1

AC Circuits 1

Capacitance in AC Circuitsi

CX

Y

Vc

Vvoltagepeak

ftcosVfC)ftsinV(dt

dC

dt

dVCi 222

VfCcurrentpeak 2

fCi

v

I

Vreactancecapacitive

rms

rms

2

1

The opposition to the flow of ac current in a capacitor is known as the capacitive reactance and is the equivalent of resistance in a resistor circuit.

capacitive reactance is measured in Ohms. Note that it is inversely proportional to the frequency of the source.

The current leads the voltage by 90o .

During the positive half-cycle of the voltage waveform, plate X of the capacitor becomes positively charged and plate Y negatively charged. During the negative half-cycle, X receives a negative charge and Y a positive one. There is therefore an alternating flow of charge or alternating current, i, through the capacitor Thus, unlike the case for a dc voltage where the capacitor eventually charges up to the value of the dc supply voltage and no more current flows, a capacitor continually conducts an ac current. Thus we may say that a capacitor passes ac and blocks dc.

ftsinV 2

CapacitorIn a purely capacitive a.c. circuit the current IC leads the applied voltage VC by 900

In a purely capacitive a.c. circuit the opposition to current flow is known as Capacitive Reactance, XC

Capacitive Reactance is measured in ohms

C

CC I

VX

fCXC 2

1

Capacitor

fCXC 2

1

Thus as frequency rises the capacitive reactance decreases non linearly

reactance

frequency

CX

Hz

Capacitor

In a purely capacitive a.c. circuit the current IC leads the applied voltage VC by 900

CI

CV

090

(Self ) Inductance

A piece of wire wound in the form of a coil possesses an electrical property known as inductance.

The property arises from the observable phenomenon that if a current flowing through the coil changes for some reason then an emf e is induced in the coil which tries to oppose the current change.

The magnitude of the induced emf is proportional to the rate of change of current. The constant of proportionality is known as the inductance of the coil L and is measured in a unit called the henry.

Mathematically we can summarise this with the equation

6

dt

diLe

Inductance in AC Circuits

7

voltage

or

current

IL (t)VL (t)

vL

iL

Icurrentpeak

ftcosIfL)ftsinI(dt

dL

dt

diLv L

L 222

IfLvoltagepeak 2

The opposition to the flow of ac current in an inductor is known as the inductive reactance and is the equivalent of resistance in a resistor circuit.

inductive reactance is measured in Ohms. Note that it is proportional to the frequency of the source.

The voltage leads the current by 90o .

fLi

v

I

Vreactanceinductive

rms

rms 2

ftsinI)t(iL 2

Let

Inductor

In a purely inductive a.c. circuit the current IL lags the applied voltage VL by 900

LV

LI

o90

Inductor

In a purely inductive a.c. circuit the current IL lags the applied voltage VL by 900

LV

L

LI

LI

LV

090

Circuit Diagram Phasor Diagram

InductorIn a purely inductive a.c. circuit the current IL lags the applied voltage VL by 900

In a purely inductive a.c. circuit the opposition to current flow is known as Inductive Reactance, XL

Inductive Reactance is measured in ohms

L

LL I

VX fLX L 2

Inductor fLX L 2

Thus as frequency rises the inductive reactance increases linearily

frequency

reactance

Resistance and Inductance in series

SISV

LVRV

SI

RV

LVSV

As the current IS flows through both components it should be used as the reference for the phasor diagram

Resistance and Inductance in series

SISV

LVRV

SI

RV

LVSV

Phasor diagram

Circuit Diagram

S

S

I flowingcurrent

V voltageapplied ratio the Is the opposition to current

flow in the circuit

However as the current and voltage are not in phase this opposition to current flow is known as Impedance Z (Ω)

As the current flowing lags the applied voltage the circuit is said to be inductively reactive

Resistance and Capacitance in series

As the current IS flows through both components it should be used as the reference for the phasor diagram

CV RV

SI

SVSI

RV

SVCV

Resistance and Capacitance in series

SI

RV

SV

CV RV

SI

SVCV

S

S

I flowingcurrent

V voltageapplied ratio the Is the opposition to current

flow in the circuit

However as the current and voltage are not in phase this opposition to current flow is known as Impedance.(Z) (Ω)

As the current flowing leads the applied voltage the circuit is said to be capacitively reactive

SI

RV

SVCV

Voltage Triangle

CV

SV

Phasor diagram for CR circuit Voltage triangle for CR circuit

RV

If each of the voltages are divided by IS then opposition to current flow of each element is obtained.

sistanceReI

V i.e.

S

R reactance capacitiveS

C

I

V impedanceS

S

I

V

SI

R

ZCV

Impedance Triangle

CX

SV

Phasor diagram for CR circuit

Impedance triangle for CR circuit

RV

Using the same steps the voltage triangle and impedance triangle for R and L in series may be obtained

SI

R

ZCV

Impedance Triangle

CX

SV

Phasor diagram for CR circuit

Impedance triangle for CR circuit

RV

2C

2 XRZ theorem s'PythagorasUsing

R

Xtry tan trigonomefrom and C

6mH46.25

HzV 5010

Find the inductive reactance XL

Find the Impedance Z

Find the current flowing

Find the voltage across the inductor

Find the voltage across the resistor

Phase angle between VS and IS

81046.255022 3HHzfLX L

1086 2222LXRZ

AV

Z

VI SS 1

10

10

VAXIV LSL 881

VARIV SR 661

013.53333.1arctan6

8arctanarctan

R

X L

6mH46.25

HzV 5010

8LX 10Z

AIS 1 VVL 8

VVR 6 013.53

The point Z could be specified to the origin by :-

86 jZ

013.5310

OR

Construct a reactance/resistance/impedance vector for the above diagram

Construct a reactance/resistance/impedance vector for the above diagram

10,15,20 LC XXR

R

LX

CX

CL XX and add

LC XX

Z

)XX(RZ LC

22

10,15,20 LC XXR

R

LX

CX

LC XX

Z

Note that as XC > XL the circuit Is said to be capacitively reactive

The series circuit could be replaced by a 20Ω resistor in series with a capacitive reactance of 5Ω

form cartesian in j5-20 Z or

j15-j1020 Zby definedbe could Z pointthe

formpolar in 14.04- 20.62or o