Low pass filter RC low-pass filter: preserves lower frequencies, attenuates frequencies above the cutoff frequency ω 0 =1/RC.
Low pass filter Break frequency ω=ω 0 =1/RC, H V =1/√2 N.B. decibels: 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 C L Measuring voltage output signal over R, V r Low frequencies, C open, L shorted, V r minimum High frequency, C shorted, L open, V r minimum so, at high and low frequencies, see an open circuit - V r minimum
Band-stop (Notch) filters Measuring voltage output signal over L and C Low frequencies, C open, L shorted, V lc maximum High frequency, C shorted, L open, V lc maximum so, at high and low frequencies, see an open circuit - V lc 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 - V 0 =0)
Charging a capacitor Integrate both sides using: Gives: During Charging (left-hand loop): Boundary condition: at t=0, v C (t)=0 so:
Charging a capacitor Solution 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 charge Larger R implies smaller current flow The 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 charge Current flows in the opposite direction to charging Larger RC means the capacitor takes longer to discharge file:///Users/jholder/teaching/phys645/2011/rc/rc.html http://www.phy.ntnu.edu.tw/ntnujava/viewtopic.php?t=48 Charged Begin discharging
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: Integrate and apply boundary condition t=0, i=0 Time constant τ=L/R. Switch to position b: Talk about "Charging a capacitor" "Current build-up" in an inductor
Time response of Inductors A battery is connected to an inductor. When the switch is opened does the light bulb: 1.Remain off 2.Go off 3. Slowly Dim out 4. Keep burning as brightly as it did before the switch was opened 5. 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 IdealRealistic