Capacitor Load

tCVdt

dqi

Cvq

tVv

CC

CC

CC

cos

sin

90sin90sin1

tIt

C

Vi C

CC

The capacitive reactance of a capacitor CXC

1

C

CC X

VI Generalized Ohm’s law: Notice Ic and VC are amplitudes

ICE

Inductive Load

tL

Vdt

L

Vi

dt

diLv

tVv

LLL

LL

LL

cossin

sin

90sin90sin tItL

Vi L

LL

The Inductive reactance of a inductive LX L

L

LL X

VI Generalized Ohm’s law: Notice IL and VL are amplitudes

ELI

The series RLC circuittdm sin

1. Same current through R, L, C

tIi dsin

Same frequency as in the source

t-

Ii2. Consider VR, VC, VL

tIRiRV dR sinVR

vR

C

IIXVtv

dCCdC

,

2sinVC

VC

vCLIIXVtv dLLdL

,

2sinVL

VL

LCR vvv

The series RLC circuit: Continuous

LCR vvv

This relation has to be maintained as phosors are rotating

Values at t.

CLRm VVV

General rules: 1. KVL and KCL still hold, but

values at the same t have to be used, i.e. vertical components in phasor diagram.

2. Vectors operation for Amplitude

vV

The series RLC circuit: Continuous

CLRm VVV

22

22

22

1R

CLI

RXXI

VVV

CL

RCLm

R

XX

RCL

CL

1

tan

ZRCLI mm

221

Z is the impedance of the circuit

=I(XL-XC)

=IR

=IZ

Examples33-43P. A coil of inductance 88 mH and unknown resistance and a 0.94 F are connected in series with an alternating emf of frequency 930 Hz. If the phase constant between the applied voltage and current is 75, what is the resistance of the coil.

f=930 Hzd=2f

=I(XL-XC)

=IR

=IZ

=I(XL-XC)

=IR

=IZRLC Resonance

R

XX

RCL

CL

1

tan

XL>XC: inductive loading

XC>XL: Capacitive loading

XL=XC: Resonance

RLC Resonance: Cont

R

XX

RCL

CL

1

tan 221 RCL

I m

LC

CL

XX

d

dd

CL

1

1

RI mmax

=I(XL-XC)

=IR

=IZConditions at Resonance

22

22

1 RCL

RXXI

m

CL

m

• I is a maximum

• Z is at minimum; Z=R; Z is purely resistive• XL=XC; inductive reactance cancels capacitive reactance; net reactance is zero• The phase angle is zero; the current is perfectly in phase with applied emf; the tangent of the phase angle is zero.• The driven frequency is identical to the natural frequency.

• The power factor is unity