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Lecture 7 Overview

• Filters

• Transient Circuits

Low pass filter

• RC low-pass filter: preserves lower frequencies, attenuates frequencies above the cutoff frequency ω0=1/RC.

01

1

1

/tan

2

tan

2

1tan

0

2

V

io

i

oV

)(1

1

)(1

1

)(1

1

1

1

1

1)(H

)(V)(V

)(V

)(V)(H

j

RCj

RCj

j

RC

C

eRC

eRC

e

e

RCRCj

RCj

Cjj

jZZ

Zj

j

jj

RC

10

Low pass filter

01

2

/tan

)(1

1

o

i

iVo

VRC

VHV

Break frequency ω=ω0=1/RC,

HV=1/√2

N.B. decibels:

db32

1log20

log10

log20

10

010

010

X

XX

X

XX

dB

dB

For voltage

For power

Build other filters by combining impedance response

Which of the following is a low-pass filter?

Answer: (c)

What happens to the output voltage when ω→0 (DC condition)?

Which of the following are high-pass or low-pass filters?

Answers: (b) and (c) are high-pass; (a) and (d) are low-pass

RLC Band-pass filters

CL

Measuring voltage output signal over R, Vr

Low frequencies, C open, L shorted, Vr minimum

High frequency, C shorted, L open, Vr minimum

so, at high and low frequencies, see an open circuit - Vr minimum

Band-stop (Notch) filters

Measuring voltage output signal over L and C

Low frequencies, C open, L shorted, Vlc maximum

High frequency, C shorted, L open, Vlcmaximum

so, at high and low frequencies, see an open circuit - Vlc maximum

Another Example:

Measuring voltage output signal over L and C, but this time in parallel (i.e. at high and low frequencies, see a short - V0=0)

Transient Circuits

Charging a capacitor

RC

dt

tv

dvR

tv

dt

dvC

R

tv

R

tvii

dt

dvC

dt

dQi

tvtv

c

C

cC

cRRC

CCC

CR

)(

)( :so

)()(

0)()(

Integrate both sides using: )ln(11

baxa

dxbax

RC

t

RC

tA

C

C

Beeetv

ARC

ttv

)(

))(ln(

Gives:

During Charging (left-hand loop):

)1()( RC

t

C etv

Boundary condition: at t=0, vC(t)=0 so: BBe 0

RC

t

c etv

)(

Charging a capacitorSolution is only true for that particular circuit (Voltage source plus resistor), but more complicated circuits can be reduced to this using Thevenin's Theorem

Time constant τ=RC. Time needed to charge capacitor to 63% of full charge

Larger RC means the capacitor takes longer to chargeLarger R implies smaller current flowThe larger C is, the more charge the capacitor can hold.

Discharging a capacitor

Time constant τ=RC. Time needed for capacitor to drop to 37% of full chargeCurrent flows in the opposite direction to charging

Larger RC means the capacitor takes longer to dischargefile:///Users/jholder/teaching/phys645/2011/rc/rc.htmlhttp://www.phy.ntnu.edu.tw/ntnujava/viewtopic.php?t=48

RC

t

etv

)(

Charged

Begin discharging

RC

t

eRdt

dvC

dt

dqi

Charging Capacitors

Which configuration has the largest final charge on the capacitor?

Answer: both the same (no current flow means no voltage drop across resistors)

Charging Capacitors

Which capacitor charges fastest?

Answer: b)

Time response of Inductors

Switch to position a:

0

0

iRdt

dIL

iRVL

iR

didt

L

dt

diLiR

1

Integrate and apply boundary condition t=0, i=0

)1( L

Rt

eR

i

Time constant τ=L/R.

Switch to position b:L

Rt

eR

i

Talk about "Charging a capacitor"

"Current build-up" in an inductor

)1( L

Rt

eR

i

)L

Rt

L ev

LL v

Ri ,

Time response of Inductors

A battery is connected to an inductor. When the switch is opened does the light bulb:

1.Remain off2.Go off3. Slowly Dim out4. Keep burning as brightly as it did before the switch was opened5. Flare up brightly, then dim and go out

Answer 5

Why care about Transient Analysis?not everything is a sine wave....

• Time dependence is very useful for some applications (e.g. making a clock)

• Also becomes important in digital applications

• Transient: refers to transition region between two states.

• e.g. at the edges of a square wave

Transient Analysis

• Transistors are used as switches in digital circuits

• Gate input used to control switching

• Physical construction of transistors leads to "Capacitor Effect"

• Switching is not instantaneous

• Delay determined by RC time constant

Ideal Realistic