Capacitors

Capacitance ($C$) is defined as the ratio of the charge ($Q$) stored on the capacitor to the potential difference ($V$) across its plates:

$C = \frac{Q}{V}$

This means the amount of charge stored increases as the voltage across the plates increases.

Units of Capacitance

The SI unit of capacitance is the Farad (F). Since a farad is a very large unit, capacitors are usually measured in: Microfarads ($\mu\text{F} = 10^{-6} \, \text{F}$) Nanofarads ($\text{nF} = 10^{-9} \, \text{F}$) Picofarads ($\text{pF} = 10^{-12} \, \text{F}$)

Definition of One Farad: A capacitor has a capacitance of 1 Farad if 1 coulomb of charge changes its potential by 1 volt.

Example 1: A 3 $\mu\text{F}$ capacitor is connected across a potential difference of 18 V. What is the total charge stored?

Solution:

$Q = C \times V = 3 \times 10^{-6} \, \text{F} \times 18 \, \text{V} = 54 \times 10^{-6} \, \text{C} = 54 \, \mu \text{C}$

Example 2: Calculate the capacitance of a capacitor if it stores a charge of 120 C when the potential difference is 1.5 V.

Solution:

$C = \frac{Q}{V} = \frac{120 \, \text{C}}{1.5 \, \text{V}} = 80 \, \text{F}$

i. Paper or plastic capacitor

Made of two tin foil sheets separated by paper or plastic as the dielectric. The structure is rolled into a cylindrical shape and enclosed in a plastic case.

Application:

1. Used in coupling and decoupling circuits.
2. Common in radio receivers and analog signal processing.

Diagram of Paper capacitor

ii. Mica capacitor

Mica capacitors use mica (a group of natural minerals) as the dielectric. There are two main types:

1. Clamped Mica Capacitors: Now considered obsolete due to poor performance.
2. Silver Mica Capacitors: More reliable and accurate; made by sandwiching a mica sheet coated with metal on both sides. The whole unit is then enclosed in epoxy for environmental protection.

Applications:

1. Used in radio frequency (RF) circuits.
2. Suitable for timing, filtering, and oscillator circuits that require high stability and reliability.
3. Often used where precision and temperature stability are critical.

Diagram of Mica capacitor

iii. Electrolytic capacitor

These capacitors are used when large capacitance values are required. They consist of:

1. A thin metal film acting as one electrode.
2. A semi-liquid electrolyte as the other electrode.
3. A very thin oxide layer (about 10 microns) as the dielectric.

They are polarized (must be connected with the correct polarity).

Applications:

1. Used in power supply filters to smooth out voltage.
2. Found in audio amplifiers to couple and decouple AC signals.
3. Common in low-frequency applications due to their large capacitance.

Diagram of Electrolytic capacitor

iv. Air-filled capacitor

This capacitor consists of two parallel metal plates separated by air. One plate (A) is charged using induction and connected to a gold-leaf electroscope. The second plate (B) is brought close to A (without touching). The presence of plate B decreases the potential of plate A, indicating an increase in capacitance.

Applications:

1. Used in radio tuning circuits and variable capacitors.
2. Suitable for high-frequency applications because of low dielectric losses.
3. Used in experimental setups in labs to demonstrate basic capacitor principles.

Diagram

v. Series combination of capacitors

In a series connection:

1. The charge (Q) on each capacitor is the same.
2. The voltage across each capacitor can be different.
3. The total capacitance $C_{\text{total}}$ is less than any individual capacitance and is given by:

$\frac{1}{C_{\text{total}}} = \frac{1}{C_1} + \frac{1}{C_2} + \dots$

Applications:

1. Used when high voltage ratings are needed.
2. Helpful when designing circuits where lower capacitance is required.

Diagram

vi. Parallel combination of capacitors

In a parallel connection:

1. The voltage (V) across each capacitor is the same.
2. The charge (Q) on each capacitor can be different.
3. The total capacitance is the sum of individual capacitances:

$C_{\text{total}} = C_1 + C_2 + \dots$

Applications:

1. Used when a larger capacitance is needed.
2. Common in power supply smoothing and filter circuits.

Diagram

i. Area of Plates (A) The capacitance is directly proportional to the area of the plates:

$C \propto A$

ii. Distance Between Plates (d) The capacitance is inversely proportional to the distance between the plates:

$C \propto \frac{1}{d}$

iii. Dielectric Material Between the Plates The type of dielectric between the plates affects the capacitance. A material with a higher dielectric constant increases capacitance.

Relative Permeability (Dielectric Constant), $\varepsilon_r$ Relative permeability is the ratio of the capacitance with a dielectric to the capacitance with a vacuum:

$\varepsilon_r = \frac{C_{\text{with dielectric}}}{C_{\text{with vacuum}}}$

It is a unitless ratio. Examples: Paraffin wax: $\varepsilon_r \approx 2$ Mica: $\varepsilon_r \approx 8$

Formula for Capacitance of a Parallel-Plate Capacitor:

$C = \frac{\varepsilon_0 \varepsilon_r A}{d}$

Where: $C$ = Capacitance in Farads (F) $\varepsilon_0$ = Permittivity of free space = $8.85 \times 10^{-12} \, \text{F/m}$ $\varepsilon_r$ = Relative permeability (dielectric constant) $A$ = Area of the plates $d$ = Distance between the plates