily as they may be constructed from fixed value resistors


But just like resistive circuits, a capacitive voltage divider network is
not affected by changes in the supply frequency even though they use
capacitors, which are reactive elements, as each capacitor in the series
chain is affected equally by changes in supply frequency.

But before we can look at a capacitive voltage divider circuit in more
detail, we need to understand a little more about capacitive reactance and
how it affects capacitors at different frequencies.

In our first tutorial about Capacitors, we saw that a capacitor consists of
two parallel conductive plates separated by an insulator, and has a positive
(+) charge on one plate, and an opposite negative (–) charge on the other.
We also saw that when connected to a DC (direct current) supply, once the
capacitor is fully charged, the insulator (called the dielectric) blocks the
flow of current through it.
polypropylene capacitor

Typical Capacitor





A capacitor opposes current flow just like a resistor, but unlike a resistor
which dissipates its unwanted energy in the form of heat, a capacitor stores
energy on its plates when it charges and releases or gives back the energy
into the connected circuit when it discharges.
 
	


This ability of a capacitor to oppose or “react” against current flow by
storing charge on its plates is called “reactance”, and as this
reactance relates to a capacitor it is therefore called Capacitive Reactance
( Xc ), and like resistance, reactance is also measured in Ohm’s.

When a fully discharged capacitor is connected across a DC supply such as a
battery or power supply, the reactance of the capacitor is initially
extremely low and maximum circuit current flows through the capacitor for a
very short period time as the capacitors plates charge up exponentially.

After a period of time equal to about “5RC” or 5 time constants, the
plates of the capacitor are fully charged equalling the supply voltage and
no further current flows. At this point the reactance of the capacitor to DC
current flow is at its maximum in the mega-ohms region, almost an open
-circuit, and this is why capacitors block DC.

Now if we connect the capacitor to an AC (alternating current) supply which
is continually reversing polarity, the effect on the capacitor is that its
plates are continuously charging and discharging in relationship to the
applied alternating supply voltage. This means that a charging and
discharging current is always flowing in and out of the capacitors plates,
and if we have a current flow we must also have a value of reactance to
oppose it. But what value would it be and what factors determine the value
of capacitive reactance.

In the tutorial about Capacitance and Charge, we saw that the amount of
charge, ( Q ) present on a capacitors plates is proportional to the applied
voltage and capacitance value of the capacitor. As the applied alternating
supply voltage, ( Vs ) is constantly changing in value the charge on the
plates must also be changing in value.

If the capacitor has a larger capacitance value, then for a given
resistance, R it takes longer to charge the capacitor as τ = RC, which
means that the charging current is flowing for a longer period of time. A
higher capacitance results in a small value of reactance, Xc for a given
frequency.

Likewise, if the capacitor has a small capacitance value, then a shorter RC
time constant is required to charge the capacitor which means that the
current will flow for a shorter period of time. A smaller capacitance
results in a higher value of reactance, Xc. Then we can see that larger
currents mean smaller reactance, and smaller currents mean larger reactance.
Therefore, capacitive reactance is inversely proportional to the capacitance
value of the capacitor, XC ∝-1 C.

Capacitance, however is not the only factor that determines capacitive
reactance. If the applied alternating current is at a low frequency, the
reactance has more time to build-up for a given RC time constant and oppose
the current indicating a large value of reactance. Likewise, if the applied
frequency is high, there is little time between the charging and discharging
cycles for the reactance to build-up and oppose the current resulting in a
larger current flow, indicating a smaller reactance.

Then we can see that a capacitor is an impedance and the magnitude of this
impedance is frequency dependent. So larger frequencies mean smaller
reactance, and smaller frequencies mean larger reactance. Therefore,
Capacitive Reactance, Xc (its complex impedance) is inversely proportional
to both capacitance and frequency and the standard equation for capacitive
reactance is given as:



Capacitive Reactance Formula
	
	capacitive reactance formula

    Where:
       Xc = Capacitive Reactance in Ohms, (Ω)
       π (pi) = a numeric constant of 3.142
       ƒ = Frequency in Hertz, (Hz)
       C = Capacitance in Farads, (F)



Voltage Distribution in Series Capacitors

Now that we have seen how the opposition to the charging and discharging
currents of a capacitor are determined not only by its capacitance value but
also by the frequency of the supply, lets look at how this affects two
capacitors connected in series forming a capacitive voltage divider
circuit.


Capacitive Voltage Divider





	

capacitive voltage divider


Consider the two capacitors, C1 and C2 connected in series across an
alternating supply of 10 volts. As the two capacitors are in series, the
charge Q on them is the same, but the voltage across them will be different
and related to their capacitance values, as V = Q/C.


Voltage divider circuits may be constructed from reactive components just as
easily as they may be constructed from resistors as they both follow the
voltage divider rule. Take this capacitive voltage divider circuit, for
instance.

The voltage across each capacitor can be calculated in a number of ways. One
such way is to find the capacitive reactance value of each capacitor, the
total circuit impedance, the circuit current and then use them to calculate
the voltage drop, for example:



Example No1
Using the two capacitors of 10uF and 22uF in the series circuit above,
calculate the rms voltage drops across each capacitor when subjected to a
sinusoidal voltage of 10 volts rms at 80Hz.

Capacitive Reactance of 10uF capacitor
	
	capacitive reactance of 10uF

Capacitive Reactance of 22uF capacitor
	
	capacitive reactance of 22uF

 Total capacitive reactance of series circuit – Note that reactance’s in
series are added together just like resistors in series.
	
	capacitive reactance of circuit
or:
	
	total capacitive reactance

Circuit current
	
	circuit current
 

 Then the voltage drop across each capacitor in series capacitive voltage
divider will be:
	
	capacitive divider voltage drop
 

When the capacitor values are different, the smaller value capacitor will
charge itself to a higher voltage than the larger value capacitor, and in
our example above this was 6.9 and 3.1 volts respectively. Since
Kirchhoff’s voltage law applies to this and every series connected
circuit, the total sum of the individual voltage drops will be equal in
value to the supply voltage, VS and 6.9 + 3.1 does indeed equal 10 volts.

Note that the ratios of the voltage drops across the two capacitors
connected in a series capacitive voltage divider circuit will always remain
the same regardless of the supply frequency. Then the two voltage drops of
6.9 volts and 3.1 volts above in our simple example will remain the same
even if the supply frequency is increased from 80Hz to 8000Hz as shown.



Capacitive Voltage Divider Example No2
Using the same two capacitors, calculate the capacitive voltage drop at
8,000Hz (8kHz).
	
	capacitive voltage divider drop

While the voltage ratios across the two capacitors may stay the same, as the
supply frequency increases, the combined capacitive reactance decreases, and
therefore so too does the total circuit impedance. This reduction in
impedance causes more current to flow. For example, at 80Hz we calculated
the circuit current above to be about 34.5mA, but at 8kHz, the supply
current increased to 3.45A, 100 times more. Therefore, the current flowing
through a capacitive voltage divider is proportional to frequency or I ∝
ƒ.

We have seen here that a capacitor divider is a network of series connected
capacitors, each having a AC voltage drop across it. As capacitive voltage
dividers use the capacitive reactance value of a capacitor to determine the
actual voltage drop, they can only be used on frequency driven supplies and
as such do not work as DC voltage dividers. This is mainly due to the fact
that capacitors block DC and therefore no current flows.

Capacitive voltage divider circuits are used in a variety of electronics
applications ranging from Colpitts Oscillators, to capacitive touch
sensitive screens that change their output voltage when touched by a persons
finger, to being used as a cheap substitute for mains transformers in
dropping high voltages such as in mains connected circuits that use low
voltage electronics or IC’s etc.

Because as we now know, the reactance of both capacitors changes with
frequency (at the same rate), so the voltage division across a capacitive
voltage divider circuit will always remain the same keeping a steady voltage
divider.