Lecture 4: R-L-C Circuits and Resonant Circuits

Lecture 4: R-L-C Circuits and Resonant Circuits

RLC series circuit:

What's VR? Simplest way to solve for V is to use voltage divider equation in complex notation:

VR

=

Vin R R + XC +

XL

= R+

Vin R 1 + jL

jC

Vin = V0 cos t

X L XC L CR

Using VR

complex notation for

=

R

+

V0e jt R j$&L -

1

' )

the

apply

voltage

Vin

=

V0cost

=

Real(V0

e

jt

):

% C (

We are interested in the both the magnitude of VR and its phase with respect to Vin.

First the magnitude:

V0e jt R

VR

=

R + j$&L -

1

' )

% C (

=

V0 R

R2 + $&L -

1

' 2 )

% C (

K.K. Gan

L4: RLC and Resonance Circuits

1

 The phase of VR with respect to Vin can be found by writing VR in purely polar notation.

For the denominator we have:

R

+

j$%&L

-

1 C

' ( )

=

R2

+

$%&L

-

1 C

' 2 ( )

0 2 exp1 2 3

j

tan

-1*,L , + ,

-1 C

R

- 4 / 2 / 5 ./62

Define the phase angle :

tan = Imaginary X

Real X

L - 1

= C

R

We can now write for VR in complex form:

VR = e j

Vo R e jt

R2 + %'L -

1

( 2 *

Depending on L, C, and , the phase angle can be

& C ) positive or negative! In this example, if L > 1/C,

= VR e j(t-)

then VR(t) lags Vin(t).

Finally, we can write down the solution for V by taking the real part of the above equation:

VR = Re al

V0 R e j(t-)

=

R

2

+

%&'L

-

1 C

( 2 ) *

V0R cos(t - )

R2

+

%&'L

-

1 C

( 2 ) *

K.K. Gan

L4: RLC and Resonance Circuits

2

R = 100 , L = 0.1 H, C = 0.1 ?F

VR ................
................

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