Transcript AC_Circuits1

AC Circuits 1
1
Capacitance in AC Circuits
i
iC

dV
d 
 C ( V sin 2ft )  2fC V cos 2ft
dt
dt

X
peak voltage V
C
Vc
Y

peak current  2fC V
The current leads the voltage by 90o .

V sin 2ft
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.
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 
V

I

v rms
1

irms 2fC
capacitive reactance is measured in
Ohms. Note that it is inversely
proportional to the frequency of the
source.
Capacitor
In 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
VC
XC 

IC
1
XC 
2fC

Capacitor
1
XC 
2fC

Thus as frequency rises the capacitive
reactance decreases non linearly
XC
reactance
frequency
Hz
Capacitor
In a purely capacitive a.c. circuit the
current IC leads the applied voltage VC
by 900
90 0
VC
IC
(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
e  L
di
dt
6
Inductance in AC Circuits
vL  L

di L
d 
 L ( I sin 2ft )  2fL I cos 2ft
dt
dt

peak current  I
voltage
iL
vL
VL (t)
or
IL (t)

peak voltage  2fL I
The voltage leads the current by 90o .
current
Let

i L ( t )  I sin 2ft
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 
V

I

v rms
 2fL
irms
inductive reactance is measured in
Ohms. Note that it is proportional to
the frequency of the source.
7
Inductor
In a purely inductive a.c. circuit the
current IL lags the applied voltage VL by
900
o
90
IL
VL
Inductor
In a purely inductive a.c. circuit the
current IL lags the applied voltage VL by
900
L
IL
VL
Circuit Diagram
VL
90 0
IL
Phasor Diagram
Inductor
In 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
VL
XL 

IL
X L  2fL 
Inductor
X L  2fL 
Thus as frequency rises the inductive
reactance increases linearily
reactance
frequency
Resistance and Inductance in series
VR
IS
VL
VS
VS
VL
As the current IS flows through
both components it should be
used as the reference for the
phasor diagram
VR
IS
Resistance and Inductance in series
VR
VL
Circuit Diagram
IS
VS
VS
VL
VR
IS
Phasor diagram
applied voltage VS
the ratio
current flowing IS
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
VC
VR
VR
IS
IS
VS
VC
VS
As the current IS flows through
both components it should be
used as the reference for the
phasor diagram
Resistance and Capacitance in series
VC
VR
VR
IS
IS
VS
applied voltage VS
the ratio
current flowing IS
VC
VS
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
Voltage Triangle
VR
VR


IS
VC
VS
VS
Phasor diagram for CR circuit
VC
Voltage triangle for CR circuit
If each of the voltages are divided by IS then opposition to
current flow of each element is obtained.
V
i.e. R  Re sistance
IS
VC
 capacitive reactance
IS
VS
 impedance
IS
Impedance Triangle
VR
R

IS
VC

Z
XC
VS
Phasor diagram for
CR circuit
Impedance triangle for CR circuit
Using the same steps the voltage triangle and impedance
triangle for R and L in series may be obtained
Impedance Triangle
VR
R

IS
VC

XC
Z
VS
Phasor diagram for
CR circuit
Impedance triangle for CR circuit
Using Pythagoras' s theorem Z  R 2  X C
XC
and from trigonome try tan  
R
2

Find the inductive reactance XL
25.46 mH
6
10 V 50Hz
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
3
X L  2fL  2    50Hz  25.46  10 H  8
Z  R 2  X L 2  62  82  10
IS 
VS 10V

 1A
Z 10
VL  I S  X L  1A  8  8V
VR  I S  R  1A  6  6V
XL
8
  arctan
 arctan
 arctan 1.333  53.130
R
6
25.46 mH
6
10 V 50Hz
X L  8
Z  10
I S  1A
VL  8V
VR  6V
  53.130
The point Z could be specified
to the origin by :-
Z  6  j8
OR
0
1053.13
Construct a reactance/resistance/impedance
vector for the above diagram
R  20,
X C  15,
X L  10
Construct a reactance/resistance/impedance
vector for the above diagram
XL
add X L and X C
R
XC  X L
Z
Z  R 2  ( X C  X L )2
XC
R  20,
X C  15,
X L  10
Note that as XC > XL the circuit Is
said to be capacitively reactive
XL
R
XC  X L
Z
XC
The series circuit could be
replaced by a 20Ω resistor in
series with a capacitive reactance
of 5Ω
the point Z could be defined by Z  20  j10 - j15 
or Z  20 - j5  in cartesian form
or 20.62  - 14.04o in polar form