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The electrical voltage, the resistance of a conductor, or the current passing through a wire are quantities that can be easily measured by using a tester. But if you need to know the capacity of a handmade capacitor or one whose plate data is not read, then you need another measuring instrument, the “capacitance meter,” which is usually expensive. There are many methods for measuring an unknown capacity value of any degree of difficulty and precision. Let’s see how these two electrical quantities can be easily measured and with the help of theory.

**Capacitors in sinusoidal alternating voltage**

When we apply a direct voltage to a capacitor, when the transient is exhausted, it behaves like an open circuit. When a capacitor is instead subjected to a sinusoidal regime, it no longer behaves like an open circuit, but it begins to absorb current, presenting a “capacitive reactance” expressed in ohms. The component is similar to an electrical resistance. By using this principle, we can easily calculate the value of the unknown capacitor, remembering that the formula of its reactance is:

Xc = 1 ÷ 2πfC

If the capacitor is subjected to a sinusoidal periodic signal, with some measures and some equations, we can calculate the value of its capacitance.

**Capacitors with square-wave voltage**

Capacitors with a square wave behave differently. There is no capacitive reactance with a square wave. The concept of reactance itself depends on the presence of a sinusoidal signal. Because a square-wave signal is the sum of infinite sine waves, the reactances of the sinusoids at different frequencies cannot be added significantly. Because the (ideal) capacitors are linear, we can decompose the square wave into sinusoidal components, find the associated sinusoidal voltage for each component, and then add the voltages to find the total voltage. However, this measurement is very complex and it is advisable to change strategy and measure their capacitive value in an alternative way.

**Strategy used**

To measure the capacitance of a capacitor, we use a simple method: We generate a square wave with an oscillator composed of a CD40106 inverting logic gate and an RC network. By changing the value of C (unknown), a different frequency is obviously obtained. It is sufficient to perform a “curve fitting” of the values to find a good formula that describes the relationship between the frequency produced and the value of the capacitor to be revealed.

**The electrical schematics**

Here are two different solutions with two electric schematics. The first diagram is dedicated to those who have a frequency meter and can measure the frequency by this instrument. It is much simpler and needs few electronic components. On the other hand, the second wiring diagram is intended for those who do not have a frequency meter but a simple tester, even a cheap one. The scheme is therefore similar to the first but uses an additional frequency/voltage converter to read the values on a normal tester.

**First wiring diagram for those who have a frequency meter**

The first wiring diagram is simpler, and it is shown in *Figure 1*. The heart is represented by the integrated circuit CD40106, which, together with C1 and R1, generates a periodic square-wave signal. The frequency is determined by C1 and R1, but because R1 is fixed, it changes in proportion to the unknown capacitor. The first logic gate (X1) generates the signal and the second gate (X2) works as an impedance buffer. In this way, any load connected to its output does not change the frequency or the amplitude of the signal produced. The latter is available on resistance R2, ready to be measured, in frequency, with a frequency meter.

*Figure 2* shows the graphs of the signals at these points on the circuit:

- Blue graph (V2) of the signal on the capacitor
- Red graph (V1) of the signal at the output of the first inverter
- Green graph (V3) of the square signal at the output of the second inverter

**Scale 1 pF/100 nF**

The table below contains all the theoretical frequency values, measured by changing only the capacitor C1. For this range of measurement, between 1 pF and 100 nF, the resistance R1 must be 470 k. The relationship graph is shown in *Figure 3*.

pF | Hertz |
---|---|

1 | 277,393 |

5 | 193,611 |

10 | 140,449 |

22 | 84,667 |

47 | 46,419 |

100 | 23,706 |

220 | 11,245 |

470 | 5,367 |

1,000 (1 nF) | 2,542 |

3,300 (3.3 nF) | 775 |

4,700 (4.7 nF) | 544 |

10,000 (10 nF) | 256 |

22,000 (22 nF) | 116 |

47,000 (47 nF) | 54 |

100,000 (100 nF) | 25 |

For this range of values, the two formulas that describe the relationship between capacitance and frequency are shown in *Figure 4*. These are two very complex formulas, obtained from an advanced process of non-linear curve fitting.

**100-nF/100-µF scale**

The table below contains all the measured theoretical frequency values, replacing the capacitor C1. For this measuring range, between 100 nF and 100 µF, the resistance R1 must be 470 Ω. The relationship graph is shown in *Figure 5*.

µF | Hertz |
---|---|

0.1 | 19,139 |

0.5 | 3,540 |

1 | 1,768 |

2.2 | 804 |

4.7 | 376 |

10 | 176 |

22 | 80 |

47 | 37 |

100 | 17 |

For this range of values, the two formulas that describe the relationship between capacitance and frequency are shown in *Figure 6*.

*Figure 7* shows the simple wiring between the square-wave generator circuit and a frequency meter. It is important for the measuring instrument to be able to read the frequency of a periodic square or rectangular wave signal.

**Second wiring diagram for those who have only a tester**

Users having only a tester can implement the second solution. An additional circuit connected to the first converts the output frequency into a negative voltage, which can be measured by a common tester. The new circuit to be connected to the previous one is a pulse repetition rate meter with a “pump” diode. The entire system (see *Figure 8*) allows us to obtain a negative voltage depending on the capacitance C1 to be measured.

The positive impulse loads at maximum voltage C2 through D1. In the interval between the pulses, with the input at 0 V, C2 discharges quickly through D2 to the large capacitor C3. The output voltage is therefore proportional to the speed of the received pulses. The condenser C3 is similar to a large tank, which is slowly emptied by R3. The following table contains the data collected from different measurements with different capacitance values for C1. The values refer to a capacitance between 100 nF and 100 µF. To obtain a stable voltage value, it is necessary to wait a few seconds for the transient, as also shown in *Figure 9*.

µF | Voltage on R3 for the tester (mV) |
---|---|

0.1 (100 nF) | –2,655 mV |

0.47 (470 nF) | –1,185 mV |

1 | –663.70 mV |

2.2 | –334.43 mV |

4.7 | –165.62 mV |

10 | –80.36 mV |

22 | –37.16 mV |

47 | –17.48 mV |

100 | –8.21 mV |

For this range of values, the formula that describes the relationship between the capacitance and the output voltage is shown in *Figure 10*.

*Figure 11* shows the wiring between the square-wave generator circuit, the frequency/voltage converter, and a normal tester configured in VDC mode. It is an extremely simple connection that requires the construction of the system within a simple PCB.

**Conclusion**

The measurements presented in the article are related to the simulations of the various SPICE models. It is advisable to collect data on your real circuit. The user can freely create his own mathematical models with reference to the capacitance intervals needed, also depending on the waiting times of the transients and the RC time constants, which could generate long waits. We recommend trying to change the values of the electronic components according to your needs. If you encounter difficulties in applying the formulas, you can simply consult the table of data collected, and then you can find real empirical data by interpolation. To perform the curve fitting of the data, it is possible to use any mathematical and statistical software with this available option. The main purpose of the article is to demonstrate how electronics and mathematics are closely linked together. The project is open to any modification or improvement for different purposes and functions.