Measuring Capacitance: Methods (fwd)

---------- Forwarded message ----------
Date: Sun, 26 Apr 1998 12:23:58 -0400
From: Thomas McGahee <tom_mcgahee-at-sigmais-dot-com>
To: tesla-at-pupman-dot-com
Cc: Steve Falco <sfalco-at-worldnet.att-dot-net>, Tesla-2 <tesla-2-at-emachine-dot-com>
Subject: Measuring Capacitance: Methods

File this away for future reference.

This FAQ has been prepared by Fr. Tom McGahee 
for use by the Tesla List and the Tesla-2 List. Copy freely.

>From time to time List members ask questions about how to measure
capacitance. This FAQ attempts to show several methods. The info is
drawn from a number of sources, including past posts by Steve Falco
sfalco-at-worldnet.att-dot-net and myself. 

The easiest method is to buy a meter that measures capacitance. You
can sometimes find a Digital Multi Meter (DMM) that includes the ability
to measure capacitance. Make sure that it covers the range that you are
interested in. Most coilers would want a meter that can measure from a 
few picofarads to at least 1 microfarad.

An LCR meter is a meter specifically designed to measure Inductance,
Capacitance, and Resistance. In general these meters cover a greater range
than the DMMs and also have a higher accuracy. They also generally cost

Some companies sell meters that ONLY measure capacitance. An LCR meter
is generally more useful to a coiler. 

You can often find such meters at hamfests and swap meets, and at very
reasonable prices. When I go looking for such meters I always carry
an assortment of batteries and capacitors of known value so that I can
test the meters before buying. BTW, a working LCR meter and/or DMM is 
a good thing to bring with you to hamfests to test components before
you buy them!

Companies like Radio Shack and B&K Precision sell reasonably priced
meters that will measure capacitances in the ranges of interest
to us coilers.

For those who do not own a capacitance meter there are several 
methods available that will allow you to measure capacitance. Some of
these methods are those used in professional capacitance meters, and
others are simply el-cheapo techniques that a coiler on a limited
budget can use to quickly find the approximate value of their
homemade capacitor.

1) You can measure the RC time constant.

     Where: T is the time in seconds required to charge to 63% 
     of the battery voltage, and
     R is resistance in ohms, and 
     C is capacitance in farads. 

Note that if you measure resistance in megohms, then the capacitance
will come out in microfarads. 

Re-arrange the formula so that 


charge your cap through a resistor, and note how long it takes the 
voltage to build from 0 volts to 63% of the battery voltage.  

Use the formula C=T/R to determine the capacitance.

The circuit might look like this:

     ----------- Switch ------ Resistor --------------------
     |                                        |            |
   Battery                                 capacitor      voltmeter
     |                                        |            |

Note that the meter used should have as high an impedance as possible
to avoid loading down the circuit. In most cases you would probably be
using a 10 Megohm DMM. You can increase the effective input impedance of
the meter by using an operational amplifier connected as a Buffer
in front of the meter. You can use an LF356 for this purpose.

Start with the switch open, and the voltmeter showing 0 volts (short out
the cap for a few seconds to discharge it.  Then close the switch, start
your stopwatch, and wait till the voltmeter hits battery voltage x 0.63.

Sadly, this only works well for large capacitors like electrolytics,
because you can do the timing with a watch.  But for small capacitors
such as are used in tesla coils, you need to use a storage oscilloscope
or electronic timer/counter to measure the time 'cause it will be in
milliseconds or even microseconds.

And don't think you can get around this by making the resistor large,
like maybe a few hundred megohms, because then the leakage in your
voltmeter or o'scope will bias the readings.  (The resistor must be no
more than say 10% of the impedence of your test equipment.)

2) You can improve on method (1) by charging the capacitor with a
constant current. In this case
     Where: C is in farads
     I is in amps, and
     T is time in seconds to charge to 1 volt.

>From a practical point of view we normally want to measure microfarads,
so we would use microamps.

This FAQ will not give details of the constant current source and
means for measuring time, but the author (Fr. Tom McGahee) published
an article in Popular Electronics, October 1976, that gives full
construction details and theory of operation of a unit suitable for
use by coilers. This unit used an analog meter for the output, but
a digital meter can be substituted for greater accuracy. Since the
article was published, much better operational amplifiers have become
readily available. I recommend substuting the 741 op amp with
an LF356 or other pin-compatible device.

A method for measuring capacitance digitally using constant current
charging was published by the author (Fr. Tom Mcgahee) in the January 23,
1975 issue of Electronics magazine, page 103. Basically a window
comparator is used to produce a pulse whose width is proportional
to capacitance. This pulse is then used to gate an oscillator's signal
into a counter. The counter will display the result digitally. I have
attained useful accuracies of more than .01% with this method.

A further method that is based on constant current charging is to create
a triangle wave generator using an operational amplifier connected as
an integrator and driving an RS Flip Flop triggered by two different
voltage levels from the output of the integrator. If the output of the
Flip Flop in turn controls the input current(s) to the integrator, then the
PERIOD of the output of the Flip Flop will be *directly* proportional
to the capacitance. The author has used the above technique to design
automatic digital capacitance meters that provide updated readings
hundreds of times per second, with an accuracy of better than .01%.
3) You can also build a simple 555 timer astable oscillator.
You can find such circuits in many hobby electronics books. You
measure the frequency of oscillation with your home-made capacitor,
then substitute known capacitors in the circuit until you get about 
the same frequency.  Then your capacitor and the known capacitor are 
roughly the same value.  This can be pretty accurate, and works 
for the values used in Tesla coils, but you need a means to
measure or compare the frequencies. The resistor in series with the
capacitor can be made variable, and then you can calibrate the
potentiometer position in terms of capacitance. That is better 
than having to match the capacitor exactly, but you do have
to go through a calibration procedure. Potentiometer position
versus cap value for a given frequency may be slightly non-linear.
You can also measure the frequency of the known and unknown
capacitance and determine capacitance based on the fact that the
lower the capacitance, the higher the resulting frequency. I
find it easier to measure the ratio of the PERIODs, as the period
is directly proportional to the capacitance. This method is accurate
to about 1%.

4) The "Guide to Electronic Measurements and Laboratory Practice"
(Stanley Wolf, Prentice-Hall, 1973) says the most accurate way is with a
bridge circuit (until the advent of digital measurement techniques): 
         |                  |             |
         |                 R1             R2
         |                  |             |
         |                  |             |
    AC generator            |--- meter ---|
         |                  |             |
         |                  |             |
         |                 R3            R?
         |                  |             |
         |                 C3            C?
         |                  |             |
The equations are not too bad:
         R? = (R2 x R3) / R1
         C? = C3 x (R1 / R2)
where R1, R2, R3, C3 are permanent parts of the bridge meter, and R? and
C? are your unknown capacitor.  (That R? is confusing, but it represents
the loss or leakage in your capacitor.  If you have a good capacitor, it
will be pretty close to 0.)

Anyway, we make R1 and R3 adjustable potentiometers, and play around
with them until the meter says we have balanced the bridge.  Then we
disconnect the potentiometers and measure the values they were set to
with our ohm-meter, do a little math, and we know our capacitor value.
For best results try to get a C3 with a value very near what you THINK the
value of <C?> is going to be. To further improve the accuracy of the
circuit, you can choose R1 and R2 to be resistors whose resistance is close
to the capacitive reactance of C3. These improvements affect the practical
*resolution* of the answer for <C?>. You will get a greater final accuracy,
because the ranges have been matched.
Also, you can leave out <R3> and <R?> in most cases. Use of R3 can cause
the experimenter to miscalculate <C?> when using just a meter. (meter
should be high impedance or it will only be about 10% accurate). If you
really want to measure the dissipation/leakage, then you need a measurement
of the relative PHASE difference between the two legs of the bridge. With a
differential oscilloscope you adjust R1 and R3 until the scope shows only a
dot in the xy mode. At that point you have all the data to calculate both
capacitance and leakage. (This assumes that the reference capacitor has
little or no leakage of its OWN, of course!)
It is much less hassle to just leave <R3> out and ignore the leakage value,
as it is not all that important to us coilers. 
5) Cheapest way: you can make a voltage divider:
                   |                              |
                   |                            resistor
                   |                              |
        120 volt AC (wall plug)                   | 
                   |                              |
                   |                            capacitor
                   |                              |
Measure the voltage across the resistor and the capacitor.  Don't get
zapped - you are playing with line voltages here and there is not much
between you and eternity.
    C = (Vr / R) / (2 x pi x f x Vc)

where Vr is the voltage across the resistor, R is the resistor in ohms,
pi is 3.141, f is 60 Hertz (or 50 in some countries - you know who you
are), and Vc is the voltage across the capacitor.

Please note that you have to actually measure Vr and Vc. You can
*not* assume that Vr+Vc=120 since there is a current phase shift
through the capacitor. 

By using the existing 120 VAC line voltage we let the power company
supply us with a rock-solid 60Hz low impedance voltage source. For
So there you are: measure two voltages, do a little math, and get a
result accurate to maybe 10%.  You need a fairly high impedence meter to
read the voltages. I recommend a DMM with a 10 Megohm impedance
if at all possible. Lower impedance meters will introduce HUGE
errors when the capacitance is small.

Pick a resistor such that Vr is somewhere around 50 volts for reasonable
accuracy.  You don't want Vr to be 1 volt or 119 volts - get it somewhere in
middle by chosing a suitable value for R.
When done making the measurements, remember to always DISCHARGE the
capacitor, as it MAY have a large voltage stored in it!

6) There is one more way that uses even simpler math. It uses two capacitors
instead of a capacitor and a resistor.

                     |                              |
                     |                            Known Capacitor           
          120 volt AC (wall plug)                   | 
                     |                              |
                     |                            Unknown Capacitor
                     |                              |
This method makes use of the fact that the smaller the capacitor value, the
greater the proportion of total voltage a given capacitor will drop across

It helps if you choose a known value that is close to the one you want to

CK=Known Capacitor  CU=Unknown Capacitor  EK= Voltage across Known
EU=Voltage across Unknown Capacitor

Formula is based on the fact that EK*CK=EU*CU  Rearranging it in terms of
CU we get:


Always use a high impedance meter, such as a digital unit or a tube VTVM to
keep errors low. As in method (5), measure each voltage with the meter.
Do not assume that the total will be 120 volts. Among other things, the
actual voltage deliverd at your wallplug may be anything from 105 to 125
volts. Or even 210 to 240 for some of our European coilers.

When done making the measurements, remember to always DISCHARGE *each* of
the capacitors separately (NOT in series), to ensure that each gets fully
discharged. If you remove the caps when the voltage is at its peak, you
can have some nasty voltages present!

Several of the techniques outlined above involve using the voltage
from your wall outlet. PLEASE BE CAREFUL! 

If substituting a signal generator for your voltage source, be aware that
the impedance of the signal generator may introduce significant errors.

7) I can think of at least one more method, which is based on the fact that
the larger a capacitor is, the greater the amount of current it will
conduct *at a given frequency*. This method assumes that the applied
frequency is a SINE WAVE, well below the frequency limit of your digital
meter, and that your digital meter has an AC current range. Some don't!

For a *given* sine wave frequency  C1/I1=C2/I2 so we re-arrange and get:

The frequency chosen is arbitrary, but I like something between 1KHz and
about 1/2 the frequency limit of my meter. The higher the frequency you
use, the smaller the capacitance you can accurately measure.

Again, try to use as a reference a value that is close to what you are
measuring. That keeps the resolution of the answer higher. Assume you want
to measure a capacitor that you think is pretty close to .01 mfd.  First
connect a *sine wave* generator in series with the known capacitor and an AC
current meter. Adjust the output of the generator to about ten volts, and
then put the meter on the scale that gives you the most resolution of the
current value. You may have to play with the frequency to find one that
works well, but stay below the frequency limit of your meter! Then, leaving
ALL things exactly the same, exchange capacitors and use the reference
capacitor. Record the new reading and apply the math to find the unknown
capacitance. Note that if your signal generator has a large impedance, then
you will have a large error unless the two capacitors are close in value...
then the error is effectively cancelled.

I don't recommend D'arsonval meters because they are not linear with
frequency. A digital meter works best. The value of the internal current
shunt will add less than .1% error to your final calculated value, provided
you made your measurement with the scale that yielded the largest number of
significant digits on the readout. In other words 1.234 is better than
01.23  or 001.2

With careful attention to detail you can get a value within 5% or better.

Hope this helps.
Fr. Tom McGahee