Download ALTERNATING CURRENT STUDY NOTES and more Schemes and Mind Maps Physics in PDF only on Docsity!
ALTERNATING CURRENT “If the direction of current in a resistor or any other element changes alternately, the current is called an alternating current”
If the current or voltage is sinusoidal than it can be expressed as i=i 0 sin(ωt+ )
i 0 Peak current or current amplitude v 0 Peak voltage or voltage amplitude
v=v 0 sin(ωt+ )
2 π = 2 πf
ω= (^) T T:Time period
f:frequency (Hz or cycle/sec)
AVERAGE AND RMS VALUE OF AC
(ωt+ ): Total phase
if i=i 0 sinωt
i=i 0 cosωt
GENERAL GRAPH
⇒ for measuring ac hot wire instruments are used
i (^) O
T/4 T/
3T/4 (^) T t
-i (^) O
-i (^) O
i (^) O
T/4 3T/
T/ T
The average value of sin or cos function for one time period or n time periods (n=1,2...) is zero
Long period is equivalent to one time period
AVERAGE VALUE OF AC FOR ONE TIME PERIOD
Keep in mind
I= Idt^ =
I (^) O sin ωt dt (^) area of I-t graph dt dt time
=
I= 0 for 0→T for a sinusoidal ac wave.
The average value of square of sin or cosine function for one time period is (^12)
Remember
Mean square current for one time Period
I 2 = I =
(^2) dt dt
I 02 2
0 ∫ T^ sin^2 kωt^ =^12 xT 0 ∫ nTsin^2 kωt^ =n 2 xT
ROOT MEAN SQUARE CURRENT
I (^) rms = I^ o 2
Vrms = Vo 2
I (^) rms = (^) I 2 = I (^) o^2 = 2
I (^) o 2
Average value of ac is defined for positive or negative half cycle I = 2I π^ o V= 2V π^ o
I = I 2 o
SAWTOOTH FUNCTION I (^) O
-I (^) O
T/2 T
For half cycle
I 2 = I^ o^
2 3
Mean square current rms current
For full cycle I = 0
I (^) rms = I^ o 3
rms current I (^) rms = Io
RECTANGULAR FUNCTION I
-Io
+Io
T/2 T
For half cycle I = Io
Mean square currentI 2 = (^) Io 2
For full cycle I = 0
1 2 Io^
2 I
rms H (^) R^2 R avg=
AVERAGE HEAT PRODUCED DURING A CYCLE OF AC = Keep in mind ⇒ rms value is also called virtual value or effective value ⇒ AC ammeter and voltmeter always measure rms value ⇒ Values printed on ac circuits are rms values ⇒ In houses ac is supplied at 220V which is the rms of voltage ⇒ Peak value is 220√ 2= 311V ⇒ Frequency in general is 50Hz ⇒ w =2nf=100π rad/sec (314 rad/sec)
AVERAGE POWER CONSUMPTION
P= VI for cos =1 or =0 o
Pavg= Vrms Irms cos
1
v (^) o
I (^) o
2
Pavg= Vrms Irms
- V=V 0 sin ωt
- V=V 0 sin ωt
V=V 0 sin ωt^ V=V^0 sin^ ωt^
V=V 0 sin ωt
ω
XC= (^) ω
_ 1
c
π/
π/
π/
π/2 (^) π/
π/
- i 0 = _VR^0 8. i 0 =^ _
V 0
XL
Ol Ol
i=i 0 sin
- i=i 0 sin
V&i are in phase
- or current leads to the voltage by
(^) 4.
ωt
(ωt-
i i^ i
)
V
R ||C
SINGLE COMPONENT CIRCUITS
Resistor only Inductor only^ Capacitor only L
I V
=0,COS =
5. O=l Ol
- Inductive reactance (XL)
- Inductive reactance
,COS =
5. O=l π/2,COS Ol=
P= ε rms I^ rms
P= 0 (wattless circuits)
- (^) P=0 (wattless circuits) I
V
V,i
V (^0) i (^0) t
- V=V 0 sin ωt
- i=i 0 sin^ π/
3.Current leads the voltage by
(ωt+ )
I
V
I
XL=L
Unit-ohm(Ω) plays role of resistance
- i 0 = _XV^0 C
Unit-ohm(Ω) plays role of resistance
SUMMARY
1. R only R 0
O
2. L only ω
3. C only XC= ω
Z (Impedance)
XL = L
_ 1
c
l
1) R-L CIRCUITS
R
v=vosinwt
L
2) R-C CIRCUITS
V=V 0 sin
Voltage phasor diagram
i=i 0 sin (
Impedance phasor Z= (^) R^2 +l^2
ωt-
ωt
_VL
VL
O= VR
_Vo i o
V_rms i rms
- i 0 = _V^0 Z
=Z=
l
Ol
O)l
SERIES AC CIRCUITS
VR VL
i
V (^) R =i 0 R, VL=i 0 X
i (^0)
L
V=√
√
VR^2 +V L^2
tan
_XL
tanO=l R
)
V
V R
- V=V 0 sin 0 0
- Voltage phasor diagram
- i=io sin(ω t+ )
ωt
VC
Ol
VR =i R, VC =i X (^) C
V=√VR^2 +V c^2 )
V
VR
XL
)Ol
z
O R
R 2 +X L^2
V
V= Vosinwt
R VC
C
||
R
[
[
_Vc
tanO= l [VR
[
∫
T 0 ∫
T 0
∫
T 0 ∫
T 0
∫
T 0 ∫
T 0
PHASOR DIAGRAM Diagram representing ac voltage or current as vectors with phase angle between them.
w t
w t
ALTERNATING
CURRENT
XC
Z = R 2 + X C^2 XC R
tan =
i (^) o =
Z
R
- Impedance phasor
Vo Z
I=I 0 sin(ωt) v=v 0 sin(ω t+ )
I 0
v 0
PHYSICS
WALLAH
& i (^) rms = Vrms XL
& i (^) rms = Vrms Xc
& i (^) rms =
Vrms Xc
=
i (^0)
or i (^) rms = Vrms Z
+
Force constant K
V = Here i = i (^) o sin ( t - ) (since VL is leading ) Impedance Triangle
R,i (^) o Z = R^2 + ( X L - XC ) 2
XL - XC R
R tan =
XL - XC (^) Z Z XL - XC
VR^2 + (V L - VC) 2
V (^) L
VC
i
V = VL ~ VC [ie, (V (^) L - VC) or (VC - VL)]
Z = X (^) L ~ XC [ (XL - XC) or (XC - XL)]
if XL > Xc , Voltage leads the current by (^2) if XC > XL , current leads the voltage by 2 if X (^) L = X
= 1
C , Z = 0, i = or, ωL ωC
XL
XC
i (^) o
i (^) o
i Z
o
VR = i R, VL = i XL , VC = i X (^) C
V = VO O O O
sin t
L-C-R Series Circuit
i
Voltage phasor diagram
Assuming V (^) L > VC for drawing phasor
VL V
V=Vosinωt
V=Vosinωt
C VR
VL
io
V io C
VR
VR
VR,
VL - VC
Tuning mechanism of a radio or TV set 1.Antenna of radio accepts signals 2.Signal acts as an AC source in tuning the radio 3.In tuning, capacitance of capacitor is varied such that the resonant frequency of the circuit becomes nearly equal to the frequency of the radio signal received. So, the simple is largely amplified and distinctly heard
APPLICATION OF RESONANT CIRCUIT
ω r L ω r
_ _
_
QUALITY FACTOR Q=
Q=
R
=
or
cR
(^1) = _ R
(^1) _ C
L
Voltage across C or L [ applied voltage
[ resonance Less sharp the resonance, less is the selectivity of the circuit.If the Quality factor is large, R is low or L is large, the circuit is more selective.
POWER IN AC CIRCUIT
Average Power P=Vrms I rms cosOl
P=I^ rms^2 Z^ cosOl
cosO=0l
cosO=1l
Case 1 Purely Resistive circuit - Maximum power dissipation Case 2
Ol
Ol
Ol
O=0l
=0 ,COS Ol=
Purely inductive or capacitive circuit-
No power is dissipated even though a current is flowing in the circuit
=90 0
Case 3 LCR Series circuit non zero in R-L,C-R,or CLR circuit.
P=Vrms I rms CosOl
Case 4 Power dissipation at resonance X (^) L-XC=0 or^ => ⇒Z=R P=I 2 Z = I 2 R
LC OSCILLATIONS
Capacitor stores
electrical energy
Inductor stores
magnetic energy
When connected, charge on the
capacitor and current in the inductor
perform electrical oscillations
between each other.
Ol
COMPARISON OF LC OSCILLATION WITH A MASS
SPRING SYSTEM
1.Displacement (x)
2.Velocity V=
Acceleration a=
Mass (m),(inertia)
Momentum p = mv
LC Circuit
Charge (q)
Current I=
Rate of change of
_dI
dt
_dv
dt
_
_
_
d 2 x
dt 2
current = (^) (
(
_dI
( dt
(
Retarding force -m
Differential equation
+ ω^2 x=0 Differential equation
_d^2 q
dt 2 +^
k ω^2 q=
√ (^) m
ω= _^1
K.E=^1 √LC 2
mv 21 _
2 LI^
2
1 _
2 kx^ Elastic U = 2
_q^2
Elastic U =2C
Inductance (L), inertia of circuit
Magnetic flux =LI
Self induced emf -L
Capacitance (C)
Magnetic energy=
Mass spring system
TRANSFORMERS
“Device which raises or lowers voltage in ac circuits through mutual induction”.Transformer can increase or decrease voltage or current but not both simultaneously.
Input N^ P^ NS output
Primary coil
secondary coil
soft iron core
Vs -Voltage in secondary Vp -Voltage in primary Ns -No of turns in secondary Np -No of turns in primary Ip -Current in primary Is -Current in secondary
EQUATIONS
_VS
VP
_NS
NP
_I^ P
= =I S
3) For ideal transformer, η=
_ _VS
VP I^ P
I S
2) Efficiency η= =
Pout Pin
TRANSFORMER TYPES
step up transformer
step down transformer
N (^) P VP I (^) P R (^) P
< N
S VS I (^) S R (^) S
N (^) P VP I (^) P R (^) P
NS VS I (^) S R (^) S
LOSSES IN TRANSFORMER
Cu loss (I^2 R loss) → To minimise, windings are made of thick Cu wires (high resistance)
Eddy current loss → To minimise Cores are laminated
Hysteresis loss → select material of narrow hysteresis loop → Cores of transformer is made of soft iron
Magnetic flux linkage → To minimise, secondary winding is kept inside the primary winding
Humming loss
L-C CIRCUIT
V = VO O O
sin t V (^) L = i XL , VC = i XC
L
VL VC
C
~
In series resonance, impedance of circuit is minimum & equal to resistance Z= R, and curent is maximum Condition for resonance XL = XC L =
=
=
rad / sec resonant frequency (angular)
LC
r =
1 C
RESONANCE IN LCR SERIES CIRCUIT
r
1
f = f (^) r
f (^) r = resonant frequency
Variation of peak current with applied frequency
In resonance V = VR (applied voltage = voltage across resistance) Z = R (impedance is minimum and equal to resistance)
Voltmeter connected across VL & VC will show the same reading Voltmeter connected commonly across inductor & capacitor shows no reading Here V = 0 VL = VC VN = VR
Vnet = VR Z R
imax =
< r, XC > XL current leads
GRAPH
H (^) z
1 2 LC
io
r > r, XL > XC
V VL VC VR
~ Vnet
current lags
i
L C R
Sharpness of Resonance
Sharpness= Q=
ω r
_
∆ ω, sharper or narrower the
; 2 ∆ ω -bandwidth
smaller
resonance.
PHYSICS WALLAH
ω = 1 =ωo
ωo
LC
w < w r w > w r
c
I
q L
_dx
dt
_dq
dt
_dv
3. dt
Impedance Phasor Diagram
Voltage phasor diagram
R 1 >R 2 >R (^3)
I/P
O/P (^) I/P O/P
Maximum power is dissipiated in a circuit at resonance.
