Chapter 21: RLC Circuits

[Pages:33]Chapter 21: RLC Circuits

PHY2054: Chapter 21

1

Voltage and Current in RLC Circuits

?AC emf source: "driving frequency" f

= m sint

= 2 f

?If circuit contains only R + emf source, current is simple

i

=

R

=

Im

sin (t )

Im

=

m

R

(current amplitude)

?If L and/or C present, current is not in phase with emf

i = Im sin (t - )

Im

=

m

Z

?Z, shown later

PHY2054: Chapter 21

2

AC Source and Resistor Only

?Driving voltage is = m sint

?Relation of current and voltage

i = /R

i = Im sint

Im

=

m

R

i

~

R

Current is in phase with voltage ( = 0)

PHY2054: Chapter 21

3

AC Source and Capacitor Only

?Voltage is

vC

=

q C

=

m

sin t

?Differentiate to find current

q = Cm sint

i

i = dq / dt = CVC cost

~

C

?Rewrite using phase (check this!)

i = CVC sin (t + 90?)

?Relation of current and voltage

i = Im sin (t + 90?)

Im

=

m

XC

( XC = 1/C)

?"Capacitive reactance": XC = 1/C

Current "leads" voltage by 90?

PHY2054: Chapter 21

4

AC Source and Inductor Only

?Voltage is vL = Ldi / dt = m sint

?Integrate di/dt to find current:

di / dt = (m / L)sint i = -(m /L)cost

?Rewrite using phase (check this!)

i = (m /L)sin (t - 90?)

i

~

L

?Relation of current and voltage

i = Im sin (t - 90?)

Im

=

m

XL

(XL =L)

?"Inductive reactance": X L = L

Current "lags" voltage by 90?

PHY2054: Chapter 21

5

General Solution for RLC Circuit

?We assume steady state solution of form i = Im sin (t - )

Im is current amplitude

is phase by which current "lags" the driving EMF

Must determine Im and

?Plug in solution: differentiate & integrate sin(t-)

i = Im sin (t - )

di dt

=

Im

cos (t

-)

Substitute

L

di dt

+

Ri

+

q C

=

m

sin t

q = - Im cos(t - )

I

m

L

cos

(t

-

)

+

Im

R

sin

(t

-

)

-

Im

C

cos

(t

-

)

=

m

sin

t

PHY2054: Chapter 21

6

General Solution for RLC Circuit (2)

Im

L

cos

(t

-

)

+

Im

R

sin

(t

-

)

-

Im

C

cos

(t

-

)

=

m

sin

t

?Expand sin & cos expressions

sin (t - ) = sint cos - cost sin cos(t - ) = cost cos + sint sin

High school trig!

?Collect sint & cost terms separately

(L -1/C )cos - R sin = 0 Im (L -1/C )sin + ImR cos = m

cost terms sint terms

?These equations can be solved for Im and (next slide)

PHY2054: Chapter 21

7

General Solution for RLC Circuit (3)

?Solve for and Im

tan = L -1/C X L - XC

R

R

Im

=

m

Z

?R, XL, XC and Z have dimensions of resistance

XL =L XC = 1/C

Inductive "reactance" Capacitive "reactance"

Z = R2 + ( X L - XC )2

Total "impedance"

?This is where , XL, XC and Z come from!

PHY2054: Chapter 21

8

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