Electric Circuits ICapacitors

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Dr. Firas Obeidat

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Capacitors

A capacitor is a passive element designed to store energy in its electricfield.

Capacitors are used in the tuning circuits of radio receivers and asdynamic memory elements in computer systems.

A capacitor consists of two conducting plates separated by an insulator (ordielectric).

When a voltage source is connected to thecapacitor, as in the figure, the source depositsa positive charge q on one plate and anegative charge on the other. The capacitor issaid to store the electric charge. The amountof charge stored, represented by q, is directlyproportional to the applied voltage v so that.

q=Cv where C is the capacitance of thecapacitor farad (F).

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Capacitors

Capacitance is the ratio of the charge on one plate of a capacitor to thevoltage difference between the two plates, measured in farads (F). 1 farad=1coulomb/volt.

Capacitance depends on the physical dimensions of the capacitor.And is given by

� =��

�

where A: is the surface area of each plate (m2), D: is the distance between the plates (m), �: is the permittivity of the dielectric material between the plates (F/m).��: is the relative permittivity (F/m).

�=����

�� = �.�� × 10−���/�

The current-voltage relationship of the capacitor

� =��

��

� = ���

��

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Capacitors The voltage-current relation of the capacitor can be obtained by

The change in energy stored in its electric field is

∫ ���(�)

�(��)=

�

�∫ �dt′ �

��

v(t)=�

�∫ idt′ �

��

+v(to)

Where v(to)=q(to)/C is the voltage across the capacitor at time to.

The instantaneous power delivered to the capacitor is

w=∫ ���′�

��

= � ∫ v′dv′ �(�)

�(��)

w(t)-w(to)=�

�� [(� �) � − � ��

�]

� = �� = ����

��

If we assume that a value of t0 is selected at which the voltage is zero;it is also assumed that the energy is zero at this time. We then have

w=�

����

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CapacitorsExample: For the capacitor of shown figure:a. Determine the capacitance.b. Find the resulting charge on each plate

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Capacitors

Example:

Example:

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CapacitorsExample: Determine the current through a capacitor whose voltage isshown in the figure

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CapacitorsExample: Find the capacitor voltage that is associated with the currentshown graphically in the figure. The value of the capacitance is 5 μF.

v(t)=�

�∫ idt′ �

��

+v(to)

a. -∞ to 0s: since the current is zero, the integral ofthe current is also zero, so, the voltage is zero inthis interval. And v(to= -∞ )=0

b. 0 to2ms:

a. 2ms to ∞: since the current is zero, the integral ofthe current is also zero, so, the voltage is 8V in thisinterval because v(to= 2m)=8V.

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CapacitorsExample: Find the maximum energy stored in the capacitor of the shownfigure and the energy dissipated in the resistor over the interval 0 < t < 0.5 s.

w=�

���� = 0.5×20×10−6 ×(100sin2πt)2

w= �.��sin22πt J

i� =�

�= 10−4sin2πt A

p� = ��� = 10610−8sin22πt

= 10−2sin22πt W

wR=∫ �����.�

�=∫ 10−2sin22πt��

�.�

�

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CapacitorsExample: Obtain the energy stored in each capacitor in Fig. 6.12(a) under dcconditions.

Under dc conditions, replace each capacitor withan open circuit, as shown in fig.(b). The currentthrough the series combination of the 2kΩ and4kΩ resistors is obtained by current division as.

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CapacitorsExample: Consider the circuit in fig(a). Under dc conditions, find: (a) i, and(b) the energy stored in the capacitor and inductor.

(a) Under dc conditions, we replace the capacitorwith an open circuit and the inductor with a shortcircuit, as in fig.(b).

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Series CapacitorsThe equivalent capacitor to N capacitors in series can be obtained byApplying KVL to the loop in shown figure.

For the special case of two capacitors in series

The equivalent capacitance of series-connected capacitors is thereciprocal of the sum of the reciprocals of the individualcapacitances.

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Parallel CapacitorsIn order to obtain the equivalent capacitor of N capacitors in parallel,Apply KCL to the shown figure.

The equivalent capacitance of N parallel-connected capacitors is the sum of theindividual capacitances.

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CapacitorsExample: Find the equivalent capacitance seenbetween terminals a and b of the circuit in shownfigure.

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CapacitorsExample: For the circuit in shown figure, find thevoltage across each capacitor.

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Properties of a Capacitors

when the voltage across a capacitor is not changing with time(i.e., dc voltage), the current through the capacitor is zero.Thus, a capacitor is an open circuit to dc.

The voltage on the capacitor must be continuous;The voltage on a capacitor cannot change abruptly

The ideal capacitor does not dissipate energy. It takes powerfrom the circuit when storing energy in its field and returnspreviously stored energy when delivering power to the circuit.

Nonideal capacitor has a parallel-model leakage resistance,. The leakage resistance may be as high as 100MΩ and can beneglected for most practical applications.

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