2018-10-01_Physics_For_You

• Resonance Condition : e current will be

maximum when ω

ω

L

C

−=

1

0 ⇒=ω

1

LC

and

corresponding frequency is υ

ω

ππ

==

2

1

2

1

LC

is frequency is known as resonant frequency of

the given circuit. At this frequency peak current

will be I

V

(^0) R

=^0

If the resistance R in the LCR circuit is zero, the

peak current at resonance is I

V

0

0

0

=

It means, there can be a nite current in pure

LC circuit even without any applied emf,

is current in the circuit is at frequency,

υ

π

=^1

2

1

LC

QUALITY FACTOR

It is a measure of sharpness of resonance. It is dened

as the ratio of reactance of either the inductance or

capacitance at the resonant angular frequency to the

total resistance of the circuit.

Q

X

R

L

R

==Lrω ; Q X

RCR

C

r

==^1

ω

; Q

R

L

C

=^1

Quality factor is also expressed in terms of bandwidth

Q=

Resonant frequency

Bandwidth

POWER IN AC CIRCUIT

In an ac circuit we may dene three types of power.

• Instantaneous power : e power in the ac circuit
  at any instant of time is known as instantaneous
  power. It is equal to the product of values of
  alternating voltage and alternating current at that
  time.


• Average power (Pav) : e power averaged over one
  full cycle of ac is known as average power. It is also
  known as true power.

PVI

VI

av==rmsrmscoscφφos

00

2

• Apparent power : e product of virtual voltage
  (Vrms) and virtual current (Irms) in the circuit is
  known as virtual power.

PVI

VI

vr==ms rms

00

2

Power Factor

It is dened as the ratio of true power to apparent power

of an ac circuit

cosφ=

Truepower

Apparentpower

• Power factor is also dened as the ratio of the
  resistance to the impedance of an ac circuit

cosφ=

R

Z

It is unitless and dimensionless quantity.

• In pure resistive circuit,
  φ = 0°; cos φ = 1.


• In pure inductive or capacitive circuit

φ

π

=

2

; cos φ = 0.

• In RL circuit,

ZRX R

L Z

=+^22 andcosφ=

• In RC circuit,

ZRX

R

C Z

=+^22 andcosφ=

• In series LCR circuit,

ZRXX

R

LC Z

=+^22 ()−=andcosφ

• At resonance, XL = XC
  ? Z = R and φ = 0°
  cos φ = 1

PARALLEL AC CIRCUITS

Let us consider an alternating

source connected across an

inductance L in parallel with

a capacitor C. e resistance

in series with the inductance

is R and with the capacitor is

zero.

Let the instantaneous value of emf applied be V and the

corresponding current is I, IL and IC. en,

I = IL + IC

or V

Z

V

RjL

V

jC

V

RjL

CV

j

=

+

−=

+

−

ωω ω

ω

/

()

=

+

−

V

RjL

jCV

ω j

()ω

2 = + +=−

V

RjL

jCVj

ω

()ω ()as^21

• Admittance :
  11
  ZRjL

= jC

+

+

ω

ω