Capacitors in AC Circuits

Capacitors

• A capacitor is used to rapidly store and release charge.

• Capacitors differ from resistors in that; resistors convert electrical energy to heat (and often light), while capacitors only store the charge (electrical energy) in the electric field between their plates, then release it again.

Capacitors in DC c.f. AC

• Once the capacitor is fully charged the bulb does not glow

• The bulb glows continuously but more dimly than without the capacitor

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Although no charge flows between the plates in either circuit, the AC means that the current is constantly changing direction, making the bulb glow.

A Capacitor in AC

• Capacitors react against a change in voltage by either supplying current (discharging) or drawing current (charging).

• In an AC circuit a capacitor alternates between charging and discharging.

• As a result, capacitors behave in a similar way to resistors in that they oppose the current in a circuit as they charge and discharge. This property is called reactance and has the symbol XC

and units Ohms

V

C

~

Voltage and Current Phase Differences

Step 1 - At point a (see diagram) the voltage is zero and the capacitor is uncharged. Initially, the voltage increases quickly. The voltage across the capacitor matches the power supply voltage, so the current is large to build up charge on the capacitor plates. The closer the voltage gets to its peak, the slower it changes, meaning less current has to flow. When the voltage reaches a peak at point b, the capacitor is fully charged and the current is momentarily zero.

Note that;

Unlike a resistor where V and Iare in phase in a capacitor V lags behind I by ¼ of a cycle (90)

V

C

~

Voltage and Current Phase Differences

Step 2 - After reaching a peak, the voltage starts dropping. The capacitor must discharge now, so the current reverses direction. When the voltage passes through zero at point c, it's changing quite rapidly; to match this voltage the current must be large and negative.

V

C

~

Voltage and Current Phase Differences

Step 3 - Between points c and d, the voltage is negative. Charge builds up again on the capacitor plates, but the polarity is opposite to what it was in step one. Again the current is negative, and as the voltage reaches its negative peak at point d the current drops to zero.

Step 4 - After point d, the voltage heads toward zero and the capacitor must discharge. When the voltage reaches zero it's gone through a full cycle so it's back to point a again to repeat the cycle.

Capacitors in AC

The larger the capacitance of the capacitor, the more charge has to flow to build up a particular voltage on the plates, and the higher the current will be. The higher the frequency of the voltage, the shorter the time available to change the voltage, so the larger the current has to be. The current, then, increases as the capacitance increases and as the frequency increases.

Usually this is thought of in terms of the effective resistance of the capacitor, which is known as the capacitive reactance, measured in ohms. There is an inverse relationship between current and resistance, so the capacitive reactance is inversely proportional to the capacitance and the frequency:

Voltage and Current in a Capacitor

• In an AC circuit the current can be altered with a variable resistor

• When the voltage and current are plotted on a graph they show a linear relationship

Remind you of Ohm’s Law (V=IR)?

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VC

A

6V AC

200F

0

1

2

3

4

5

6

7

0 0.1 0.2 0.3 0.4

Cap

acit

or

Vo

ltag

e (

V)

Current (A)

Capacitor Voltage and Current

Reactance (XC ) -the maths

• In the same way that V=IR the “opposing” of AC current by a capacitor, reactance (XC ) can be calculated by;

and so;CC IXV

I

VX C

C

Examples

1. Find the voltage of a capacitor with a reactance of 1.2 and a current of 0.80A

0.96V

2. A capacitor with 8V AC across it has a reactance of 45. Calculate the current of the circuit.

0.17A

3. Calculate the reactance of a capacitor with a RMS voltage of 6V and a current of 1.8A

3.3

Worksheet

Factors Affecting Reactance (XC )• Increasing the size of the capacitor means that more current is

required to charge and discharge the capacitor (decreasing XC)

• Increasing frequency increases current (decreasing reactance).This is because more frequent charging and discharging means more current must flow to charge the capacitor in less time

• The reactance of a capacitor with a supply frequency f;

CX C

1

fX C

1

CX

fCX CC

1

2

1 or

The reactance Xc is large at low frequencies and small at high

frequencies. For steady DC which is zero frequency, Xc is infinite

(total opposition), hence the rule that capacitors pass AC but

block DC.

Examples

1. A 200F capacitor is connected to a 6V 50Hz AC supply.

a) Calculate the reactance of the capacitor16

b) The RMS current in the circuit0.38A

2. What size capacitor is needed to give an reactance of 50 in a 12V 60Hz circuit?

5 3F

RC Phase Differences

• In an AC circuit with a resistor and a capacitor (RC circuit) the voltages across each component are out of phase by ¼ of a cycle

-1.5

-1

-0.5

0

0.5

1

1.5

0 200 400 600 800

Vo

ltag

e (

mV

)

Time (ms)

Resistor and Capacitor Phase Differences

Resistor

Capacitor

The Effect of Phase Differences in RC Circuits

• In DC circuits the voltages across components in a circuit add up to the supply voltage

• In AC circuits the same does not appear to apply (at first

glance)

VS

VCVR

200F75

6.0

5.9 1.2

The Effect of Phase Differences in RC Circuits

• However if we consider the phase differences, we see that this is a vector problem

VS

VCVR

200F75

VC

VR

CRS VVV~~~

VC

VR

VS

6.0

5.9 1.2

22

222

CRS VVV

BAC

From

s; Pythagoru

VS

In an RC circuit;

• At any instant

Note the graph

• But when considering the rms voltages the phase differences are important

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 200 400 600 800

Vo

ltag

e (

mV

)

Time (ms)

Supply Voltage of Resistor/Capacitor Circuit

Resistor

Capacitor

SupplyVoltage

The Effect of Phase Differences in RC Circuits

CRS VVV

CRS VVV~~~

Exercises1. Find the AC supply voltage of an RC circuit where the

resistor voltage is 3.4V and the capacitor voltage is 1.5V

3.7V

2. Calculate the voltage across the resistor in an AC circuit with a supply voltage of 8.5V and a capacitor voltage of 2.4V

8.2V

3. Calculate the voltage across the capacitor in an 12V AC circuit with a voltage of 8.5V across the resistor.

8.5V

4. Find the supply voltage of an 60Hz AC circuit with a 120V across a 2k resistor and a capacitor voltage of 0.80V

120V

The Effect of RC Circuits on Current

• In an RC circuit both the resistor and the capacitor oppose the current so V=IR wont work

• Any calculation of the current will have to involve both resistance (R) and reactance (XC

)and allow for the phase differences between them

VS

VCVR

200F75A

Impedance in an RC Circuit

• Impedance is a measure of the combined opposition to alternating current of the components of a circuit.

• It describes not only the relative amplitudes of the voltage and current, but also the relative phases thecomponents in the circuit.

• Impedance has the symbol Z and units Ohms

-1.5

-1

-0.5

0

0.5

1

1.5

0 200 400 600 800

Vo

ltag

e (

mV

)

Time (ms)

Resistor and Capacitor Phase Differences

Resistor

Capacitor

IZV

• Impedance in an RC circuit involves both the resistance and the reactance

• because of the phase differences they must be added as vectors so;

Impedance –the maths

22

222

CXRZ

BAC

s; PythagoruFrom

XC

R

Z

Examples

1. Calculate the impedance of an RC circuit with a resistance of 75 and a reactance of 15

76

2. An RC circuit has an impedance of 65 and has a resistance of 24 . What is the reactance of the circuit?

60

3. Find the resistance of an RC circuit with 25 impedance and 12 reactance.

22

Exercises

ESA Pg 269

Activity 16B,16C,16D

ABA

Pg 180-185