E40M. RC Filters. M. Horowitz, J. Plummer, R. Howe 1

Transcription

1 E40M RC Filters M. Horowitz, J. Plummer, R. Howe

2 Reading Reader: The rest of Chapter is about log-log plots 7.4 is about filters A & L M. Horowitz, J. Plummer, R. Howe 2

3 EKG (Lab 4) Concepts Amplifiers Impedance Noise Safety Filters Components Capacitors Inductors Instrumentation and Operational Amplifiers In this project we will build an electrocardiagram (ECG or EKG). This is a noninvasive device that measures the electrical activity of the heart using electrodes placed on the skin. M. Horowitz, J. Plummer, R. Howe 3

4 RC Circuit Analysis Approaches. For finding voltages and currents as functions of time, we solve linear differential equations or run EveryCircuit. 2. For finding the response of circuits to sinusoidal signals, * we use impedances and frequency domain analysis * superposition can be used to find the response to any periodic signals M. Horowitz, J. Plummer, R. Howe 4

5 Key Ideas on RC Circuit Frequency Analysis - Review All voltages and currents are sinusoidal So we really just need to figure out What is the amplitude of the resulting sinewave And sometimes we need the phase shift too (but not always) These values don t change with time This problem is very similar to solving for DC voltages/currents M. Horowitz, J. Plummer, R. Howe 5

6 Key Ideas on Impedance - Review Impedance is a concept that generalizes resistance: For sine wave input Z for a resistor is just R Z mag( V ) mag( i) Add j to represent 90 o phase shift It does not depend on frequency, it is simply a number. What about a capacitor? Z C V i V CdV /dt ( ) ( ) V O sin 2πFt 2πFCV O cos 2πFt Z C V i j 2πFC M. Horowitz, J. Plummer, R. Howe 6

7 Analyzing RC Circuits Using Impedance - Review The circuit used to couple sound into your Arduino is a simple RC circuit. This circuit provides a DC voltage of V dd /2 at the output. For AC (sound) signals, the capacitor will block low frequencies but pass high frequencies. (High pass filter). For AC signals, the two resistors are in parallel, so the equivalent circuit is shown on the next page. M. Horowitz, J. Plummer, R. Howe 7

8 Analyzing RC Circuits Using Impedance Review (High Pass Filter) v in C0.µF v out V out V in R + R j 2πFC j 2πFRC + j 2πFRC R0kW RC ms; 2pRC about 70ms V out /V in F (Hz) M. Horowitz, J. Plummer, R. Howe 8

9 RC FILTERS M. Horowitz, J. Plummer, R. Howe 9

10 RC Circuits Can Make Other Filters Filters are circuits that change the relative strength of different frequencies Named for the frequency range that passes through the filter Low pass filter: Passes low frequencies, attenuates high frequency High pass filter Passes high frequencies, attenuates low frequencies Band pass filter Attenuates high and low frequencies, lets middle frequencies pass M. Horowitz, J. Plummer, R. Howe 0

11 RC Low Pass Filters v in R kw v out Let s think about this before we do any math C0. µf Very low frequencies à RC x 0 3 x 0. x 0-6 s. ms 2π RC 6.9 ms /(2π RC ) 45 Hz Very high frequencies à M. Horowitz, J. Plummer, R. Howe

12 RC Low Pass Filters v in RKW C0.µF v out V out V in j 2πFC R + j 2πFC + j 2πFRC + jf/fc F C /[2pRC] 0.8 V out /V in RC. ms F c /[2pRC] 45 Hz F (Hz) M. Horowitz, J. Plummer, R. Howe 2

13 RC Filters Something a Little More Complicated v in C v out Let s think about this before we do any math R 0C Very low frequencies à V out /V in Very high frequencies à capacitive divider F M. Horowitz, J. Plummer, R. Howe 3

14 RC Filters Something More Complicated v in R C 0C v out Z 2 R Z j 2πFC R + j 2πF0RC + j 2πF0C Z Z 2 V out V in R + j 2πF0RC R + j 2πF0RC + j 2πFC j 2πFRC ( ) j 2πFRC j 2πFRC + + j 2πF0RC + j 2πFRC M. Horowitz, J. Plummer, R. Howe 4

15 RC Filters Something More Complicated V out V in à Simplify using Fc / [2π RC] 3 Hz V out /V in C 0.µF, R kw F (Hz) M. Horowitz, J. Plummer, R. Howe 5

16 What If We Combine Low Pass and High Pass Filters? v in R K C 2 0.µF 0.µF R 4 0K v out What do you think it will do? We ll use a filter that operates like this in the ECG lab project. V out /V in F M. Horowitz, J. Plummer, R. Howe 6

17 Analysis Options: Nodal Analysis i v in Z v i 3 Z 3 v out Let s first solve it using Z -Z 4 and nodal analysis Z 2 i 2 Z 4 i 4 i 3 i 4 V V out Z 3 V out Z 4 V out V Z 4 Z 3 + Z 4 i i 2 + i 3 V in V Z V Z 2 + V V out Z 3 We have 2 equations in 2 unknowns (V and V out ). So we could solve this for V out /V in in terms of the impedances. M. Horowitz, J. Plummer, R. Howe 7

18 Analysis Options: Using R, C and Voltage Dividers v in R K C 2 0.µF v 0.µF R 4 0K v out For convenience, let s j*2πf V out V R 4 R 4 + s sr 4 + sr 4 We can replace R 4, and C 2 with Z eqv Z eqv R 4 + s + sc 2 s + sr 4 + sc 2 + sr 4 sc 2 C 3 ++ sr C 4 2 V V in + sr 4 sc 2 C 3 ++ sr C sr R + 4 sc 2 C 3 ++ sr C sr 4 + sr 4 + sr C 2 C 3 ++ sr C 4 2 M. Horowitz, J. Plummer, R. Howe 8

19 Output Response v in R K C 2 0.µF v 0.µF R 4 0K v out V V in V out V R 4 R 4 + s sr 4 + sr 4 + sr 4 sc 2 C 3 ++ sr C sr 4 + sr R + 4 C 3 sc 2 C + sr 4 + sr C 2 C 3 ++ sr 3 C 4 ++ sr C 4 C V out sr 4 V in + sr 4 + sr C 2 C 3 ++ sr C 4 2 V out V V V in sr 4 + s(r 4 +R C 2 +R ) + s 2 R C 2 R 4 Or, V out V in j 2πFR 4 + j 2πF(R 4 +R C 2 +R ) + ( j 2πF) 2 R C 2 R 4 M. Horowitz, J. Plummer, R. Howe 9

20 Output Response v in R K C 2 0.µF v 0.µF R 4 0K v out Vout V in j 2πFR 4 + j 2πF(R 4 +R C 2 +R ) + j 2πF ( ) 2 R C 2 R 4.2 Gain of Filters 0.8 V out /V in Low Pass High Pass Band Pass F (Hz) M. Horowitz, J. Plummer, R. Howe 20

21 So What Are The Answers To These Questions? How do we design circuits that respond to certain frequencies? What determines how fast CMOS circuits can work? Why do we often put a 200µF bypass capacitor between Vdd and Gnd? M. Horowitz, J. Plummer, R. Howe 2

22 Learning Objectives Become more comfortable using impedance To solve RC circuits Understand how to characterize RC circuits Which are low pass, high pass and bandpass filters Be able to sketch the frequency dependence of an RC circuit by reasoning about how capacitors behave at low and high frequencies M. Horowitz, J. Plummer, R. Howe 22