Capacitors Capacitor Construction

Capacitor Charging

Definition of Capacitance Capacitance is a measure of a capacitor’s ability to store charge, and is defined as the amount of charge it can store per unit of voltage applied across its plates: C=

Q V

the unit of capacitance is the farad (F)

Force of attraction between two point-source charges: F=

kQ1Q2 d2

where F = the force in Newtons (N)

Energy stored by a capacitor: W =

1 CV 2 2

where W = energy in Joules

Dielectric Strength and Voltage Ratings Breakdown voltage of a capacitor is determined by the dielectric strength of the dielectric (insulating) material used in the capacitor, and is expressed in volts/mil. The larger the dielectric strength of a material, the more voltage can be applied to the plates of a capacitor for a given plate separation before breakdown occurs. Every capacitor has a maximum voltage rating based on the dielectric strength of the insulating material used, which if exceeded, may result in permanent damage to the device. This voltage is commonly referred to as the breakdown voltage.

note: mil = 0.001 in.

Dielectric Constant The dielectric constant (or relative permittivity) is a measure of a material’s ability to establish an electric field. Capacitance is directly related to the dielectric constant. Therefore a material with a higher dielectric constant results in a capacitor which can store a greater charge per unit volts applied to its plates.

These dielectric constants are actually a relative measure of the permittivity of a material (ε) compared to the ε permittivity of a vacuum ( ε 0 ): ε r = ε0

Physical Characteristics of a Capacitor Capacitance is directly proportional to the area of the plates, and indirectly proportional to the distance between the plates. This follows logically since the larger the plate size, the more charge can be stored per unit volts. Also, the closer the plates are in relation to each other, the greater the force of attraction between the charges on each plate (and therefore a greater electric field strength).

Capacitor Equation Capacitance can be calculated exactly in terms of the three physical quantities previously discussed: namely, the dielectric constant, the plate size, and the plate separation. Aε rε 0 Aε r (8.85 x10 −12 F / m) = d d where A is in meters squared ( m 2 ), d is in meters (m) and C is in farads (F) C=

Example: determine the capacitance of a parallel plate capacitor having a plate area of 0.01 m 2 and a plate separation of 0.02 m. The dielectric is mica. (answer: 22.1pF)

Capacitor Types Mica Capacitors (stacked-foil and silver-mica) Available capacitance range: 1pF to 0.1uF Available voltage ratings: 100 Vdc to 2300 Vdc

Ceramic Capacitors Available capacitance range: 1pF to 2.2uF Available voltage ratings: up to 6000 Vdc

Ceramic capacitors have exceptionally good temperature coefficients

Plastic-Film Capacitors (polycarbonate, propylene, mylar, polystyrene etc) Available capacitance range: up to 100uF depending on dielectric material used Available voltage ratings: depends on dielectric material used

Electrolytic Capacitors Available capacitance range: 1uF to 200,000uF+ Available voltage ratings: lower than those previously mentioned (350V)

Notes: electrolytic capacitors use two dissimilar plates – one made of metal (most often aluminum, but tantalum are also common) and one made by applying a conducting electrolyte to a material like plastic film. The dielectric is formed from the oxide which forms on the surface of the aluminum plate due to the electrolyte. Electrolytic capacitors are available in much larger capacitance values than other types, but at reduced voltages. Also, due to their method of construction, electrolytic capacitors are polarized, and correct polarity must be observed when installing these devices in order to prevent their destruction.

Variable Capacitors • • •

Variable capacitors are used in circuits when there is a need to adjust the capacitance value. Ceramic or mica is a common dielectric . Capacitance is changed by plate separation.

Capacitor Labeling •

Capacitors use several standard labeling methods; we will consider a small ceramic capacitor: – values marked as .001 or .01 have units of microfarads. – values marked as 50 or 330 have units of picofarads. – a value of 103 or 104 would be 10x103 (10,000 pF) or 10x104 (100,000 pF) respectively. – The units may be on the capacitor as pF or µF (µF may be written a MF or MFD).

Parallel Capacitors The total parallel capacitance is the sum of all capacitors in parallel. CT = C1 + C2 + C3 + … + Cn

Example: Find the total capacitance of the following circuit. Also find the charge and energy stored on each capacitor:

Answer: CT = 550pF; Q1 = 1.65nC; Q2 = 1.1nC

Series Capacitors When capacitors are connected in series, the total capacitance is less than the smallest capacitance value since the effective plate separation increases. 1/CT = 1/C1 + 1/C2 + 1/C3 + … + 1/Cn

Since current is defined as the rate of flow of charge (Q/t), the same amount of charge must flow at each point in the circuit (since the current in a series circuit is constant). This means that each capacitor stores the same amount of charge which equals the total charge, i.e. QT = Q1 = Q2 = . . . . = Qn. However, the voltage across each capacitor depends on its capacitance according to the formula Q=CV. These voltages must always add up to the total applied voltage in accordance with Kirchoff’s voltage law (KVL). The voltage across each capacitor may be found by calculating the total capacitance, and then either (a) calculating the total charge Q and then using the fact that V=Q/C to find the voltage across each capacitor, or (b) using the following voltage divider law in order to find the voltage across each capacitor1: C  Vx =  T VT  CX  Example: find the voltage across each capacitor in the following circuit:

Answer: V1 = 15V; V2 = 3.19V; V3 = 6.82V

1

Note that the form of this voltage divider is similar to the current divider law presented in the analysis of parallel circuits. In all these divider equations, it is worth noting that the quantity in brackets must ALWAYS be a fraction less than one.

Capacitors in DC Circuits • • • •

A capacitor will charge up when it is connected to a dc voltage source. When a capacitor is fully charged, there is no current. There is no current through the dielectric of the capacitor because the dielectric is an insulating material. A capacitor blocks constant dc.

RC Time Constant • •

The buildup of charge across the plates occurs in a predictable manner that is dependent on the capacitance and the resistance in a circuit. The time constant of a series RC circuit is a time interval that equals the product of the resistance and the capacitance. τ = RC

Charging and Discharging • • •

The charging curve is an increasing exponential. The discharging curve is an decreasing exponential.

Important Notes: • The voltage across a capacitor CANNOT change instantaneously • The current in a purely capacitive circuit CAN change instantaneously • A fully charged capacitor appears as an OPEN to CONSTANT current • An uncharged capacitor appears as a SHORT to a CHANGING current

General Exponential Formulas: −t

v(t ) = VF + (VI − VF )e τ −t

i (t ) = I F + ( I I − I F )e τ

Example: determine the capacitor voltage 50us after the switch is closed if the capacitor is initially uncharged. Repeat the calculation if the capacitor has an initial charge of 12V.

Answer: 22.8V; and 32.65V

Example: determine the capacitor voltage in the following circuit 6ms after the switch is closed.

Answer: 7.61V

Solving for Time In order to solve the previous exponential equations for the variable t (time), it is necessary to understand the following property of logarithms:

( )

log N N X = X Example: how long will it take the capacitor in the following circuit to discharge to 25V when the switch is closed?

Answer: 3.05ms

Example: how long would it take the capacitor in the previous example to charge to 25V if the switch were replaced with a 48V source. Assume the capacitor has an initial voltage of zero. Answer: 60.3us

Example: repeat the previous problem if the capacitor has an initial voltage of 12V. Answer: 36.7us