RC Filters

B. Furman

02SEP2014

Mechatronics Concept Map

System toControl

Sensor

SignalConditioning

Controller(Hardware & Software)

PowerInterface

Actuator

UserInterface

PowerSource

BJ Furman 15JAN11

ME 106ME 154ME 157ME 195

ME 120ME 297A

ME 106ME 120

ME 106ME 190ME 187

ME 110ME 136ME 154ME 157

ME 182ME 189ME 195

ME 106ME 120

ME 106

INTEGRATION

How to Handle Noisy Signals?

Inset from: http://cache.national.com/ds/LM/LM35.pdf

Filter!

Impedance of a Capacitor

Derive the impedance of a capacitor

(1) )()( tCVtQ

(2) )(Let stAetV

(3) )(

)()( stCsAe

dt

tdVCti

dt

tdQ

(4) )(

)()( Define

ti

tVtZ

(5) 1

)(CsCsAe

AetZ

st

st

Physics for a capacitor

A time varying function

Differentiate both sides

The impedance – the ratio of voltage to current

CV(t)

What is s?

What is s?

Previously(2) )( stAetV A time varying function

rad/sin frequency,circular and

1 where

(6) Let

j

js

Why?

(7) )cos()sin()( tjtAAetV tj A sinusoidal function, which can also be thoughtof as rotating vector

jCZC

1

Complex Numbers and Vectors

Think of a complex number as a vectorVectors have:

Magnitude (length) Direction (angle)

(7) )cos()sin()( tjtAAetV tj

jba

j

Realaxis

(imaginary axis)

a

b

22 ba

a

b1tan

Magnitude =

Direction =

is of the form:

Sinusoidal Function Visualize the connection between the

vector and the sinusoidal function of time Suppose the imaginary

component is plotted as a function of time

http://upload.wikimedia.org/wikipedia/commons/8/89/Unfasor.gif

(7) )cos()sin()( tjtAAetV tj

(8) )cos()( tAtVimag

Generalized Voltage Divider

What is Vo in terms of Vi, Z1, and Z2?

Vi

+

Z2

Z1

Vo (9) ZZ

Z

21

2

IO VV

RC Filters - 1 Frequency dependent voltage divider

Impedance of a resistor, R Impedance of a capacitor, ZC

R

Vi

+

C Vo

jCZC

1

function" transfer the" i

o

V

V

1j

jRCR

jC

jCV

V

i

o

1

11

1

RC Filters - 2 Transfer function

Complex number jRCV

V

i

o

1

1

Magnitude (Magnitude of numerator)/(Magnitude of denominator)

Angle (“phase angle”) (angle of num) – (angle of denom) How much the output is out of time

synchronization with input

21

1

RCV

V

i

o

RCV

V

i

o 1tan

R

Vi C Vo

RC Filters - 3

BehaviorHow does the magnitude and angle of the

transfer function change with frequency? Ex. R=10k, C=1uF

R

Vi C Vo

Filter ?

Bode Plots (the frequency response of the filter)

Magnitude vs. frequency

Phase angle vs. frequency

Mechatronics Concept Map

System toControl

Sensor

SignalConditioning

Controller(Hardware & Software)

PowerInterface

Actuator

UserInterface

PowerSource

BJ Furman 26JAN06

ME 106ME 154ME 157ME 195

ME 120ME 297A

ME 106ME 120

ME 106ME 190ME 187

ME 110ME 136ME 154ME 157

ME 182ME 189ME 195

ME 106ME 120

ME 106

INTEGRATION