Post on 11-Jan-2016
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