JEE
Mains and Advanced
.
PHYSI
, TABLE OF CON
Chapter 01 AL
Chapter 02 EL
Chapter 03 EL
Chapter 04 EM
,ALTERNATING CURRENT ROOT MEAN Average value of
SQUARE CURRENT ac is defined for SINGLE COMPONENT CIRCUITS
“If the direction of current in a
resistor or any other element changes positive or negative
alternately, the current is called an half cycle
Irms= I2 = Io 2 = Io
alternating current” 2
2Io 2Vo
2 I= V=
Irms= Io Vrms= Vo
π π
Resistor only Inductor only Capacit
2 2
AVERAGE AND RMS VALUE OF AC R L
i i
i
If the current or voltage is SAWTOOTH FUNCTION
sinusoidal than it can be expressed as Io
IO For half cycle I=
i=i0sin(ωt+ ) 2
v=v0sin(ωt+ ) For full cycle I= 0
i0 Peak current or current amplitude T/2 T Io 2
Mean square current I2 =
v0 Peak voltage or voltage amplitude 3
Irms= Io
-IO V=V0sin ωt
rms current V=V0sin ωt
2π = 2πf T:Time period 3
1. V=V0sin ωt 1. V=V0sin ω
ω= 1. V=V0sin ωt
T
RECTANGULAR FUNCTION 2. i=i0sin (ωt- π/2 ) 2. i=i0sin (ω
f:frequency (Hz or cycle/sec)
I 2. i=i0sin ωt
(ωt+ ): Total phase +Io For half cycle I= Io 3. or current leads to the 3.Current le
3. V&i are in phase voltage by π/2 voltage b
For full cycle I= 0
T/2 T
GENERAL GRAPH Mean square current I2= Io2 4. 4.
-Io
V 4. V
)
rms current Irms= Io I π/2
if i=i0 sinωt
l =0,COS Ol =1
5. O
iO
AVERAGE HEAT PRODUCED DURING A P= εrmsIrms
CYCLE OF AC V
3T/4 T
1
T/4 T/2 t Havg= 2
Io R = Irms R
2 2 V,i )
π/2
I l π/2,C
5. O=
-iO Keep in mind V 6. P=0 (wat
V0 l π/2,COS Ol =0
5. O=
⇒ rms value is also called virtual value or effective value I
i=i0 cosωt i0 6. P= 0 (wattless circuits) 7. Inductive
⇒ AC ammeter and voltmeter always measure rms value
iO
t 7. Inductive reactance (XL) XC= c_
1
⇒ Values printed on ac circuits are rms values ω
T/2 XL=L ω Unit-oh
⇒ In houses ac is supplied at 220V which is the rms Unit-ohm(Ω)
T/4 3T/4 T plays ro
of voltage
plays role of resistance resista
⇒ Peak value is 220√2= 311V V
-iO
6. i0= _0 & irms = Vrms V V
8. i0= _0 & irms = rms V
⇒ Frequency in general is 50Hz R
XL XL 8. i0= _0 &
⇒ for measuring ac hot wire ⇒ w=2nf=100π rad/sec (314 rad/sec) X c
XC
instruments are used
AVERAGE VALUE OF AC FOR ONE TIME PERIOD PHASOR DIAGRAM SERIES AC CIRCUITS
T T
∫ Idt ∫ I sin ωt dt Diagram representing ac voltage or 1) R-L CIRCUITS
I= 0
T =
0 O
T
= area of I-t graph current as vectors with phase angle 5. Impedance phasor
∫ dt ∫ dt time v R L XL
0 0 between them. 0
Z= R2+l2
I0
I= 0 for 0→T for a sinusoidal ac wave.
VR VL X
The average value of sin or cos function for I=I0sin(ωt)
l _L
tan O=
, 3) L-C CIRCUIT RESONANCE IN LCR SERIES CIRCUIT
CO
QUALITY FACTOR
R1>R2>R3
ωr L 1 1 L
i L C In series resonance, impedance of circuit is Q= _ = _ = _ _
R ωr cR R C
minimum & equal to resistance = Z= R, and Mas
[
V = VO sin t
curent is maximum Voltage across C or L [
VL VC or Q= _
~ VL = iO XL , VC = iO XC applied voltage 1.Displace
Condition for resonance resonance
V=Vosinωt
Voltage phasor diagram
1 Less sharp the resonance, less is the selectivity 2.Velocity
XL = XC L =
C of the circuit.If the Quality factor is large, R
V = VL ~ VC [ie, (VL - VC) or (VC - VL)] 1
3. Acceler
= r = rad / sec is low or L is large, the circuit is more selective.
VL LC 4. Mass (m
XL Impedance Phasor Diagram
r resonant frequency (angular) 5. Force co
Sharpness of Resonance
i
io
1 6. Momen
io f = fr = ωr
Sharpness= Q= _
Hz 7. Retardi
Z = XL ~ XC [ (XL - XC) 2 LC ; 2∆ω -bandwidth
VC or (XC - XL)] 2∆ω
XC fr = resonant frequency smaller∆ω, sharper or narrower the 8. Differe
if XL > Xc , Voltage leads the current by GRAPH
resonance. _
d 2x +
2
io dt2
if XC > XL , current leads the voltage by
2
if XL = XC , Z = 0, i = POWER IN AC CIRCUIT ω=
w<wr
or, r, XC > XL
i Z w>wr <
ωL = 1 current leads
l
Average Power P= Vrms Irms cos O
ωC 9. K.E=
r
r, XL > XC Case 1 l
P= Irms Zcos O
2
ω = 1 = ωo Variation of peak current with applied >
Elast
LC
frequency
current lags Purely Resistive circuit - Ol =0 l =1
,COS O
o ωo
In resonance Maximum power dissipation
L-C-R Series Circuit Case 2 TRANSFORM
L C R V = VR (applied voltage = voltage across
resistance) Purely inductive or capacitive circuit- “Device whi
VL VC VR Z = R (impedance is minimum and equal to l =90
O 0
l
cos O=0 voltage in ac
resistance) No power is dissipated even though a induction”.Tr
i V=Vosinωt or decrease
Voltmeter connected across VL & VC will show current is flowing in the circuit current but
~
the same reading Case 3
V = VO sin t
Voltmeter connected commonly across inductor LCR Series circuit
VR = iOR, VL = iO XL , VC = iOX C & capacitor shows no reading l non zero in R-L,C-R,or CLR circuit.
O
Assuming VL > VC for drawing phasor Here V = 0
P=Vrms Irms Cos O
l EQUATIONS
V
Voltage phasor diagram VL = VC
VL VC VR
V N
VL
VL - VC VN = VR Case 4 1) _S = _S
Power dissipation at resonance VP NP
VR
io Vnet = VR 2) Efficiency
VR imax = l
XL-XC=0 or O=0 l
=> cos O=1 ⇒Z=R
~ Z R
P=I2Z = I 2 R 3) For idea
VC
VR, io Vnet Maximum power is dissipiated in a circuit
Mains and Advanced
.
PHYSI
, TABLE OF CON
Chapter 01 AL
Chapter 02 EL
Chapter 03 EL
Chapter 04 EM
,ALTERNATING CURRENT ROOT MEAN Average value of
SQUARE CURRENT ac is defined for SINGLE COMPONENT CIRCUITS
“If the direction of current in a
resistor or any other element changes positive or negative
alternately, the current is called an half cycle
Irms= I2 = Io 2 = Io
alternating current” 2
2Io 2Vo
2 I= V=
Irms= Io Vrms= Vo
π π
Resistor only Inductor only Capacit
2 2
AVERAGE AND RMS VALUE OF AC R L
i i
i
If the current or voltage is SAWTOOTH FUNCTION
sinusoidal than it can be expressed as Io
IO For half cycle I=
i=i0sin(ωt+ ) 2
v=v0sin(ωt+ ) For full cycle I= 0
i0 Peak current or current amplitude T/2 T Io 2
Mean square current I2 =
v0 Peak voltage or voltage amplitude 3
Irms= Io
-IO V=V0sin ωt
rms current V=V0sin ωt
2π = 2πf T:Time period 3
1. V=V0sin ωt 1. V=V0sin ω
ω= 1. V=V0sin ωt
T
RECTANGULAR FUNCTION 2. i=i0sin (ωt- π/2 ) 2. i=i0sin (ω
f:frequency (Hz or cycle/sec)
I 2. i=i0sin ωt
(ωt+ ): Total phase +Io For half cycle I= Io 3. or current leads to the 3.Current le
3. V&i are in phase voltage by π/2 voltage b
For full cycle I= 0
T/2 T
GENERAL GRAPH Mean square current I2= Io2 4. 4.
-Io
V 4. V
)
rms current Irms= Io I π/2
if i=i0 sinωt
l =0,COS Ol =1
5. O
iO
AVERAGE HEAT PRODUCED DURING A P= εrmsIrms
CYCLE OF AC V
3T/4 T
1
T/4 T/2 t Havg= 2
Io R = Irms R
2 2 V,i )
π/2
I l π/2,C
5. O=
-iO Keep in mind V 6. P=0 (wat
V0 l π/2,COS Ol =0
5. O=
⇒ rms value is also called virtual value or effective value I
i=i0 cosωt i0 6. P= 0 (wattless circuits) 7. Inductive
⇒ AC ammeter and voltmeter always measure rms value
iO
t 7. Inductive reactance (XL) XC= c_
1
⇒ Values printed on ac circuits are rms values ω
T/2 XL=L ω Unit-oh
⇒ In houses ac is supplied at 220V which is the rms Unit-ohm(Ω)
T/4 3T/4 T plays ro
of voltage
plays role of resistance resista
⇒ Peak value is 220√2= 311V V
-iO
6. i0= _0 & irms = Vrms V V
8. i0= _0 & irms = rms V
⇒ Frequency in general is 50Hz R
XL XL 8. i0= _0 &
⇒ for measuring ac hot wire ⇒ w=2nf=100π rad/sec (314 rad/sec) X c
XC
instruments are used
AVERAGE VALUE OF AC FOR ONE TIME PERIOD PHASOR DIAGRAM SERIES AC CIRCUITS
T T
∫ Idt ∫ I sin ωt dt Diagram representing ac voltage or 1) R-L CIRCUITS
I= 0
T =
0 O
T
= area of I-t graph current as vectors with phase angle 5. Impedance phasor
∫ dt ∫ dt time v R L XL
0 0 between them. 0
Z= R2+l2
I0
I= 0 for 0→T for a sinusoidal ac wave.
VR VL X
The average value of sin or cos function for I=I0sin(ωt)
l _L
tan O=
, 3) L-C CIRCUIT RESONANCE IN LCR SERIES CIRCUIT
CO
QUALITY FACTOR
R1>R2>R3
ωr L 1 1 L
i L C In series resonance, impedance of circuit is Q= _ = _ = _ _
R ωr cR R C
minimum & equal to resistance = Z= R, and Mas
[
V = VO sin t
curent is maximum Voltage across C or L [
VL VC or Q= _
~ VL = iO XL , VC = iO XC applied voltage 1.Displace
Condition for resonance resonance
V=Vosinωt
Voltage phasor diagram
1 Less sharp the resonance, less is the selectivity 2.Velocity
XL = XC L =
C of the circuit.If the Quality factor is large, R
V = VL ~ VC [ie, (VL - VC) or (VC - VL)] 1
3. Acceler
= r = rad / sec is low or L is large, the circuit is more selective.
VL LC 4. Mass (m
XL Impedance Phasor Diagram
r resonant frequency (angular) 5. Force co
Sharpness of Resonance
i
io
1 6. Momen
io f = fr = ωr
Sharpness= Q= _
Hz 7. Retardi
Z = XL ~ XC [ (XL - XC) 2 LC ; 2∆ω -bandwidth
VC or (XC - XL)] 2∆ω
XC fr = resonant frequency smaller∆ω, sharper or narrower the 8. Differe
if XL > Xc , Voltage leads the current by GRAPH
resonance. _
d 2x +
2
io dt2
if XC > XL , current leads the voltage by
2
if XL = XC , Z = 0, i = POWER IN AC CIRCUIT ω=
w<wr
or, r, XC > XL
i Z w>wr <
ωL = 1 current leads
l
Average Power P= Vrms Irms cos O
ωC 9. K.E=
r
r, XL > XC Case 1 l
P= Irms Zcos O
2
ω = 1 = ωo Variation of peak current with applied >
Elast
LC
frequency
current lags Purely Resistive circuit - Ol =0 l =1
,COS O
o ωo
In resonance Maximum power dissipation
L-C-R Series Circuit Case 2 TRANSFORM
L C R V = VR (applied voltage = voltage across
resistance) Purely inductive or capacitive circuit- “Device whi
VL VC VR Z = R (impedance is minimum and equal to l =90
O 0
l
cos O=0 voltage in ac
resistance) No power is dissipated even though a induction”.Tr
i V=Vosinωt or decrease
Voltmeter connected across VL & VC will show current is flowing in the circuit current but
~
the same reading Case 3
V = VO sin t
Voltmeter connected commonly across inductor LCR Series circuit
VR = iOR, VL = iO XL , VC = iOX C & capacitor shows no reading l non zero in R-L,C-R,or CLR circuit.
O
Assuming VL > VC for drawing phasor Here V = 0
P=Vrms Irms Cos O
l EQUATIONS
V
Voltage phasor diagram VL = VC
VL VC VR
V N
VL
VL - VC VN = VR Case 4 1) _S = _S
Power dissipation at resonance VP NP
VR
io Vnet = VR 2) Efficiency
VR imax = l
XL-XC=0 or O=0 l
=> cos O=1 ⇒Z=R
~ Z R
P=I2Z = I 2 R 3) For idea
VC
VR, io Vnet Maximum power is dissipiated in a circuit
