[PDF] Chapter 21: RLC Circuits



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Chapter 21: RLC Circuits

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PHY2054: Chapter 21

1

Chapter 21: RLC Circuits

PHY2054: Chapter 21

2

Voltage and Current in RLC Circuits

AC emf source: "driving frequency" f

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

If L and/or C present, current is

not in phase with emf

Z, shown later

sin mmm iI t I Z sin m t 2f sin current amplitude mmm iI tI R R

PHY2054: Chapter 21

3

AC Source and Resistor Only

Driving voltage is

Relation of current and voltage

Current is

in phase with voltage (= 0) i R sin mmm iI tI R sin m t /iR

PHY2054: Chapter 21

4

AC Source and Capacitor Only

Voltage is

Differentiate to find current

Rewrite using phase (check this!)

Relation of current and voltage

"Capacitive reactance":

Current "leads" voltage by 90

sin m qC t i C /cos C idqdt CV t sin Cm qvtC sin 90 C iCV t sin 90 mmm C iI t IX 1/ C XC 1/ C X C

PHY2054: Chapter 21

5

AC Source and Inductor Only

Voltage is

Integrate di/dt to find current:

Rewrite using phase (check this!)

Relation of current and voltage

"Inductive reactance":

Current "lags" voltage by 90

//sin m di dt L t i L /cos m iLt /sin Lm v Ldi dt t /sin 90 m iLt sin 90 mmm L iI t IX L XL L XL

PHY2054: Chapter 21

6

General Solution for RLC Circuit

We assume steady state solution of form I m is current amplitude is phase by which current "lags" the driving EMF

Must determine I

m and

Plug in solution: differentiate & integrate sin(

t- sin m iI t cos sin cos sin mmm m I I

LtIRt t tC

Z sin m di qLRi tdt C sin m iI t cos m diItdt cos m I qt Z

Substitute

PHY2054: Chapter 21

7

General Solution for RLC Circuit (2)

Expand sin & cos expressions

Collect sin

t&cos tterms separately

These equations can be solved for I

m and (next slide)

1/ cos sin 0

1/ sin cos

mmm LC R

IL C IR

Z ZI IH sin sin cos cos sin cos cos cos sin sintt t ttt I Z IZIZ I

High school trig!

cos tterms sin tterms cos sin cos sin mmm m I I

LtIRt t tC

Z

PHY2054: Chapter 21

8

Solve for

and I m R, X L , X C and Z have dimensions of resistance

This is where

, X L , X C and Z come from!

General Solution for RLC Circuit (3)

1/tan LC XXLC R R mm I Z 22
LC ZRXX L XL 1/ C XC

Inductive "reactance"

Capacitive "reactance"

Total "impedance"

PHY2054: Chapter 21

9

AC Source and RLC Circuits

tan mm LC IZ XX R

Phase angleMaximum current

= angle that current "lags" applied voltage 2 1/ L C XL f XC

Inductive reactance

Capacitive reactance

22LC
ZRXX

Total impedance

PHY2054: Chapter 21

10

What is Reactance?

Think of it as a frequency-dependent resistance

Shrinks with increasing

1 C XC L X L R X R

Grows with increasing

Independent of

PHY2054: Chapter 21

11

Pictorial Understanding of Reactance

tan LC XX R 22LC
ZRXX cos RZ

PHY2054: Chapter 21

12 Summary of Circuit Elements, Impedance, Phase Angles 22LC
ZRXX tan LC XX R

PHY2054: Chapter 21

13 Quiz Three identical EMF sources are hooked to a single circuit element, a resistor, a capacitor, or an inductor. The current amplitude is then measured as a function of frequency. Which one of the following curves corresponds to an inductive circuit? (1) a (2) b (3) c (4) Can't tell without more info f I max a c b max max 2 L L XL f IX

For inductor, higher frequency gives higher

reactance, therefore lower current

PHY2054: Chapter 21

14

RLC Example 1

Below are shown the driving emf and current vs time of an RLC circuit. We can conclude the following

Current "leads" the driving emf (<0)

Circuit is capacitive (X

C > X L I t

PHY2054: Chapter 21

15

RLC Example 2

R = 200

, C = 15

F, L = 230mH,

max = 36v, f = 60 Hz

2 60 0.23 86.7

L X 6

1/ 2 60 15 10 177

C X 22

200 86.7 177 219Z

max max / 36/219 0.164AIZ X C > X L

Capacitive circuit

1

86.7 177tan 24.3200

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