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Capacitance

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What is a capacitor?

A capacitor is an electronic component that stores electric charge (and electric energy)

A simple version of a capacitor is the parallel-plate capacitor; it consists of two conducting plates separated by an insulating material – The insulating material is called a dielectric in

this context.

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Parallel-plate capacitor

dielectric

+++++++

-------

I I

conducting plates

2. Positive charges accumulates on first plate; dielectric does not let them through

1. Battery (not shown) pushes charges out

3. Neutral atoms on other plate are made up positive and negative charges

4. Negative charges are attracted and stay behind, while positive charges are repelled and move out giving rise to the current on other side

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We can’t go on like this

dielectric

+++++++

-------

I I

conducting plates

2. There are positive charges on the first plate. Like charges repel.

1. There are positive charges coming from the battery

3. When the push from the battery is equal to the push back from the plate, the capacitor stops charging

4. The bigger the push from the battery (i.e. voltage), the more charge goes onto the plate. The voltage and charge are proportional.

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Q = VC

When we talk about the charge on a capacitor, we mean the charge on one of the plates.The charge is proportional to the voltage: Q V.The proportionality constant C is called the capacitance.Solving for V, the equation becomes V =

Q

C

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Farad

The standard unit of capacitance is the farad (F).

A farad is quite large, usually you see – millifarad mF (1 mF = 10-3 F)– microfarad F (1 F = 10-6 F) – nanofarad nF (1 nF = 10-9 F)– picofarad pF (1 pF = 10-12 F)

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Capacitance Q=VC

The capacitance is a measure of how easy it is to put charge on the plates, it is – directly proportional to the surface area of the plates,

bigger plates can hold more charge

– inversely proportional to the distance between the plates, the interaction between the positive and negative charges is greater when they are closer

– dependent on the material (dielectric) separating the plates, having a good insulator between them is like their being further apart

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Capacitive keyboards

Capacitive keyboards detect which key was pressed by looking at changes in capacitance.

Under each key are two oppositely charged plates ( a capacitor).

Pressing a key moves its upper plate closer to its lower plate, changing its capacitance, and hence changes the amount of charge the plates can hold for a given voltage.

The keyboard circuitry detects this change and sends the appropriate information (interrupt request and ASCII code) to the CPU.

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DRAM

Capacitors can be put on integrated circuit (IC) chips. Together with transistors, they are used in dynamic random access memory (D-RAM). The charge or lack thereof of the capacitor corresponds to a stored bit. Since these capacitors are small, their capacitance is pretty low. DRAM has to be recharged (refreshed) thousands of times per second or it loses its data.

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Power Supply

Large capacitors are used in the power supplies of computers and peripherals.

The capacitors (along with diodes) are used in rectifying: turning the alternating current (AC) into smoothed out direct current (DC).

Capacitors do the smoothing part.

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Monitors

Monitors (CRT’s) are another place that large capacitors are found.

Since capacitors store charge, monitors and power supplies can be dangerous even when the power is off. – If you don’t know what you’re doing, don’t

even open them up.

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Equivalent Capacitance

When a combination of capacitors can be replaced by a single capacitor, which has the same effect as the combination, the capacitance of the single capacitor is called the equivalent capacitance.

Having the same effect means that same voltage results in the same amount of charge being stored.

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Capacitors in parallel

Recall that things in parallel have the same voltage

And the charge is split between the capacitors. The charge has a choice, some will go onto one capacitor, the rest on the other.

V C1

C2

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Capacitors in parallel

Individual capacitors obey basic equationQ1 = C1 V and Q2 = C2 V

Equivalent capacitor obeys basic equation Qtotal = Ceq V

The total charge is the sum of the individual charges Qtotal = Q1 + Q2

Solve basic equations for Q’s and substitute Ceq V = C1 V + C2 V

Divide by common factor V

Ceq = C1 + C2

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Capacitors in series

Capacitors in series have the same charge– When the capacitors are uncharged, the region including the

lower plate of C1 and the upper plate of C2 is electrically neutral. This region is isolated and so this remains true; the negative charge on the lower plate of C1 when added to the positive charge on the upper plate of C2 would give zero

The voltage is split between them

VC1

C2

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Capacitors in series

Individual capacitors obey basic equation

V1 = Q1 / C1 and V2 = Q2 / C2

Equivalent capacitor obeys basic equation

Vtotal = Qtotal / Ceq

The total voltage is the sum of the individual voltages

Vtotal = V1 + V2 Substitute basic equations into voltage equation

Qtotal / Ceq = Q1 / C1 + Q2 / C2

All of the charges are the same (Qtotal = Q1 = Q2 = Q), divide out the common factor

1 / Ceq = 1 / C1 + 1 / C2

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Example

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Example (Cont.)

Something must take the brunt of the voltage when the battery is first connected, that’s why the 0.2-k resistor is there.

The 2.5-F and 1.5-F are in parallel, so they can be replaced with one 4.0-F capacitor.

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Example (Cont.)

The 3.5-F and 4.0-F capacitors are in series, so they can be replaced with one 1.87-F capacitor.

The charge on it Q=CV would be 9.33 C

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Example (Cont.)

Going back to the 3.5-F and 4.0-F capacitors are in series, capacitors in series have the same charge, so Q3.5 = 9.33 C

Then V3.5 = Q3.5 / C3.5 or V3.5 = 2.67 V

And V4.0 = Q4.0 / C4.0 or V4.0 = 2.33 V

(Note that V3.5 + V4.0 = 5)

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Example (Cont.)

The 4.0-F capacitor was really a 1.5-F and a 2.5-F in parallelThings in parallel have the same voltage, so V1.5 = V2.5 = 2.33 V

Since Q1.5 = V1.5 C1.5, Q1.5 = 3.50 C

Similarly, Q2.5 = 5.83 C

(Note Q1.5 + Q2.5 = 9.33 C)

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Testing in Electronics Workbench

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Two Cautions

When the switches are closed, there is an easy path from one side of the capacitor to another, this makes sure that the capacitors are discharged.

Even when the switches are open there is a path from one side to another for the upper capacitors, it is through the voltmeter. The voltmeter must be made extremely ideal (very high resistance) to get agreement with theory.

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RC circuits: Charging

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What happens

When you connect an uncharged capacitor and a resistor in series to a battery, the voltage drop is initially all across the resistor. – Because the voltage drop across a capacitor is

proportional to the charge on it and there is not charge on it at the beginning.

But charge starts to build up on the capacitor, so some voltage is dropped across the capacitor now. – Capacitors have a gap and while current gets all the

way around the circuit, individual charges are trapped on one side of the capacitor.

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What happens (Cont.)

Since some voltage is dropped across the capacitor, less voltage (than before) is dropped across resistor. With less voltage being dropped across the resistor, the current drops off. – V = IR (smaller V smaller I)

With less current, the rate at which charge goes onto the capacitor decreases. The charge continues to build up, but the rate of the build up continues to decrease. In mathematical language, the charge as a function of time Q(t) increases but its slope decreases. Theory says the charge obeys Q(t) = C V (1 - e- t / )

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Time Constant

Q(t) = C V (1 - e- t / )

in that equation is known as the “time constant” and is given by = RC

Note that since R = V / I , resistance has units volt/ampere

Since C = Q / V, capacitance has units coulomb/volt

RC = (V / I) (Q / V) = Q / I

Then RC has units coulomb/ampere but an ampere is coulomb/second

RC = Q / (Q / T) = T

So RC has units of second

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Charging Capacitor

RC Circuit (Charging)

Time

Ch

arg

e o

n c

apac

ito

r

1

63%

3

95%

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Voltage is similar

If the charge on the capacitor varies according to the expression

Q(t) = C V (1 - e- t / )

then since the voltage across a capacitor is V=Q/C the voltage is

V(t) = V (1 - e- t / )the voltage approaches its “saturation” value, which in a simple RC circuit in the battery’s voltage.

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Time Constant

The time constant ( = RC) is the time required for a certain percentage (63%) of the saturation charge (the charge after a very long time) to be put on the capacitor. If the resistance is large, the currents are small, even from the start and it requires more time to charge up the capacitor ( R)If the capacitance is large, the capacitor can take a lot of charge and again it requires more time to charge up the capacitor ( C)

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RC circuits: Discharging

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What happens

This time we imagine that there are already charges on the capacitor, but there is no battery.With no battery to "push" the charges around, the opposite charges on the two capacitor plates would prefer to be together. They must pass through the resistor before they can reunite. With all those like charges on one plate, there is a strong incentive for charges to leave the plate.

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What happens (Cont.)

However, as charges leave the plate, the voltage across the capacitor decreases (V = Q C) and the incentive for charges to leave the capacitor decreases, thus the rate at which charges leave decreases as well. In mathematical language, this time the charge as a function of time Q(t) decreases and its slope decreases.

Theory says the charge obeys Q(t) = Q0 e- t / .Same time constant as before.

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Discharging Capacitor

RC Circuit (Discharging)

Time

Ch

arg

e o

n t

he

cap

acit

or

1 3

37%

5%