Capacitance is a basic electrical property

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Capacity is a common electrical property that can be used in circuits to perform certain functions that include storing and unloading of charge. Capacity estimation is subject to a variety of considerations and requires a sophisticated instrument. In this experiment, the capacitance is determined using a digital multimeter and the measured capacitance is used to calculate the thickness of the dielectric material within the two parallel condenser plates. The key object of the experiment is to match the experimentally obtained value of the thickness of the dielectric material with that suggested by the producer.
Objective
The objective of the experiment is to observe the variance of capacitance of six electronic devices with the applied voltages and to find apparent dielectric thickness associated with the capacitance found at different voltage level. It further explores the level of voltage when the capacitance becomes constant.

Background Theory

Capacitance is an electrical property of materials or devices that characterizes the ability to store electrical charges. Capacitors are devices designed to store and release charges in and from electric field (Floyd). Two conductive materials separated by a dielectric material can form a capacitance. In parallel plate capacitors, two parallelly set conductive plates are used with a dielectric material between them. The value of a capacitance can be approximated by following equation:

C = ——– (i) (Kaiser)

Here,

C = Capacitance in Farads

= Permittivity of free space =8.854×10-12 Farad/ meter

= Relative permittivity of the dielectric medium

A = Cross sectional area of the plates in square meter

d = Thickness of the dielectric medium in meter

Rearranging equation (i), we get,

d = —– (ii)

Equation (ii) can give the thickness of the dielectric material used in the capacitor if capacitance, relative permittivity of the dielectric material and the area of cross section of the conductive plates are known.

The relationship between the capacitance of a capacitor and voltage across it, is given by,

C = ——- (iii) (Smith and Alley)

It is obvious that the relationship is inverse, which means the more is the capacitance the less is the voltage across it and vice versa. Since the amount of charge flowing is the product of current and time of flow. We can write the above equation as: C = —– (iv)

With the application of a voltage across the plates of a capacitor, charge accumulates across the plates exponentially i.e. at the beginning, small amount of charge accumulates, which goes on increasing with time and after a certain time, the charges across the capacitor plates rises very quickly and reach a point of maximum. Beyond that maximum point, charge does not increase despite increase. The charging phase is assumed to be complete at this point in time. The reason is obvious; at the beginning the charges start moving slowly. As the time progresses, charges accelerate and hit and release more electrons from other atoms. This increases charge flow. After a certain period of time, the accumulation of charges in the two plates of the capacitor creates a voltage difference between them, which is in direct opposition to the applied voltage (Boylestad). When this opposing voltage across the plates of the capacitor approaches a value closer to the applied voltage, the charge accumulation and release get slower until practically stop at some point. The exponential growth of charge is given by the following equation:

q = q0 (1- )

where,

q = Accumulated charge across the capacitor plates,

q0 = Initial charge,

t = time,

τ = time constant = Product of Capacitance and Resistance = R× C

The exponential growth can be shown in the figure below:

Figure 1: Exponential Growth of Charges across Capacitor Plates

Equipment

Voltmeter

Digital multimeter

Voltage supply

Method

At first the capacitor and the voltage supply is connected in series. For the measurement of the capacitance, the multimeter is connected in parallel to the capacitor.

Figure 2: Circuit Setup

The voltage supply can supply variable voltages. At first the voltage will set to 0 volt and the capacitance will be measured. With gradual increase in voltages, the capacitance will be measured and the record will be listed in a workbook. At voltage level of 50 Volt the experiment ends.

Predictions

According to the supplier data,

Relative permittivity of the dielectric material, = 12.9

Diameter of each circular plate of four devices, D = 200 µm

Area of cross section for first four devices, A = π × () ² square meter

= 3.14159 × ()² meter²

= 3.1415 × 10-4 Square meter

Dielectric material thickness, d= 10 µm

The capacitance, C=

= F

= 0.358821728 pF

Following table exhibits the theoretical values for the four devices with cross sectional diameter of 200 µm ± 5 µm.

Table 1: Theoretical values of capacitance for first four devices

Diameter (m)

Area of Cross Section

Permittivity of free space

Relative Permittivity of dielectric material

Thickness of dielectric material (m)

Capacitance (pF)

0.000195

2.98647E-08

8.85E-12

12.8

0.000010

0.338460681

0.000200

3.14159E-08

8.85E-12

12.8

0.000010

0.356040165

0.000205

3.30063E-08

8.85E-12

12.8

0.000010

0.374064698

0.000195

2.98647E-08

8.85E-12

12.9

0.000010

0.341104906

0.000200

3.14159E-08

8.85E-12

12.9

0.000010

0.358821728

0.000205

3.30063E-08

8.85E-12

12.9

0.000010

0.376987078

0.000195

2.98647E-08

8.85E-12

13.0

0.000010

0.34374913

0.000200

3.14159E-08

8.85E-12

13.0

0.000010

0.361603292

0.000205

3.30063E-08

8.85E-12

13.0

0.000010

0.379909459

Following table exhibits the theoretical values for the four devices with cross sectional diameter of 400 µm ± 5 µm.

Diameter (m)

Area of Cross Section

Permittivity of free space

Relative Permittivity of dielectric material

Thickness of dielectric material (m)

Capacitance (pF)

0.000395

1.22542E-07

8.85E-12

12.8

0.000010

1.388779167

0.000400

1.25664E-07

8.85E-12

12.8

0.000010

1.424160658

0.000405

1.28825E-07

8.85E-12

12.8

0.000010

1.4599872

0.000395

1.22542E-07

8.85E-12

12.9

0.000010

1.399629004

0.000400

1.25664E-07

8.85E-12

12.9

0.000010

1.435286914

0.000405

1.28825E-07

8.85E-12

12.9

0.000010

1.47139335

0.000195

2.98647E-08

8.85E-12

13.0

0.000010

0.34374913

0.000200

3.14159E-08

8.85E-12

13.0

0.000010

0.361603292

0.000205

3.30063E-08

8.85E-12

13.0

0.000010

0.379909459

Answer to Question No 2

Introduction

The thickness of the dielectric material used in a capacitor is an important factor to determine the quality of the capacitor. This experiment is designed to measure the capacitance of six individual electronic devices in which circular gold plate materials have been used as the dielectric medium. Four of the devices has a dielectric circle of radius 200 µm ± 5 µm and the rest two has a diameter of 400 µm ± 5 µm. The supplier suggests that the devices have a strange property of changing its capacitance with applied voltage, at least at the smaller voltage ranges. The main objective of the experiment is the apply supply voltage, varying from 0 Volt to 50 Volt to observe the change in capacitance due to the inherent property of the electronic devices. Using the formula of parallel plate capacitor, the thickness of the dielectric material will be computed and compared to that of the theoretical values found earlier. However, this report will take the associated uncertainty into consideration.

Experimental Method

The voltage supply is connected across the capacitor in order to supply voltages of different magnitude gradually. For the measurement of the capacitance, the multimeter is connected in parallel to the capacitor.

Figure 3: Circuit Setup

The voltmeter can supply variable voltages. At first the voltage will set to 0 volt and the capacitance will be measured. With gradual increase in voltages, the capacitance will be measured and the record will be listed in a workbook. At voltage level of 50 Volt the experiment ends.

Observations and Measurements

Here, 60 capacitance values have been measured for each of the devices applying 60 different voltages. Ten voltages are applied between 0 to 1 Volt at an increasing step of 0.1 Volt. 50 voltages have been applied between 1 to 50 Volts at an increasing step of 1 Volt. We have 360 capacitance values for a total of six electronic devices and 60 different voltage levels. The table presented below lists all the voltage levels and device capacitance values.

Table 2: Capacitance for different devices measured at different voltage levels

Device number:

1

2

3

4

5

6

Diameter (µm):

400

200

200

200

200

400

Voltage applied (V)

Device Capacitance (pF)

Device Capacitance (pF)

Device Capacitance (pF)

Device Capacitance (pF)

Device Capacitance (pF)

Device Capacitance (pF)

0.00

1.49

0.37

0.41

0.39

0.39

1.51

0.10

1.49

0.37

0.41

0.38

0.39

1.50

0.20

1.48

0.37

0.41

0.38

0.39

1.50

0.30

1.48

0.36

0.40

0.38

0.39

1.49

0.40

1.48

0.36

0.40

0.38

0.38

1.49

0.50

1.48

0.36

0.40

0.38

0.38

1.49

0.60

1.47

0.36

0.40

0.38

0.38

1.48

0.70

1.47

0.36

0.40

0.38

0.38

1.48

0.80

1.47

0.36

0.40

0.38

0.38

1.48

0.90

1.47

0.36

0.40

0.37

0.38

1.48

1.00

1.47

0.36

0.40

0.37

0.38

1.48

2.00

1.44

0.35

0.39

0.37

0.37

1.46

3.00

1.43

0.34

0.39

0.36

0.37

1.45

4.00

1.43

0.34

0.39

0.36

0.37

1.44

5.00

1.42

0.34

0.38

0.36

0.37

1.43

6.00

1.42

0.34

0.38

0.36

0.36

1.43

7.00

1.42

0.34

0.38

0.36

0.36

1.43

8.00

1.41

0.34

0.38

0.36

0.36

1.42

9.00

1.41

0.34

0.38

0.36

0.36

1.42

10.00

1.41

0.34

0.38

0.36

0.36

1.42

11.00

1.41

0.34

0.38

0.36

0.36

1.42

12.00

1.41

0.34

0.38

0.36

0.36

1.42

13.00

1.41

0.34

0.38

0.36

0.36

1.42

14.00

1.41

0.34

0.38

0.36

0.36

1.42

15.00

1.41

0.34

0.38

0.36

0.36

1.42

16.00

1.41

0.34

0.38

0.36

0.36

1.42

17.00

1.41

0.34

0.38

0.36

0.36

1.42

18.00

1.41

0.34

0.38

0.36

0.36

1.41

19.00

1.40

0.34

0.38

0.36

0.36

1.41

20.00

1.40

0.34

0.38

0.36

0.36

1.41

21.00

1.40

0.34

0.38

0.36

0.36

1.41

22.00

1.40

0.34

0.38

0.36

0.36

1.41

23.00

1.40

0.34

0.38

0.36

0.36

1.41

24.00

1.40

0.34

0.38

0.36

0.36

1.41

25.00

1.40

0.34

0.38

0.36

0.36

1.41

26.00

1.40

0.34

0.38

0.36

0.36

1.41

27.00

1.40

0.34

0.38

0.36

0.36

1.41

28.00

1.40

0.34

0.38

0.36

0.36

1.41

29.00

1.40

0.34

0.38

0.36

0.36

1.41

30.00

1.40

0.34

0.38

0.36

0.36

1.41

31.00

1.40

0.34

0.38

0.36

0.36

1.41

32.00

1.40

0.34

0.38

0.36

0.36

1.41

33.00

1.40

0.34

0.38

0.36

0.36

1.41

34.00

1.40

0.34

0.38

0.36

0.36

1.41

35.00

1.40

0.34

0.38

0.36

0.36

1.41

36.00

1.40

0.34

0.38

0.36

0.36

1.41

37.00

1.40

0.34

0.38

0.36

0.36

1.41

38.00

1.40

0.34

0.38

0.36

0.36

1.41

39.00

1.40

0.34

0.38

0.36

0.36

1.41

40.00

1.40

0.34

0.38

0.36

0.36

1.41

41.00

1.40

0.34

0.38

0.36

0.36

1.41

42.00

1.40

0.34

0.38

0.36

0.36

1.41

43.00

1.40

0.34

0.38

0.36

0.36

1.41

44.00

1.40

0.34

0.38

0.36

0.36

1.41

45.00

1.40

0.34

0.38

0.36

0.36

1.41

46.00

1.40

0.34

0.38

0.36

0.36

1.41

47.00

1.40

0.34

0.38

0.36

0.36

1.41

48.00

1.40

0.34

0.38

0.36

0.36

1.41

49.00

1.40

0.34

0.38

0.36

0.36

1.41

50.00

1.40

0.34

0.38

0.36

0.36

1.41

It is obvious from the tabular data that the capacitance values change rapidly at voltage levels up to 20 Volts and becomes constant afterwards. Again, devices of same circular dielectric plates of equal dimeter exhibit different capacitance values at the same level of voltages. For devices with 400 µm dimeter of circular plate dielectric, the capacitance values are higher than those with 200 µm diameter of circular plate dielectric.

Calculations

The apparent thicknesses of the dielectric materials can be measured using the formula of thickness presented in equation (ii), which can be written as:

d =

Let us calculate the thickness of dielectric material for the first 200 μm diameter device with a dielectric permittivity of 12.9.

Here,

Permittivity of the free space, = 8.854×10-12 F/m

Relative permittivity of the dielectric material, = 12.9 ± 0.1

Area of cross section, A = π × (D/2) ²

= 3.14159 × ()² meter²

= 3.1415 × 10-4 Square meter

Capacitance, C= 0.37 pF

Dielectric thickness, d =

=

= m

Using the same formula and procedure in excel worksheet, the dielectric thickness is calculated for all devices at all voltage levels and is presented in the table below:

Device number:

1

2

3

4

5

6

Diameter (µm):

400

200

200

200

200

400

Voltage applied (V)

Thickness (m)

Thickness (m)

Thickness (m)

Thickness (m)

Thickness (m)

Thickness (m)

0.00

9.6328E-18

9.69788E-18

8.75175E-18

9.20056E-18

9.20056E-18

9.50521E-18

0.10

9.6328E-18

9.69788E-18

8.75175E-18

9.44268E-18

9.20056E-18

9.56858E-18

0.20

9.69788E-18

9.69788E-18

8.75175E-18

9.44268E-18

9.20056E-18

9.56858E-18

0.30

9.69788E-18

9.96727E-18

8.97054E-18

9.44268E-18

9.20056E-18

9.6328E-18

0.40

9.69788E-18

9.96727E-18

8.97054E-18

9.44268E-18

9.44268E-18

9.6328E-18

0.50

9.69788E-18

9.96727E-18

8.97054E-18

9.44268E-18

9.44268E-18

9.6328E-18

0.60

9.76386E-18

9.96727E-18

8.97054E-18

9.44268E-18

9.44268E-18

9.69788E-18

0.70

9.76386E-18

9.96727E-18

8.97054E-18

9.44268E-18

9.44268E-18

9.69788E-18

0.80

9.76386E-18

9.96727E-18

8.97054E-18

9.44268E-18

9.44268E-18

9.69788E-18

0.90

9.76386E-18

9.96727E-18

8.97054E-18

9.69788E-18

9.44268E-18

9.69788E-18

1.00

9.76386E-18

9.96727E-18

8.97054E-18

9.69788E-18

9.44268E-18

9.69788E-18

2.00

9.96727E-18

1.0252E-17

9.20056E-18

9.69788E-18

9.69788E-18

9.83073E-18

3.00

1.0037E-17

1.05536E-17

9.20056E-18

9.96727E-18

9.69788E-18

9.89853E-18

4.00

1.0037E-17

1.05536E-17

9.20056E-18

9.96727E-18

9.69788E-18

9.96727E-18

5.00

1.01077E-17

1.05536E-17

9.44268E-18

9.96727E-18

9.69788E-18

1.0037E-17

6.00

1.01077E-17

1.05536E-17

9.44268E-18

9.96727E-18

9.96727E-18

1.0037E-17

7.00

1.01077E-17

1.05536E-17

9.44268E-18

9.96727E-18

9.96727E-18

1.0037E-17

8.00

1.01793E-17

1.05536E-17

9.44268E-18

9.96727E-18

9.96727E-18

1.01077E-17

9.00

1.01793E-17

1.05536E-17

9.44268E-18

9.96727E-18

9.96727E-18

1.01077E-17

10.00

1.01793E-17

1.05536E-17

9.44268E-18

9.96727E-18

9.96727E-18

1.01077E-17

11.00

1.01793E-17

1.05536E-17

9.44268E-18

9.96727E-18

9.96727E-18

1.01077E-17

12.00

1.01793E-17

1.05536E-17

9.44268E-18

9.96727E-18

9.96727E-18

1.01077E-17

13.00

1.01793E-17

1.05536E-17

9.44268E-18

9.96727E-18

9.96727E-18

1.01077E-17

14.00

1.01793E-17

1.05536E-17

9.44268E-18

9.96727E-18

9.96727E-18

1.01077E-17

15.00

1.01793E-17

1.05536E-17

9.44268E-18

9.96727E-18

9.96727E-18

1.01077E-17

16.00

1.01793E-17

1.05536E-17

9.44268E-18

9.96727E-18

9.96727E-18

1.01077E-17

17.00

1.01793E-17

1.05536E-17

9.44268E-18

9.96727E-18

9.96727E-18

1.01077E-17

18.00

1.01793E-17

1.05536E-17

9.44268E-18

9.96727E-18

9.96727E-18

1.01793E-17

19.00

1.0252E-17

1.05536E-17

9.44268E-18

9.96727E-18

9.96727E-18

1.01793E-17

20.00

1.0252E-17

1.05536E-17

9.44268E-18

9.96727E-18

9.96727E-18

1.01793E-17

21.00

1.0252E-17

1.05536E-17

9.44268E-18

9.96727E-18

9.96727E-18

1.01793E-17

22.00

1.0252E-17

1.05536E-17

9.44268E-18

9.96727E-18

9.96727E-18

1.01793E-17

23.00

1.0252E-17

1.05536E-17

9.44268E-18

9.96727E-18

9.96727E-18

1.01793E-17

24.00

1.0252E-17

1.05536E-17

9.44268E-18

9.96727E-18

9.96727E-18

1.01793E-17

25.00

1.0252E-17

1.05536E-17

9.44268E-18

9.96727E-18

9.96727E-18

1.01793E-17

26.00

1.0252E-17

1.05536E-17

9.44268E-18

9.96727E-18

9.96727E-18

1.01793E-17

27.00

1.0252E-17

1.05536E-17

9.44268E-18

9.96727E-18

9.96727E-18

1.01793E-17

28.00

1.0252E-17

1.05536E-17

9.44268E-18

9.96727E-18

9.96727E-18

1.01793E-17

29.00

1.0252E-17

1.05536E-17

9.44268E-18

9.96727E-18

9.96727E-18

1.01793E-17

30.00

1.0252E-17

1.05536E-17

9.44268E-18

9.96727E-18

9.96727E-18

1.01793E-17

31.00

1.0252E-17

1.05536E-17

9.44268E-18

9.96727E-18

9.96727E-18

1.01793E-17

32.00

1.0252E-17

1.05536E-17

9.44268E-18

9.96727E-18

9.96727E-18

1.01793E-17

33.00

1.0252E-17

1.05536E-17

9.44268E-18

9.96727E-18

9.96727E-18

1.01793E-17

34.00

1.0252E-17

1.05536E-17

9.44268E-18

9.96727E-18

9.96727E-18

1.01793E-17

35.00

1.0252E-17

1.05536E-17

9.44268E-18

9.96727E-18

9.96727E-18

1.01793E-17

36.00

1.0252E-17

1.05536E-17

9.44268E-18

9.96727E-18

9.96727E-18

1.01793E-17

37.00

1.0252E-17

1.05536E-17

9.44268E-18

9.96727E-18

9.96727E-18

1.01793E-17

38.00

1.0252E-17

1.05536E-17

9.44268E-18

9.96727E-18

9.96727E-18

1.01793E-17

39.00

1.0252E-17

1.05536E-17

9.44268E-18

9.96727E-18

9.96727E-18

1.01793E-17

40.00

1.0252E-17

1.05536E-17

9.44268E-18

9.96727E-18

9.96727E-18

1.01793E-17

41.00

1.0252E-17

1.05536E-17

9.44268E-18

9.96727E-18

9.96727E-18

1.01793E-17

42.00

1.0252E-17

1.05536E-17

9.44268E-18

9.96727E-18

9.96727E-18

1.01793E-17

43.00

1.0252E-17

1.05536E-17

9.44268E-18

9.96727E-18

9.96727E-18

1.01793E-17

44.00

1.0252E-17

1.05536E-17

9.44268E-18

9.96727E-18

9.96727E-18

1.01793E-17

45.00

1.0252E-17

1.05536E-17

9.44268E-18

9.96727E-18

9.96727E-18

1.01793E-17

46.00

1.0252E-17

1.05536E-17

9.44268E-18

9.96727E-18

9.96727E-18

1.01793E-17

47.00

1.0252E-17

1.05536E-17

9.44268E-18

9.96727E-18

9.96727E-18

1.01793E-17

48.00

1.0252E-17

1.05536E-17

9.44268E-18

9.96727E-18

9.96727E-18

1.01793E-17

49.00

1.0252E-17

1.05536E-17

9.44268E-18

9.96727E-18

9.96727E-18

1.01793E-17

50.00

1.0252E-17

1.05536E-17

9.44268E-18

9.96727E-18

9.96727E-18

1.01793E-17

Discussion and Interpretation

A plot between the device capacitance and voltage levels is presented below:

Figure 4: Capacitance Vs. Voltage

The plot indicates that the value of capacitance drops rapidly at lower voltages and becomes fairly constant after 20 Volts. This is the cause for getting varying thickness of dielectric material at lower voltage levels. However, it is not possible to have various thickness of the same dielectric material inside a capacitor. Therefore, we can assume that the apparent thickness value is coming a result of non-saturation across the capacitor plates. In other words, a lower voltage, the measured capacitance is lower than the actual value because the applied voltage cannot saturate the plates of the capacitor using the charge accumulated. On the other hand, as the voltage rises, more charges gather across the plates and create a greater potential difference between them, which is reflected in higher values of capacitance. After a certain value of applied voltage, the saturated plates cannot gather anymore charges and the capacitance becomes constant irrespective of increased level of voltages.

Conclusion

The thickness of the dielectric material is measured using the capacitance value. As seen in the experiment, the capacitance value varies with the applied voltage, which may lead to a wrong value of the thickness as well as other capacitance related parameters. In order to get the capacitance value right, it is essential to apply the rated voltage of the capacitor. The rated value can be obtained from the device datasheet or the value inscribed on the body of the capacitor.

Works Cited

Boylestad, Robert L. Introductory Circuit Analysis. Pearson Education, 2016.

Floyd, Thomas L. Electronics Fundamentals: Circuits, Devices, and Applications. Pearson/Prentice Hall, 2004.

Kaiser, Cletus J. The Capacitor Handbook. CJ Publishing, 1995.

Smith, K. A., and R. E. Alley. Electrical circuits: an introduction. Cambridge University Press,

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