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Announcements
• Assignment 1 due now.
• Assignment 2 posted, due on Thursday
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.
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Low pass filter
01
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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
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dvC
R
tv
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dt
dvC
dt
dQi
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CCC
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)()(
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Integrate both sides using: )ln(11
baxa
dxbax
RC
t
RC
tA
C
C
Beeetv
ARC
ttv
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))(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
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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
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