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The Art of Electronics SM Transistors II

The Art of Electronics SM Transistors II - 100 Class 5...

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Unformatted text preview: 100 Class 5: Transistors II: Corrections to the first model: .Ebers-Moll: re; applying this new view Topics: -° old: — Our first transistor model: 0 Simple: IC= BXIB 0 Simpler: Ic~ ~ IE; V BE = 0.6V 0 new: - transistor is controlled by VBE: Ebers-Moll view — applications: circuits that baffle our earlier view: ‘ 0 current mirror . ~0" common-emitter amp with no RE 4 Rout of follower driven by mee of moderate value — complications: 0 temperature effects 0 Early Effect Our first View of transistors held that two truths were pretty much sufficient to describe what was going on— 1. VBE=0'67 and 2. IE~IC= BXIB This account can take us a long way; VBE = constant = 0.6V is often a good enough approximation to allow understanding a schematic or, say, designing a not-bad current source. . Sometimes, however, we cannot settle for this first view. Some circuits require that we recognize that 1n fact VBE varies with IC. In fact, the relation looks just like the diode curve already familiar to you (it differs only in slope): N5 — 2 > Class 5: Tr n: Corrections to the first model: Ebers-Mon 101 * Ic . 13.9»: v i/Slgpe : J/sla : v 60m V/decade 2: 956420 mV/c/ecaa‘e i VBE VDIODE ! Figure N5.1: V35 does vary with IC, after all. In fact. IC vsl V35 looks a lot like a diode’s curve You knew, anyway, that the transistor had limited gain, so you would guess that a circuit like the one just below has a“ gain limited by the properties of the transistor—the IC vs VBE curve shown above. V +Vcc Pc db M , G = _ gs Figure N5.2: lnfinitegain amplifier? ' A naive application of the rule G = — RC/RE would imply infinite gain; but you know better. Wiggling the input = wiggling V135, and that produces a limited variation of IC, which in turn produces a limited variation in Vout. “Intrinsic emitter resistance:” r, You can describe this effect handily by drawing it as a little resistor in series with the emitter (for a derivation of re, see end of these notes): V ‘ Text sec. 2.10 * "Rule of Thumb No. 2" Figure N53: “Little re’i—the intrinsic emitter resistance This “resistance” we call “little rcf’ please note that it is not a resistor planted in the transistor; it only models the limited gain of the device. Another way to say this is to say that re is the slope of the gain curve—but with the curve plotted on its side, with VBE vertical (just so it will have the conventional units of resistance) : 102 Class 5: Tr II: Corrections to the first model: Ebers-Moll N5 -— 3 12.5.0. i 19.5st 1mA EMA 1c Figure N5.4: re is slope of transistor gain curve, if you turn the plot on its side Evidently the value of re varies with IC. Specifically, here’s our rule of thumb: r— — 25 ohms/(IC (in mA)) Watch out for the denominator: you must write “1 mA” as 1, not 1010‘3 If you forget this, your answers will be off by even more than what we tolerate in this course! re, “little I e,” expresses the Ebers-Moll equation in a convenient form, and you will use this simplifying model more often than you will use the equation. But we should notice what the Ebers-Moll equation says, before we. go on: T ext sec. 2.10 Ebers-Moll V35 [C = IS(eVBE/(kT/q) _ 1) ‘ 2 (treat Is as a constant, ,for any (negligible) particular transistor, exceptthat Is grows fast with temperature; more on this later) Ignoring the “-1” term, we can say simply that [C grows exponentially with VBE. In addition, we might as well plug in the room— —temperature value for that complicated ' expression “kT/q: ” 25 mV. Then Ebers-Moll doesn’ t look so bad: Ebers—Moll: (slightly simplified) , [c ~15 CVBE/zsmv This equation is most often useful to reveal the relative values of [C as VBE changes. What happens, for example, if you increase V35 by 18mV? Let’s call the old [C “101:” the new one “I02: 10; / Ic1={Is e“Ex/25M} / {Is eVie/25W} But that is just e(18ran25mV) ~ 2 This is a number perhaps worth remembering: 18mV AVBE for a doubling .of Ic; also sometimes handy; 60mV AVBE per decade (that is, 10X) change in IC. « N5 — 4 Class 5: Tr II: Corrections to the first model: Ebers-Moll 103 Applying the Ebers-Moll view to circuits _ Here, for example, is a circuit that makes no sense without the help of this View of transistors: * current mirror Text sec. 2.14 Lab 5-3 V8527 applieJ [3] (215 d 7‘? cal 25 e mm }’ evokes affrofriai'e 1"ch ohm’s Law, cvaltas afpmfriaie V551 ’ Figure N55: Current mirror: EbersFMoll view required Why is a mirror Useful? It makes it easy to link currents in a circuit, matching one to another. It also shows very wide output compliance. But for our present purposes it is most useful as a device to demonstrate the power of the Ebers—Moll view. It’s easy to make Icflogmn = Ic_om, and only a little harder to scale IC-out relative to IC-program' You will find in Lab 5 that the mirror departs rather far from this ideal model. Early efiect and temperature effects both disturb it. We will learn later how to fight these problems; for now, let’s leave the mirror in its simplest form, as shown above. Other consequences of this amended view of transistor operation: It brings some circuits down to earth: * a ceiling on gain (a recapitulation): no infinite-gain amps Text sec. 2.12. ' V0; = +10v Re 5.1k ‘ ‘Ar- +Sv M (bias not shown) Figure N16: What gain? Not infinite * a floor under ZMt Text sec. 2.11 vim. (bias not shown) m = allege a) x 25.0. (n: dommai-es) Figure N5.7: What Rm? Not 0.5 ohms 104 Class 5: Tr II: Corrections to the first model: Ebers-Moll N5 — 5 Let’s look closely at the problem of the grounded-emitter amplifier. You knew, anyway, that its gain was not infinite. Now, with the help of r,, we can evaluate the gain. '+Vcc =1Dv A Rt: 5.1:: W M = -32 (bias not shown) _ r‘e Figure N58: Grounded emitter amp again We can use our familiar rule to evaluate gain here, simply drawing in re (at least in our heads). To evaluate r,3 we need a value of 1C. There is no single right answer; the best we can do is specify IC at the quiescent point—where Vout is centered. Roughly, then, . G = - 5.1k§2/25§2 z —200 That’s high. But evidently the gain is not constant, since 1c must vary as vcult moves (indeed, it is variation in Ic that causes v0m to move.) v Here’s the funny, “barn-root" distortion you see (this name is not standard, incidentally) if you feed this circuit a small triangle: Text sec. 2.12,- compare fig. 2.36; Lab” ‘ Vaut : Ic => Q ¢Gain ’ 7.5V 0.5 MA son too 5v} 1mA .251 200 0.2V 2 mA 12. 5:1 4 oo ————\ {25mg Figure N53: Gain of grounded-emitter amp varies duringoutput swing (call it “barn-roof” distortion): Gain evaluated at 3 points in output swing The plots below. show how gain varies (continuously) during the output swing: [n W \ ZOWJBI'JX high gain , $3325 0 time Figure N510: During swing of Vom, 1c and thus re and gain vary This is bad distortion: —50% to +100%! What is to be done? ( i N5 — 6 Class 5: Tr 11: Corrections to the first model: Ebers-Moll 105 Remedy: emitter resistor One cannot eliminate this variation in re (—can one?), but one can make its effects negligible. Just add a constant resistance much larger than the varying re. That will hold the denominator of the gain equation nearly constant. Text sec. 2.12 With emitter resistor added, gain variation shrinks sharply: G _ RE+ r‘e ’\/‘ varies From 521‘: ' (bias noi- ‘EZI; = ' 94° Shah/n) "95:49; _ 51k = _ 40 "Fee 9.75 Figure N5.11: Emitter resistor cuts gain, but also cuts gain variation re still varies as widely as before; but its variation is buried by the big constant in the denominator. . Circuit gain new varies only from a low of —9.1 to a high of -9.75: a —4%, +3% variation about the midpoint gain of 9.5. Punchline: emitter resistor greatly reduces error (variation in gain, and consequent distortion). This we .get at the price of giving up some gain. (This is one of many instances ' of Electronic Justice: here, those greedy for gain will be punished: their output waveforms will be rendered grotesque.) . ' We will see shortly that the emitter resistor helps solve other problems as well: the problem of temperature instability, and even distortion caused by Early efiect. How can a humble resistor do so much? It can because in the latter two cases the resistor is applying ' negative feedback, a design remedy of almost magical power. Later in these notes, we will .look more closely at how theemitter resistor does its job. And in Chapter 4 we will see negative feedback blossom from marginal remedy to central technique. Negative feedback is lovely to watch. Many sudh treats lie‘ahead. . ' If you are in the mood to find negative feedback at work in today’s lab, you can find it in the simple-looking circuit fragment: the program side of the current mirror: +15 15k Figure N5.12: Subtle negative feedback: programming side of the current mirror See if you can explain to yourself how this circuit works. Hint: nearly all of the current flows not in the base path, but from collector to emitter. 106 ' Class 5: Tr II: Corrections to the first model: Ebers-Moll N5 — 7 Complications: Temperature effects; Early Effect Temperature Effects Semiconductor junctions respond so vigorously to temperature changes that they often are used as temperature sensors. If you hold VBE fixed, for example, you can watch 1c , which varies exponentially with temperature. But in any circuit not designed to .measure'temperature, the’response' of a transistor to temperature is a nuisance. Most of the time, the simple trick of adding an emitter resistor will let you forget about temperature effects. We will see below how this remedy works. at 25‘c ‘I I if hoiier I I I . V35 Figure N5.13: Transconductance of bipolar transistor varies rapidly with temperature Preliminary warning. Do not look for a description of this dependence in the Ebers-Moll equation. That equation (mis-read) will point you .in‘exactly the wrong direction: increasing T should shrink the exponent: ' ' [C = [Sam/(km) _ 1) Don’t be fooled: Ebers-Moll equation seems to say Ic falls with temperature. Not so. But, to the contrary, increasing T increases 1C, and fast. :Solution to the riddle: [s grows very fast with temperature, overwhelming the effect of the shrinking exponent. Here are two formulations 'for the way a transistor responds to temperature: Text sec. 2.10 V ' Temperature Effects: two equivalent formulations — IC grows at about 9%/°C, if you hold VBE constant. — VBE falls at 2mV/°C, if you hold Ic constant (This is the text’s formulation.) The first formulation is the easier to grasp intuitively: heat the device and it gets more vigorous, passes more current. The second formulation often makes your calculation easier. If you use the second formulation, just make sure that you don’t get the feeling that the way to calm your circuits is to build small fires under them! Example: current mirror The current mirror misbehaves if the temperatures of the two transistors become unequal. This you will see in the lab. If you heat one of the two uansistors, the current out rises; if you heat the other, the current out falls. Thinking through why this happens will help you get used to the two formulations of temperature effects, because this is the rare circuit that invites one to use both formulations, since it illustrates both cases: IC fixed, and VBE fixed. Let’s try it. N5 — 8 Class 5: Tr lI: Corrections to the first model: Ebers—Moll 107 +45 V W an hot Q1 5’» 1m \éE ‘91" V8: 9 6255 V35, $1“de i; Q; mt hot Figure N514: Current mirror again: consequence of temperature differences between Q1 and Q In Q1, current is held essentially constant by the large voltage across the 15k resistor. So, if we heat Q1 what happens? The second formulation says VBE falls. In effect, the curve shifts as shown above: at 1 mA, Q1 now finds a smaller VBE suffices. - Meanwhile, Q2 feels someone reducing its VBE. We assume Qz’s temperature is unchanged; So what does Q2 do? It delivers the lower IC that its curve says is appropriate to the lower VBE. . You can easily talk your way through a similar argument to explain why heating Q2 while leaving Q1 at room temperature causes Iout to increase. Remedies: making circuits stable despite temperature changes; 1. Compensation . The mirror becomes wonderfully immune to temperature variation if the two transistors stay at the same temperature. One can arrange that by building them on one piece of silicon. The argument that this works is simple: both transistors ‘look at the same curve,’ if they are heated together. We don’t care what those curves look like, only that they match. Text sec. 2.14 Lab 5 -3 1'th Q1 7? law V8; ------ . but inf Q2 1‘? LMA air Low VBE Figure N5.15: Example of temperature compensation: Current mirror is indifferent to temperature. ’ if transistor temperatures track 108 Class 5: Tr II: Corrections to the first model: Ebers-Moll N5 - 9 Here is another example, from the text. This circuit compensates for changes in one direction by planting a circuit element that tends to change at the same rate in the opposite direction. Text sec. 2.12 +20v raising temp / C lowers Ql's V55 Signal _I [It . . and that turns down "value", Q1 Figure N5.16: A second example of temperature compensation This circuit (from Text’s fig. 2.39) lets the fall of Ql’s VBE squeeze down Q2 as both get hotter. (The 10k resistor on the base of Q2 makes the biasing circuit not too stiff: the signal source (presumed to be of impedance « 10k) can have its way, as usual.) 2. Feedback: emitter resistor The remedy described here is simpler, and more widely used. It is also subtler. Text sec. 2.12 v +15v +15V cak ask \ snakg nega‘HVe . . ' £3453“? '_. - cl$~cuit reaches back {aagyusé valve! Figure N5.17: An unstable circuit stabilized by emitter resistor The left-hand circuit is so unstable that it is useless. An 8°C rise in temperature saturates the transistor. Text sec. 2.12, ex. 2.9 Why does is the right-hand circuit work better? How does the emitter resistor help, as Ic grows? Here is feedback at work: the circuit senses trouble as it begins: — [c begins to grow in response to increased temperature; _ — VB rises, as aresult of increased 1c (this is just Ohm’s Law at work); — But this rise of VE diminishes VBE, since VB-is fixed. Squeezing VBE tends to close the transistor “valve.” Thus the circuit slows itself down. The remedy is not quite perfect: some growth of 1c with temperature is necessary in order to. generate the error signal. But the emitter resistor prevents wide movement of the quiescent point. - ' N5 — 10 Class 5: Tr II: Corrections to the first model: Ebers-Moll 109 2a. Temperature stability M high gain Text sec. 2.13; , Lab 5-2 - If you want stability and high gain, you can have that combination, by including R5 for DC biasing, but making it disappear at AC. You make it “disappear” by bypassing or paralleling RE with a capacitor, thus: +15V Re R 6.6k G = - 7::- => hvjh gain fl PE; (ulna-11's bypassed [37 C, _ R: + C 680 I toflF Figure N548: Bypassed—emitter resistor: high gain plus temperature stability , This circuit still distorts; note that RB here remedies only the temperature instability problem. Here is feedback in a more obvious form, but used to similar effect: this is from Lab 5: Text sec. 2.12 Lab 5 -5 Figure N5.19: DC feedback protects against temperature effects V _ This looks a lot like operational amplifier circuits that you will see in Chapter 4 (called “Chapter .3” in ‘81 edition). When you get there you may want to look back at this circuit. Then you will appreciate what now is obscure: the feedback affects DC levels, but not circuit gain—not what happens to the signal, in other words. It does not affect the signal because the low output impedance of the function generator overwhelms the relatively feeble feedback signal (that is, high—impedance feedback), It is meant to work that way, to keep things simple for us at this stage. 110 - Class 5: Tr II: Corrections to the first model: Ebers-Moll N5 —— 11 Early Effect . Here is a picture of this effect, which just describes the transistor’s departure from the ideal view that it is a current source: Ear/g Effect 1: Ic ('5 defermz'neo’, 53.9 bath offlur. transis for mode/s [determined by I; ”153%3... AIC {or Aw; =2ov, at fixed VBE _ z: ‘7. _ 84‘ R cs 3 ‘1’ models — — - Ear/7 efich W—I (fig/“1' {47: (7/085 u.. m az‘i er a Ari/e, sags Earl -: 1—: grows burr/z 1/55, as if c-Ewere a very /arge remn‘ance. 0 10 20 VCE, VDHS Figure N520: Early Effect: graphically, the curve’s departure from horizontal A transistor is a pretty good current source, but it also acts a little like a resistor: Ic grows as the voltage across the device (VCE ) grows. _ Like the temperature-dependence in the Text’s formulation, Early Effect looks like an effect on Vbe , because [c is assumed fixed: ‘ Text sec. 2.11 Early Effect: AVBE=_aAVCE where or = 0.0001v(or 10“) (IC is assumed constant.) Despite the assumption used in this formulation, that IC is fixed, usually you will see Early Effect causingvariation in _IC, while VBE is fixed. The current mirror provides a good example of the problem, and of ways to beat it. +ISv Q1 Q2 . r/cE1 =0.év‘ V“: vanes gram 0.2V 19: g! Rivas. LOAD . Figure N521: Current mirror: Early Effect predicts I“: will not match [prop-m The flaw in the simple mirror is the likely difference between the voltages across the two transistors: to the extent that the VCE’s differ, Early says the currents will differ. ' Quantitatively: try an example: the mirror, powered from +15v. What happens if Rload is small, so that most of the 15—volt supply appears across VCE of Q2? Early Effect predicts Io“t a good deal larger than I pmgmm. In N5 — 12 Class 5: Tr 11: Corrections to the first model: Ebers-Moll 111 How much larger? The quantitative argument is a bit convoluted, as it can be for temperature effects. The difficulty is the assumption that IC remains constant, while we know that in this case the result is just the contrary: Io“t is going to vary as VCE varies. You need to do a sort of thought—experiment contrary to fact: assume VBE on 02 (1065 change, then later recognize that it can’t; see what change in IC must have occurred to have held VBE constant. Here’s the .argument. — Assume IC constant; — The extreme difference between the VCE’s on 01 and Q2 would be about 15 volts (this would occur for Rload close to zero 9). In that case the mismatch of VBE’s predicted by Early Effect would be about 1.5 mV. Now we are ready to admit this is not possible: the two VBE’sare equal. One can see that with a glance at the circuit diagram. _ — Admitting this is not possible, we ask what difference in Ic’s must occur instead. Ebers-Moll gives the answer: 4 - IC = 13(eVBr/(KT/q) __ 1) — The ratio of the higher to lower currents is eCVBsz/ 7-5 mV) / eWBm / 25 mV) — And this is the same as 30735.2 47354) / 25 mV — In this case that is about 61.5 mV /25mV _ com -~ 1.06 This is smaller than the mismatch we saw when we tried this expeiiment: we saw a ratio of ' about 1.15. Probably our estimate of 0.0001 (or 10‘4)for the Early Effect (1 is low (our experiments suggests 0t ~0.0002 to 0.0003 (or 2 to 3404).). ‘ But the central point is sound: Early Effect introduces a .disappointing error into the otherwise-admirable mirror. We need a remedy. Here are two: . , 1. clamp VCE for Q2, so that the VCR difference does not occur. This the clever Wilson Mirror does. Textsec. 2.13 Lab 5 —3 large 1:.ch -V but harmless Figure N 5.22: Beating early effect: one way: wilson mirror clamps VCE 112 Class 5: Tr II: Corrections to the first model: Ebers-Moll N5 — 13 2; Or Add an emitter resistor that tends to fight the growth of Qz’s 1c. (This works against temperature instability, too, incidentally—in case Q2 gets hotter than Q, as it tends to in the simple mirror above). neja‘h've «Feedback 33min! Figure N523: Beating Early effect: another way: emitter resistorsenscs and corrects mots Using Early Efi’ect to estimate R0,” for a current source Usually we settle for calling Rout for ...
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