Lecture 16. More RC Circuits and Impedance

Transcription

1 Lecture 16 More RC Circuits and Impedance Copyright 2016 by Mark Horowitz 1

2 Reading Reader: Chapter 6 Capacitance (if you haven t read it yet) Section 7.3 Impedance You should skip all the parts about inductors We will talk about them at the end of the class 2

3 Roadmap In the last lecture we learned about capacitors: a device where its current is proportional to dv/dt and not V. We showed how to use nodal analysis to derive the solution of a simple RC circuits in the time domain which leads to an exponential V vs. time behavior. Now we generalize that solution a little bit to allow you to compute the output for a circuit with many resistors and sources, and one capacitor by using a Thevenin equivalent circuit. But then we try to figure out what happens when sounds flows through the circuit and get a differential equation a computer can solve, but doesn t mean much to us. So we take a step back and remember that sound can be represented by the sum of sine waves. We show that RC circuits with sinewaves looks just like our old resistor circuit. A capacitor acts just like a resistor, but its resistance depends on the frequency of the sinewave! 3

4 Learning Objectives for Today Generalize RC circuit analysis in the time domain Impedance is the relationship between voltage and current For a sinusoidal input Z = V/I so for a capacitor, Z = 1/2πFC Understand how to use impedance to analyze RC circuits 4

5 PREVIOUSLY IN E40M 5

6 Capacitors Capacitors store charge The voltage across the capacitor is proportional to Q V = Q/C; or Q = CV Q in Coulombs, V in Volts, and C in Farads But like all devices it is charge neutral Stores +Q on one terminal; stores Q on the other Sometimes we purposely use capacitors in circuits; Other time we use them to model the capacitance of wires These are sometime called parasitic capacitance Resulting i-v relation: i = C dv/dt 6

7 Simple RC Circuit Demo EveryCircuit Demo CMOS Inverter 7

8 8

9 Keys to Analyzing RC Circuits The voltage across a capacitor can t change instantaneously That means the voltage across a capacitor won t change the instant after any switches/transistors flip Write the equations in terms of the voltage across the resistor Since that voltage will go to zero eventually Across R1 (Vout) for falling transition Across R2 (Vdd-Vout) for rising transition Just write the nodal equations: We just have one voltage, V R i R = V R /R (1 or 2) C 1 dv R /dt = i C i R + i C = 0 V out C1 9

10 Why V= Makes Sense We know that the current flowing into a node must be zero So the cap current and resistor current must be opposite The device currents are: Resistor i = V(t)/R 2 Capacitor i=c dv(t)/dt Need a function where is the only function that works M. Horowitz, J. Plummer E40M Lecture 17 10

11 RC Circuits in the Time Domain 5V V out 5V 1 e t/r C 1 V out 5V e t/r C 1 In capacitor circuits, voltages change slowly, while currents can be instantaneous. 11

12 Work Out Example 12

13 MORE COMPLEX CIRCUITS 13

14 What To Do Here? 14

15 Find The Voltage Before Switch Connects That is easy, just ignore the capacitor! 15

16 Convert to a Solved Problem 16

17 Solution 17

18 But How To Solve This Circuit? The input is sound from your computer; the output is going to go to your Arduino Now V in is a complex waveform How are we going to find V out? Two approaches Simulate the differential equation on a computer Decompose the input into sine waves (frequency analysis) 18

19 Time Domain vs. Frequency Domain Directly solving for the output to this: Requires a computer And the output will just be another squiggly line But This waveform is the sum of sinewaves 19

20 IMPEDANCE 20

21 Too Bad The Input Isn t An Exponential If V in = Then V R might be And V C = -V const If this was the case: i R = ; i C = 21

22 For Exponential Input A capacitor acts like a resistor! But the value of the resistance depends on The current through a capacitor is -V C * So R eff = - 22

23 Unfortunately the Input is NOT a Sum of Exponentials* But we know that the input can be represented by tones And tones are just sine waves. What happens if we use sine wave inputs? * Actually it is, but that is a little advanced for this class. I will get to it in some bonus material 23

24 But We Know it is a Sum of Sine Waves We also know that R, C are linear elements So superposition holds Superposition says The output is the sum of the response from each source So the output from a sound waveform Is the sum of the outputs generated from each sinewave 24

25 Properties of Sinewaves The problem with capacitors is that they take derivatives This makes the problem solution a differential equation Exponential waveforms are nice since d t e 1 e dt Sine waves have a similar property d dt t sin 2 Ft 2 F cos 2 Ft d dt cos 2 Ft 2 F sin 2 Ft 25

26 What This Means If you drive a R, C, circuit with sin(2 F t) All the waveforms in the circuits will be sin(2 F t) At different amplitudes, and with a phase shift We will mark terms that are phase shifted by a j 26

27 Sinewave Driven Circuits 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 In fact we can solve it exactly the same way 27

28 Impedance Impedance is a concept that generalizes the resistance of a resistor. In the resistance case, R V i R (the impedance of a resistor) does not depend on frequency, it is simply a number. What about a capacitor? If we limit V and i to sine waves, then Z C V i V CdV /dt Z C V i V O sin 2 Ft 2 FCV O cos 2 Ft 1 j *2 FC Z C V i 1 2 FC If we ignore phase shift 28

29 Impedance of a Capacitor The impedance of a capacitor depends on frequency Z C At low frequencies (F 0) and a capacitor behaves like an open circuit. Thus if we are doing a DC analysis of a circuit (voltages and currents), capacitors are modeled as open circuits. Z C 0 At very high frequencies (F infinity) and a capacitor behaves like a short circuit. At intermediate frequencies the capacitor has an impedance given by Z C Z C V i 1 j *2 FC 29

30 USING IMPEDANCE 30

31 Using Impedance Makes Everything an R Circuit Find v out / v in First, note that v out will have a DC voltage, but the capacitor Z C = at F = 0, so it becomes an open circuit. v out DC We can now use superposition. Assume we have a sine wave input at v in Use the Thevenin Equivalent connected to the capacitor to simply the circuit 31

32 RC Circuit Analysis Using Impedance The circuit becomes just a voltage divider, and we can analyze it the same way we have analyzed resistor only circuits. That s the power of using impedance! 32

33 Analyzing RC Circuits Using Impedance v in C R v out Z C 1 2 FC Z R R If the circuit had two resistors then we would know how to analyze it V V out in R 1 R 2 R 2 What if we do the same thing but use impedances 33

34 Analyzing RC Circuits Using Impedance v in C R v out V V out in R R j *2 FRC 1 1 j *2 FRC j *2 FC At low frequencies, (F 0), V out = 0 which means that low frequencies are not passed to the output. The capacitor blocks them. Recall that we used this idea earlier to calculate the DC voltage at the output. At high frequencies (F large), V out = V in 34

35 Frequency Dependence of RC Circuit v in C v out V out /V in R This circuit passes high frequencies but blocks low frequencies. F Sometimes called a high pass filter. V V out in j*2 FRC 1 j *2 FRC 35

36 Learning Objectives for Today Generalize RC circuit analysis in the time domain Impedance is the relationship between voltage and current For a sinusoidal input Z = V/I so for a capacitor, Z = 1/2πFC Understand how to use impedance to analyze RC circuits Convert capacitors to effective resistors Compute the voltage divider ratio to find output voltage 36