The Essence of Chapter 3

Transcription

1 The Essence of Chapter 3 A short summary of the essence of Chapter 3 in MEDICAL INSTRUMENTATION, APPLICATION AND DESIGN, 3 rd edition By Fred- Johan Pettersen, Oslo University Hospital This is primarily meant as notes for the lecturer, but if the students find it useful, that s fine, but keep in mind that it is not defining the syllabus in FYS4250/FYS9250 in any way Analog signal processing We have a selection of building blocks available when we re about to make a circuit for linear signal processing Most of these are based on the opamp It is mostly used in one of three basic configurations: figure 1 a) non- inverting amplifier, figure 1 b) inverding amplifier, and figure 1 c) differential amplifier In addition, there are two special variants: figure 1 d) unity- hain amplifier/buffer, and figure 1 e) summation amplifier a) Non- inverting amplifier b) Inverting amplifier c) Difference amplifier d) Unity- gain amplifier, a special case of the non- inverting amplifier 1

2 e) Summation amplifier, a special case of the inverting amplifier Figur 1: Lineære opamp circuits The ideal opamp The ideal opamp has a number of characteristics: " "# 0! "# =!"! "! This makes our analysis a bit simpler There are some requirements, though The opamp should not go into saturation (try to give an output voltage higher than it is supposed to), one must stay within legal input voltage range, and a (somewhat) linear circuit with feedback is normally required, like the circuits in figure 1 The simplifications are: The voltage on both inputs are the same It will not flow any current at all into the inputs Figure 2: Opamp 2

3 The non- inverting amplifier Have a look at figure 1 a) Since this is an idealistic circuit, we know that = And since the two resistors form a voltage divider, we have enough to work with, and we end up with V IN = V UT R 2 R 1 + R 2! V UT V IN = R 1 + R 2 R 2 =1+ R 1 R 2 In the special case where =, ie removes it, and/or let " = 0, ie shorts it, we get the circuit in figure 1 e) We can easily fint the gain for this amplifier by using the expression above, and get V UT V IN =1+ 0 R 2 =1+ R 1! =1+ 0! =1 The inverting amplifier Shave a peek at figure 1 b) Again, the opamp is ideal, and the voltage on both inputs are 0 V This means that " = " " And since IIN cannot enter the opamp, it has to find it s way through RF which causes a voltage drop And due to the idealistic opamp, the left side of the RF is 0 V, the right side voltage is given by wich means that = 0 " = " " " = " A special variant is shown in figure 1 d) Instead of only one input voltage & resistor, there is a bunch of them IIN will now be a sum of the currents through all input reisitors, and given by " = "! "! + "! "! + + "# "# So if all input resistor are identical, we have made a circuit that sums the input voltages! Neat, don t you think? The difference amplifier The difference amplifier in 1 c) won t be analysed to death, but I simply state that where AD is differential gain =!! =!! If =! =!, =! =! and the opamp is really ideal, this is correct But unfortunately, such idealism is more common in political programs than in real life So 3

4 we introduce a new and important quantity called ACM, or common- mode gain if you prefer Inserted into the previous equation, we get =!! + "!! The fun thing now is that with a new quantity, we may define even more quantities (and it s not for bugging you, even it may seem so) CMRR, Common Mode Rejection Ratio is defined as!"#! = 20 log " The circuit has a number of drawbacks: Denne kretsen har flere ulemper: Relatively low CMRR, especially at high AD Input impedance is low (equal to RIN) IIN will vary with VCM Let s try to fix this We can fix the low imput impedance by adding a couple of unity- gain amplifiers to each of the inputs (shown in figure 3 a) This is nice, but what we have not fixed is the low CMRR, and the requirements on resistor matching is still very high In other words: Close, but no cigar Let s try again The circuit in figure 3 b) has it all The input impedance is OK since the inputs are connected directly to the opamp inputs And since we are living in a (semi- )ideal world, we know that the voltage across R1 is!!, and then we get =!!! And since this current must go through the two R2s, we can find the voltage at the output of the opamps:!!!! = 2 + =!!!! In other words, the diferential signal is amplified by!!, while the common- mode signal is the same, 1 We got =!! and " = 1 for this stage The next stage is known: We can choose between low AD & low ACM or high AD & high ACM If we are smart, and I dare say that we are, we go for low AD & low ACM Since we have introduced a gain in the input stage, we don t need much gain in the second stage, and that gives us the opportunity to make a difference amplifier (or a subtractor, if you prefer) with low ACM So what do we end up with? Differential gain is taken care of by the input stage + a small contribution from the second stage The first stage does not contribute to the common- mode gain, but ACM of the second stage is lowered So all- in- all, we have the same gain as we got in a simple difference amplifier, but the common- mode gain is lower, giving us an improved CMRR! Oh, and lets not forget that we also get high- impedance inputs The expression for the differential gain is! =!! 4

5 To make your life easier, some companies have made small ICs that contain one or more instrumentation amplifiers These will normally miss R1, but let the user connect a suitable resistor, and thus let the user control gain How nice In other words: The Instrumentation Amplifier is a difference amplifier on steroids! a) Differanseforsterker med forbedret inngangsimpedans b: Instrumenteringsforsterker Figur 3: Differanse- og instrumenteringsforsterkere Frequency domain If we expand our view a bit, we can replace all resistances in the above equations with impedances Frequency dependent impedances that are complex numbers Signals: o Voltages:! " o Currents:! " Resistors: o Resistors:!!" =! o Capacitors:!!" =!!"# o Inductors:!!" =!"# The!" is the imaginary number times omega which is frequency given in radians An example of how we can alter our basic inverting amplifier is shown in figure 4 5

6 Figur 4: Lavpassfilter / integrator The gain of the circuit in figure 4 is given by " = "!"#$!"#$%&!" "!" =!"!" =!"#! =!!"! We can see that for high frequencies, the equation goes towards zero, while it is very high for low frequencies, and for DC, it goes towards infinity This means that we must change it a bit to get a circuit that will work We add RFF The gain can now be re- calculated, and we get Quick verification:!" = " =!"#! =! "!"!"!!!!" a) Set frequency to 0, and evaluate the equation above You ll see that gain is the same as for a normal noninverting amplifier This makes sense since the capacitor can be considered a open connection at DC b) Set frequency to, and evaluate the equation above You ll see that gain is zero This makes sence since the capacitor is effectively a short for very high frequencies, and the output is now shorted to the +- input of the opamp which is equal to zero thanks to it s idealism This means that we can create a number of circuits that has frequency- dependent characteristics Examples are low- pass filters like the one above, or we can move the capacitor to the input, and get a high- pass filter, or we can make band- pass or band- stop filters Just for the fun of it: Let s try to have a look at the circuit in figure 4 in the time- domain You (might or might not) remember that the voltage- current relationship for a capacitor is given by! = 1!" Let s disregard the RFF for simplicity, and let s see what we got Well, we know that the output voltage is minus whatever voltage drop we got across the feedback device (the capacitor), and we know the current that is goind to flow through it, se let s put this into a equation: 6

7 = 1!"!" = 1! " = 1!!"! We got an integrator! How cool is that? And since you are so incredible col too, I m sure you can imagine what opportunities we have if we play around with inductors as well Or capacitors in other positions Non- linear analogue functions Fun with diodes It was fun replacing the resistors in figure 1 with inductors and capacitors Bt what happens if we replace a resistor by a nonlinear device such as a diode? The diode has voltage- current relations that is more or less given by =!! 1!! or! ln So if we replace RI by a diode in forward- direction as shown in figure 5 a, we get This is an exponential amplifier! Thrilling =!"!!! And is we put the diode where RF used to be, we get an logarithmic amplifier described by = ln!! ln "! Fun facts: 1 If we let VIN be constant for the circuit in figure 5 b, we will have a circuit with an output propotional to temperature since VT is, indeed, propotional to temperature 2 Want to do multiplication? Check out this, and consider the two circuits in figure 5!" = "!!!"! Witsome imagination, it is possible to do division as well Det var vel fint? Morsom fakta: Med konstant VIN vil VO være direkte proposjonal med diodetemperaturen temperatursensor! 7

8 a) Exponential amplifier b) Logaritmic amplifier Figure 5: Non- linear amplifiers Rectifiers This sections is not about amplifiers, and we will just explain the principles It is possible to refine these circuits, but that is beyond the scope of this piece of text Have a look at the circuit in figure 6 a) Add a sinus- ike the one in figure 6 b) Since the diode conduct current in only one direction, the output is something like the one shown in figure 6 b) If we refine the circuit, we get something like the circuit in figure 6 c) The new waveforms is shown in figure 6 d a) Half- wave rectifier b) Waveform from half- wave rectifier c) Full wave rectifier d) Waveform from full- wave rectifier Figure 6: Rectifiers and waveforms Comparator If you want to behave really unlinearly, you may use a comparator As the name implies, it compares It is basically an opamp with an output stage that gives a logical output Figure 7 says it all Comparators may be made with or without hysteresis 8

9 a) Symbol b) Waveforms Figure 7: Comparator Phase- sensitive demodulators If we should feel the desire of picking only one frequency component out of a signal, then a phase- sensitive demodulator is our choise It can be made both in hardware and software, so we ll only look at the principle here It is really simple: Multiplicate a reference signal with values 1 and - 1 to the input signal as shown in figure 8 a) Then use a low- pass filter to remove the higher frequency components Figure 8 b) shows three set of waveforms illustrating different cases Signals A show what happens if the reference signal and input signal has different frequency Average output is 0 Signals B show what happens if the reference signal and input signal has same frequency and phaseaverage output is max Signals C show what happens if the reference signal and input signal has same frequency, but phase difference is 90 degreesaverage output is 0 For those of you without anything else to do, go ahead and draw more examples and/or phases! Be my guest! a) Principle for a phase- sensitive demodulator 9

10 b) Waveforms Figure 8: Phase- sensitive demodulator Last words Open- loop gain tells us something about circuit stability Closed- loop gain tells us how a circuit amplifies our signal Gain- Bandwidt, GBW, is simply gain multiplied by frequency for an opamp at a given frequency For simplicity, choose a frequency where gain is 1, and GBW is equal to that frequency Slew- rate tells us how fast an opamp can alter it s output voltage Important for large signals since it may limit the opamps high- frequency performance Offset voltage is how far away from desired voltage a signal is For instance, opamps typically has 1 mv offset on the inputs This may be amplified Noise is both generated within most devices and is received from the outside Rule of thumb: Place as much gain as possible as early as possible in a chain of amplifiers Bode- plot is useful, and you may read more here: 10