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THE ART OF ELECTRONICS

Third Edition

Paul Horowitz HARVARD U N IV ER SIT Y

Winfield Hill R O W L A N D I N S T I T U T E AT H A R V A R D

H I Ca m br id g e
UNIVERSITY PRESS
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A rt o f E lectronics Third Edition
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.c 30 1.5. Inductors and transformers .d o
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.c

connected to each (a “50% duty cycle” ). But the defin-

ing equation V = L d l /d t requires that the average voltage
across an inductor must be zero, otherwise the magnitude
of its average current is rising without limit. (This is some-
[
times called the volt-second balance rule.) From this it fol- Figure 1.53. Transformer.
lows that the average output voltage is half the input volt-
age (make sure you understand why). In this circuit C2
Transformers are quite efficient (output power is very
acts as a storage capacitor for steadying the output voltage
nearly equal to input power); thus, a step-up transformer
(more on this later, and in Chapter 9).
gives higher voltage at lower current. Jumping ahead for
Producing an output that is half the voltage of an input
a moment, a transformer of turns ratio n increases the
is not very exciting; after all, a simple voltage divider can
impedance by n2. There is very little primary current if the
do that. But, unlike a voltage divider, this circuit does not
secondary is unloaded.
waste any energy; apart from non-idealities of the compo-
Power transformers (meant for use from the 115 V pow-
nents, it is 100% efficient. And in fact this circuit is widely
erline) serve two important functions in electronic instru-
used in power conversion; it’s called a “synchronous buck
ments: they change the ac line voltage to a useful (usually
converter.”
lower) value that can be used by the circuit, and they “iso-
But look now at Figure 1.52B, which is just a turned-
late” the electronic device from actual connection to the
around version of Figure 1.52A. This time, volt-second
powerline, because the windings of a transformer are elec-
balance requires that the output voltage be twice the in-
trically insulated from each other. They come in an enor-
put voltage. You can’t do that with a voltage divider! Once
mous variety of secondary voltages and currents: outputs
again, the output capacitor (Cj this time) serves to hold the
as low as 1 volt or so up to several thousand volts, current
output voltage steady by storing charge. This configuration
ratings from a few milliamps to hundreds of amps. Typical
is called a “synchronous boost converter.”
transformers for use in electronic instruments might have
These and other switching converters are discussed ex-
secondary voltages from 10 to 50 volts, with current rat-
tensively in Chapter 9, where Table 9.5 lists some fifty rep-
ings of 0.1 to 5 amps or so. A related class of transformers
resentative types.
is used in electronic power conversion, in which plenty of
power is flowing, but typically as pulse or square wave-
forms, and at much higher frequencies (50 kHz to 1 MHz
is typical).
Transformers for signals at audio frequencies and ra-
dio frequencies are also available. At radio frequencies you
sometimes use tuned transformers if only a narrow range
of frequencies is present. There is also an interesting class
of transmission-line transformers. In general, transformers
for use at high frequencies must use special core materi-
50 % als or construction to minimize core losses, whereas low-
frequency transformers (e.g., ac powerline transformers)
Figure 1.52. Inductors let you do neat tricks, such as increasing a
dc input voltage.
are burdened instead by large and heavy cores. The two
kinds of transformers are in general not interchangeable.

1.5.2 Transformers
A. Problems, pro blem s...
A transformer is a device consisting of two closely coupled This simple “first-look” description ignores interesting -
coils (called primary and secondary). An ac voltage applied and important - issues. For example, there are inductances
to the primary appears across the secondary, with a voltage associated with the transformer, as suggested by its cir-
multiplication proportional to the turns ratio of the trans- cuit symbol: an effective parallel inductance (called the
former, and with a current multiplication inversely propor- magnetizing inductance) and an effective series inductance
tional to the turns ratio. Power is conserved. Figure l .53 (called the leakage inductance). Magnetizing inductance
shows the circuit symbol for a laminated-core transformer causes a primary current even with no secondary load;
(the kind used for 60 Hz ac power conversion). more significantly, it means that you cannot make a “dc
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c u -tr a c k Art o f Electronics Third Edition 1.6.2. Rectification 31d o c u - t r a c k

transformer.” And leakage inductance causes a voltage which is measured in the nanoamp range for a general-
drop that depends on load current, as well as bedeviling cir- purpose diode (note the hugely different scales in the graph
cuits that have fast pulses or edges. Other departures from for forward and reverse current), is almost never of any
ideal performance include winding resistance, core losses, consequence until you reach the reverse breakdown volt-
capacitance, and magnetic coupling to the outside world. age (also called the peak inverse voltage, P1V), typically
Unlike capacitors (which behave nearly ideally in most cir- 75 volts for a general-purpose diode like the 1N4I48. (Nor-
cuit applications), the deficiencies of inductors have signif- mally you never subject a diode to voltages large enough to
icant effects in real-world circuit applications. We’ll deal cause reverse breakdown; the exception is the zener diode
with these in Chapter Ix and Chapter 9. we mentioned earlier.) Frequently, also, the forward volt-
age drop of about 0.5 to 0.8 V is of little concern, and the
diode can be treated as a good approximation to an ideal
1.6 D iod es a n d diode circuits
one-way conductor. There are other important characteris-
We are not done with capacitors and inductors! We have tics that distinguish the thousands of diode types available,
dealt with them in the time domain (RC circuits, exponen- e.g., maximum forward current, capacitance, leakage cur-
tial charge and discharge, differentiators and integrators, rent, and reverse recovery time; Table 1.1 includes a few
and so on), but we have not yet tackled their behavior in popular diodes, to give a sense of the capabilities of these
the frequency domain. little devices.
We will get to that soon enough. But this is a good time
to take a break from “RLC” and put our knowledge to use
with some clever and useful circuits. We begin by intro-
ducing a new device, the diode. It’s our first example of a
nonlinear device, and you can do nifty things with it.

1.6.1 Diodes
The circuit elements we’ve discussed so far (resistors, ca-
pacitors, and inductors) are all linear, meaning that a dou-
bling of the applied signal (a voltage, say) produces a dou-
bling of the response (a current, say). This is true even for
the reactive devices (capacitors and inductors). These com-
ponents are also passive, as opposed to active devices, the
latter exemplified by transistors, which are semiconductor
devices that control the flow o f power. And they are all
two-terminal devices, which is self-explanatory. Before jumping into some circuits with diodes, we
should point out two things; (a) a diode doesn’t have a re-
anode cathode sistance (it doesn’t obey Ohm ’s law), (b) If you put some
A K
diodes in a circuit, it won’t have a Thevenin equivalent.
Figure 1.54. Diode.

1.6.2 Rectification
The diode (Figure l.54) is an important and useful two-
terminal passive nonlinear device. It has the V - l curve A rectifier changes ac to dc; this is one of the simplest and
shown in Figure 1.55. (In keeping with the general phi- most important applications of diodes (which are some-
losophy of this book, we will not attempt to describe the times called rectifiers). The simplest circuit is shown in
solid-state physics that makes such devices possible.) Figure l .56. The “ac” symbol represents a source of ac
The diode’s arrow (the anode terminal) points in the di- voltage; in electronic circuits it is usually provided by a
rection of forward current flow. For example, if the diode transformer, powered from the ac powerline. For a sine-
is in a circuit in which a current of 10 mA is flowing from wave input that is much larger than the forward drop
anode to cathode, then (from the graph) the anode is ap- (about 0.6 V for silicon diodes, the usual type), the out-
proximately 0.6 V more positive than the cathode; this is put will look like that in Figure l.57. If you think of the
called the “forward voltage drop.” The reverse current, diode as a one-way conductor, you won’t have any trouble
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32 1.6. D iod es and diode circu its Art o f Electronics Third Edition

Table 1.1 Representative Diodes

Vp (max) / R (typ, 25°C) VF @ /F C a pa citan ce
P art # (V) (A @ V) (m V) (m A) (pF @ VR) SM T a p /n C o m m e n ts
S ilico n
PAD5 45 0.2 5pA 20V 800 1 0.5pF 5V SSTPAD5 m etal + glass can
1N4148 75 10nA 20V 750 10 0.9pF 0V 1N4148W jellybean sig diode
1N4007 1000 5 0 nA 800V 0.8 250 12pF 10V DL4007 1N4004 low er V
1N 5406 600 <10pA 600V 1.0 V 10A 18pF 10V none heat through leads
S c h o ttk v b
1N6263 60 7nA 20V 400 1 0.6pF 10V 1N6263W see also 1N5711
1N5819 40 10 mA 32V 400 1000 150pF 1V 1N5819HW jellybean
1N5822 40 40p A 32V 480 3000 450 pF 1V none pow er S cho ttky
M B R P40045 45 500 pA 40V 540 400A 3500pF 10V you jest! M oby dual S chottky

N otes: (a) SMT, s urfa ce-m ount technology, (b) S ch ottky diod es have low er forw ard voltage and zero
reverse-recovery time, but m ore capacitance.

understan ding ho w the circu it w ork s. T h is circuit is called w h ich the diode drop b ecom es sig n ificant, yo u have to re-
a h a lf-w a ve rectifier , because on ly h a lf o f the input w a ve - m em b er that.29
form is used.

<3-----M— o —

o-------------o— 1

F ig u re 1.56. H alf-w ave rectifier.

1.6.3 P o w e r-s u p p .y filte rin g

T h e precedin g rectified w avefo rm s aren’ t g ood for much as

th ey stand. T h e y ’re “ d c” on ly in the sense that they don ’ t
ch an g e polarity. But they still h ave a lot o f “ ripple” (peri-
o dic variations in vo ltag e about the steady value) that has
to be smoothed out in order to generate genuine dc. Th is
w e do b y attaching a relatively large value capacito r (F ig -
ure l . 6 0 ); it charg es up to the peak output voltage during
the diode condu ction , and its stored charg e (Q = C V ) pro-
vid es the output current in betw een ch argin g cyc le s. Note
that the diodes prevent the capa cito r from disch arging back
through the a c source. In this application y ou should think
o f the capacitor as an e n ergy storage d evice, with stored
en e rg y U = j C V 2 (recall § 1 . 4 . 1 ; for C in farads and V in

F ig u re 1.58. F ull-w ave bridge rectifier. volts, U co m es out in jou les, or eq uivalently, watt secon ds).
Th e capacito r value is chosen so that

F ig u re 1 . 5 8 sh o w s another rectifie r circu it, a “ fu ll-w a ve ^loudC ^ I/ / 1

b ridg e.” F ig u re 1 . 5 9 sh o w s the voltag e acro ss the load;
note that the entire input w a ve fo rm is used. T h e g aps at 29 The diode drop can be eliminated with active switching (or syn-
ze ro vo lta ge o ccu r b ecause o f the d io d e s’ fo rw a rd vo ltag e chronous switching, a technique in which the diodes are replaced by
dro p. In this circuit, tw o dio des are a lw a y s in series with transistor switches, actuated in synchronism with the input ac wave-
the input; w hen y ou de sig n lo w -vo lta g e p o w e r supplies, fo r form (see §9.5.3B).
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c u -tr a c k A rt o f E lectro n ics T h ird E dition 1.6.4. Rectifier configurations for power supplies c u -tr a c k

o f accuracy, though, for tw o reasons, (a) The discharge is

an exponential only if the load is a resistance; many loads
C load
are not. In fact, the m ost com m on load, a voltage regula-
tor. looks like a constant-current load, (b) Power supplies
are built w ith capacitors with typical tolerances o f 20%
or more. Realizing the m anufacturing spread, you design
F ig ure 1.60. Full-wave bridge with output storage (“filter” ) capaci- conservatively, allow ing for the worst-case com bination o f
tor. com ponent values.
In this case, viewing the initial part o f the discharge as
(w here / is the ripple frequency, here 120 Hz) in order to a ram p is in fact quite accurate, especially if the ripple is
ensure sm all ripple by m aking the time constant for dis- sm all, and in any case it errs in the direction o f conservative
charge m uch longer than the tim e between recharging. We design - it overestim ates the ripple.
m ake this vague statem ent clearer now.
Exercise 1.20. Design a full-wave bridge rectifier circuit to de-
liver 10 V dc with less than 0.1 V (pp) ripple into a load drawing
A. C alcu lation o f ripple voltage
up to 10 m A . Choose the appropriate ac input voltage, assuming
It is easy to calculate the approxim ate ripple voltage, par- 0.6 V diode drops. Be sure to use the correct ripple frequency in
ticularly if it is small com pared with the dc (see Fig- your calculation.
ure 1. 6 1). The load causes the capacitor to discharge som e-
w hat betw een cycles (or half-cycles, for full-wave rectifi-
cation). If you assume that the load current stays constant
(it w ill, for small ripple), you have

AV = ^A t ( from I = C ^ j- (1.25)
V dt
Just use 1I f (or 1 /2 / for full-wave rectification) for At (this
estim ate is a bit on the safe side, because the capacitor be- F igu re 1.62. Bridge rectifier circuit. The polarity m arking and
curved electrode indicate a polarized capacitor, which m ust not be
gins charging again in less than a half-cycle). You get30
allowed to charge with the opposite polarity.
Aoad
AV = (half-w ave),
7c
/load
AV' = (full - wave).
2fC 1.6.4 R e ctifie r c o n fig u ra tio n s fo r po w er s u pp lie s
A. F ull-w ave bridg e
A dc pow er supply w ith the bridge circuit we just discussed
looks as show n in Figure l . 62. In practice, you generally
buy the bridge as a prepackaged module. The sm allest ones
com e w ith m axim um current ratings o f l A average, with
a selection o f rated m inim um breakdow n voltages going
from 100 V to 600 V, or even 1000 V. G iant bridge rectifiers
are available w ith current ratings o f 25 A or more.

F igure 1.61. Power-supply ripple calculation. B. C enter-tapped fu ll-w a ve re c tifie r

The circuit in Figure 1.63 is called a center-tapped full-
If you w anted to do the calculation without any approxi- wave rectifier. The output voltage is half w hat you get if
m ation, you w ould use the exact exponential discharge for- you use a bridge rectifier. It is not the most efficient cir-
m ula. You would be m isguided in insisting on that kind cuit in term s of transform er design, because each half o f
the secondary is used only h alf the tim e. To develop som e
10 W h ile te ac h in g electro n ics w e 'v e n o ticed that stud en ts lo ve to m em o - intuition on this subtle point, consider two different con-
rize th ese eq u atio n s! A n in form al poll o f the a u th o rs sh o w ed that tw o figurations that produce the sam e rectified dc output volt-
o u t o f tw o en g in eers d o n 't m em o rize them . P lease d o n 't w aste b rain age: (a) the circuit o f Figure 1.63, and (b) the sam e tran s-
c ells th a t w ay - in stead , learn h ow to d erive th em . form er, this tim e with its secondary cut at the center tap and
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rewired with the two halves in parallel, the resultant com - I

bined secondary winding connected to a full-wave bridge.
2 .0 A
Now, to deliver the same output power, each half winding
in (a), during its conduction cycle, must supply the same
current as the parallel pair in (b). But the power dissipated ~1ms
in the winding resistances goes like I 2R, so the power lost
to heating in the transformer secondary windings reduced F ig u re 1.64. Illustrating greater l 2R heating with discontinuous
current flow.
by a factor of 2 for the bridge configuration (b ).
Here’s another way to see the problem: imagine we use
the same transformer as in (a), but for our comparison cir- +v
cuit we replace the pair of diodes with a bridge, as in Fig-
ure 1.62, and we leave the center tap unconnected. Now,
to deliver the same output power, the current through the
winding during that time is twice what it would be for a true
full-wave circuit. To expand on this subtle point: heating in
the windings, calculated from Ohm ’s law, is I 2R, so you
have four times the heating for half the time, or twice the
average heating o f an equivalent full-wave bridge circuit.
You would have to choose a transformer with a current rat-
ing 1.4 (square root o f 2) times as large compared with the
(better) bridge circuit; besides costing more, the resulting
supply would be bulkier and heavier.

Exercise 1.21. This illustration o f /2/? heating may help you un-
derstand the disadvantage of the center-tapped rectifier circuit.
What fuse rating (minimum) is required for passing the current
waveform shown in Figure 1.64, which has 1 amp average cur-
rent? Hint: a fuse “blows out" by melting a metallic link (I2R
heating), for steady currents larger than its rating. Assume for D. Voltage m ultipliers
this problem that the thermal time constant of the fusible link The circuit shown in Figure 1.66 is called a voltage dou-
is much longer than the time scale of the square wave, i.e., bler. Think o f it as two half-wave rectifier circuits in se-
that the fuse responds to the value of / 2 averaged over many ries. It is officially a full-wave rectifier circuit because both
cycles.
halves o f the input waveform are used - the ripple fre-
quency is twice the ac frequency (120 Hz for the 60 Hz line
C. S plit supply voltage in the United States).
A popular variation o f the center-tapped full-wave circuit Variations of this circuit exist for voltage triplers,
is shown in Figure 1.65. It gives you split supplies (equal quadruplers, etc. Figure 1.67 shows doubler, tripler, and
plus and minus voltages), which many circuits need. It is an quadrupler circuits that let you ground one side o f the trans-
efficient circuit, because both halves of the input waveform former. You can extend this scheme as far as you want, pro-
are used in each winding section. ducing what’s called a Cockcroft-W alton generator; these
are used in arcane applications (such as particle accelera-
tors) and in everyday applications (such as image intensi-
fiers, air ionizers, laser copiers, and even bug zappers) that
require a high dc voltage but hardly any current.

1.6.5 Regulators
By choosing capacitors that are sufficiently large, you can
reduce the ripple voltage to any desired level. This brute-
F igure 1.63. Full-wave rectifier using center-tapped transformer. force approach has three disadvantages.
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-H -

Figure 1.68. Regulated dc power supply.

A. doubler
10% of the dc voltage), then use an active feedback circuit
to eliminate the remaining ripple. Such a feedback circuit
“looks at” the output, making changes in a controllable se-
ries resistor (a transistor) as necessary to keep the output
voltage constant (Figure 1.68). This is known as a “linear
regulated dc power supply.”32
These voltage regulators are used almost universally as
power supplies for electronic circuits. Nowadays complete
voltage regulators are available as inexpensive ICs (priced
B under $1). A power supply built with a voltage regulator
can be made easily adjustable and self-protecting (against
short circuits, overheating, etc.), with excellent properties
as a voltage source (e.g., internal resistance measured in
milliohms). We will deal with regulated dc power supplies
in Chapter 9.

1.6.6 C irc uit app lications o f diodes

A. S ignal rectifier
C. qu adrupler
There are other occasions when you use a diode to make
a waveform o f one polarity only. If the input waveform
F ig u re 1.67. Voltage multipliers; these configurations don’t require
a floating voltage source. isn’t a sinewave, you usually don’t think of it as a recti-
fication in the sense o f a power supply. For instance, you
• The required capacitors may be prohibitively bulky and might want a train of pulses corresponding to the rising
edge o f a square wave. The easiest way is to rectify the dif-
expensive.
• The very short interval of current flow during each cy- ferentiated wave (Figure 1.69). Always keep in mind the
0.6 V(approximately) forward drop o f the diode. This cir-
cle31 (only very near the top of the sinusoidal waveform)
produces more l 2R heating. cuit, for instance, gives no output for square waves smaller
• Even with the ripple reduced to negligible levels, you still than 0.6 V pp. If this is a problem, there are various tricks
to circumvent this limitation. One possibility is to use hot
have variations of output voltage that are due to other
causes, e.g., the dc output voltage will be roughly pro- carrier diodes (Schottky diodes), with a forward drop of
portional to the ac input voltage, giving rise to fluctua- about 0.25 V.
A possible circuit solution to this problem o f finite diode
tions caused by input line voltage variations. In addition,
changes in load current will still cause the output volt- drop is shown in Figure 1.70. Here D\ compensates T Vs
forward drop by providing 0.6 V o f bias to hold D i at the
age to change because of the finite internal resistances of
threshold of conduction. Using a diode (D i) to provide
the transformer, diode, etc. In other words, the Thevenin
equivalent circuit of the dc power supply has R > 0. the bias (rather than, say, a voltage divider) has several

A better approach to power-supply design is to use 32 A p op ular variant is the regulated s witc hing p ow er converter. A lthough
enough capacitance to reduce ripple to low levels (perhaps its o peration is q uite d ifferen t in d etail, it u ses the same feed back p rin-
ciple to maintain a con stan t o utpu t voltage. See C hap ter 9 for much
31 C alled the conduction angle. more on both techniques.
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A rt o f Electronics Third Edition

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l i i +5V (when tu rn ed on)

D2
JU T T L o,
- N — A- real-tim e clock
'JT NXP PC F8563,

Figure 1.69. Signal rectifier applied to differentiator output.

lithium
“co in ”
cell
I X2
Seik o S-353 90A, e tc

digital
SDA ---- >- link to
~L- y SCL -- ---
*1 G microprocessor
C
100 pF
I
F ig ure 1.71. Diode o r gate: battery backup. The real-time clock
chips are specified to operate properly with supply voltages from
+1.8 V to +5.5 V. They draw a paltry 0.25 pA, which calculates to a
1-million-hour life (a hundred years) from a standard CR2032 coin
cell!

+5V

Figure 1.70. Compensating the forward voltage drop of a diode J i 1N4148

signal rectifier.
1.0k
■ou t

F igu re 1.72. Diode voltage clamp.

advantages: (a) there is nothing to adjust, (b) the com pen-
sation will be nearly perfect, and (c) changes of the for-
ward drop (e.g., with changing temperature) will be com- +15V
pensated properly. Later we will see other instances of
matched-pair compensation of forward drops in diodes,
transistors, and FETs. It is a simple and powerful trick.

B. Diode gates
Another application of diodes, which we will recognize
later under the general heading o f logic, is to pass the
higher o f two voltages without affecting the lower. A good
example is battery backup, a method of keeping some-
thing running (e.g, the “real-time clock” chip in a com - Figure 1.73. Voltage divider providing clamping voltage.
puter, which keeps a running count of date and time) that
must continue running even when the device is switched clamping action; however, a side effect is that it adds 1 kQ
off. Figure 1.71 shows a circuit that does the job. The bat- o f series resistance (in the Thevenin sense) to the signal,
tery does nothing until the +5 V power is switched off; then so its value is a com promise between maintaining a desir-
it takes over without interruption. able low source (Thevenin) resistance and a desirable low
clamping current. Diode clamps are standard equipm ent
C. Diode clam ps on all inputs in contemporary CM OS digital logic. W ith-
Sometimes it is desirable to limit the range o f a signal (i.e., out them, the delicate input circuits are easily destroyed by
prevent it from exceeding certain voltage lim its) some- static electricity discharges during handling.
where in a circuit. The circuit shown in Figure 1.72 will ac-
Exercise 1.22. Design a symmetrical clamp, i.e., one that confines
complish this. The diode prevents the output from exceed-
a signal to the range —5.6 to +5.6 V.
ing about + 5.6 V, with no effect on voltages less than that
(including negative voltages); the only limitation is that A voltage divider can provide the reference voltage for
the input must not go so negative that the reverse break- a clamp (Figure 1.73). In this case you must ensure that the
down voltage o f the diode is exceeded (e.g., —75 V for a resistance looking into the voltage divider (/fVd) is small
1N4148). The series resistor limits the diode current during compared with R because what you have looks as shown
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c u -tr a c k A rt o f Electronics Third Edition 1.6.6. Circuit applications of diodes 37 c u -tr a c k

+5V ■
667 0 JUL
signal
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F igu re 1.77. dc restoration.
F ig u re 1.74. Clamping to voltage divider: equivalent circuit.

1.0k - o ut

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+5.6V
F igure 1.78. Diode limiter.
Figu re 1.75. Poor clamping: voltage divider not stiff enough.

the frequency domain, a subject we’ll take up shortly. For

now we’ll simply say that you can put a capacitor across
the lk resistor, and its stored charge acts to maintain that
point at constant voltage. For example, a 15 )iF capacitor to
ground would make the divider look as if it had a Thevenin
resistance of less than 10C2 for frequencies above 1 kHz.
(You could similarly add a bypass capacitor across D\ in
Figure 1.70.) As we’ll learn, the effectiveness of this trick
F ig u re 1.76. Clamping waveform for circuit of Figure 1.73. decreases at low frequencies, and it does nothing at dc.
One interesting clamp application is “dc restoration” of
a signal that has been ac coupled (capacitively coupled).
in Figure 1.74 when the voltage divider is replaced with its
Figure 1.77 shows the idea. This is particularly important
Thevenin equivalent circuit. When the diode conducts (in-
for circuits whose inputs look like diodes (e.g., a transistor
put voltage exceeds clamp voltage), the output is really just
with grounded emitter, as w e’ll see in the next chapter);
the output of a voltage divider, with the Thevenin equiva-
otherwise an ac-coupled signal will just fade away, as the
lent resistance o f the voltage reference as the lower resistor
coupling capacitor charges up to the signal’s peak voltage.
(Figure 1.75). So, for the values shown, the output o f the
clamp for a triangle-wave input would look as shown in
Figure 1.76. The problem is that the voltage divider doesn’t D. Lim iter
provide a stiff reference, in the language o f electronics. A One last clamp circuit is shown in Figure 1.78. This cir-
stiff voltage source is one that doesn’t bend easily, i.e., it cuit limits the output “swing” (again, a common electronics
has low internal (Thevenin) resistance. term) to one diode drop in either polarity, roughly ± 0 .6 V.
In practice, the problem of finite impedance o f the That might seem awfully small, but if the next stage is an
voltage-divider reference can be easily solved by use o f a amplifier with large voltage amplification, its input will al-
transistor or an op-amp. This is usually a better solution ways be near 0 V; otherwise the output is in “saturation”
than using very small resistor values, because it doesn’t (e.g., if the next stage has a gain o f 1000 and operates from
consume large currents, yet it provides a voltage reference ± 1 5 V supplies, its input must stay in the range ± 1 5 m V
with a Thevenin resistance o f a few ohms or less. Further- in order for its output not to saturate). Figure 1.79 shows
more, there are other ways to construct a clamp, using an what a limiter does to oversize sinewaves and spikes. This
op-amp as part of the clamp circuit. You will see these clamp circuit is often used as input protection for a high-
methods in C hapter 4. gain amplifier.
Alternatively, a simple way to stiffen the clamp circuit
of Figure 1.73, for time-varying signals only, is to add a so- E. Diodes as nonlinear elements
called bypass capacitor across the lower (1 k£2) resistor. To To a good approximation the forward current through a
understand this fully we need to know about capacitors in diode is proportional to an exponential function o f the
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F ig ure 1.81. Approximate log converter.

Voul« lo g V in

Figu re 1.82. Diode drop compensation in the logarithmic con-

+0.6V verter.

1 point A at about - 0 .6 V. Point B is then near ground (mak-

ing /in accurately proportional to Vjn, incidentally). As long
C. -0.6V
as the two (identical) diodes are at the same temperature,
Figure 1.79. A. Limiting large-amplitude sinewaves; B. details; and there is good cancellation of the forward drops, except, of
C. spikes. course, for the difference owing to input current through
D i, which produces the desired output. In this circuit, R \
should be chosen so that the current through £>2 is signif-
icantly larger than the maximum input current in order to
keep D 2 in conduction.
v0u, « log /in
In Chapter 2x we will examine better ways o f construct-
ing logarithmic converter circuits, along with careful meth-
ods of temperature compensation. With such methods it is
possible to construct logarithmic converters accurate to a
few percent over six decades or more of input current. A
Figure 1.80. Exploiting the diode's nonlinear V - / curve: logarith-
better understanding o f diode and transistor characteristics,
mic converter.
along with an understanding of op-amps, is necessary first.
This section is meant to serve only as an introduction for
voltage across it at a given temperature (for a discussion
things to come.
of the exact law, see §2.3.1). So you can use a diode to
generate an output voltage proportional to the logarithm
of a current (Figure 1.80). Because V hovers in the region 1.6.7 Inductive loads and d iode pro te ctio n
of 0.6 V, with only small voltage changes that reflect input
current variations, you can generate the input current with W hat happens if you open a switch that is providing current
a resistor if the input voltage is much larger than a diode to an inductor? Because inductors have the property
drop (Figure 1.81). V = L d l/d t ,
In practice, you may want an output voltage that isn’t
offset by the 0.6 V diode drop. In addition, it would be it is not possible to turn off the current suddenly, because
nice to have a circuit that is insensitive to changes in tem- that would imply an infinite voltage across the inductor’s
perature (a silicon diode’s voltage drop decreases approxi- terminals. What happens instead is that the voltage across
mately 2 mV/°C). The method o f diode drop compensation the inductor rises abruptly and keeps rising until it forces
is helpful here (Figure 1.82). R i makes D 2 conduct, holding current to flow. Electronic devices controlling inductive
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c u -tr a c k A rt o f Electronics Third Edition 1.6.8. Interlude: inductors as friends c u -tr a c k
39

+20V

BOOM! Figure 1.85. RC “sn u b b e f (or suppressing inductive kick.

tor, because the rate of change of inductor current is pro-

Figure 1.83. Inductive “kick” portional to the voltage across it. For applications in which
the current must decay quickly (high-speed actuators o r re-
lays, camera shutters, magnet coils, etc.), it may be better
to put a resistor across the inductor, choosing its value so
that Supply + IR is less than the maximum allowed voltage
across the switch. For the fastest decay with a given max-
imum voltage, a zener (or other voltage-clamping device)
can be used instead, giving a linear ramp-down o f current
JonJ f (at
I turnoff)
rather than an exponential decay.

{ For inductors driven from ac (transformers, ac relays),

the diode protection just described will not work, because
the diode will conduct on alternate half-cycles when the
Fig ure 1.84. Blocking inductive kick. switch is closed. In that case a good solution is an RC
“snubber” network (Figure 1.85). The values shown are
typical for small inductive loads driven from the ac power-
loads can be easily damaged, especially the component
line. Such a snubber should be included in all instruments
that “breaks down” in order to satisfy the inductor’s crav-
that run from the ac powerline, because the power trans-
ing for continuity of current. Consider the circuit in Fig-
former is inductive.33
ure 1.83. The switch is initially closed, and current is flow-
An alternative to the RC snubber is the use of a bidirec-
ing through the inductor (which might be a relay, as de-
tional zener-like voltage-clamping element. Among these
scribed later). When the switch is opened, the inductor tries
the most common are the bidirectional “TVS” (transient
to keep current flowing from A to B, as it had been. In other
voltage suppressor) zener and the metal-oxide varistor
words, it tries to make current flow out of B, which it does
(“MOV"); the latter is an inexpensive device that looks
by forcing B to a high positive voltage (relative to A). In a
something like a disc ceramic capacitor and behaves elec-
case like this, in which there’s no connection to terminal B,
trically like a bidirectional zener diode. Both classes are de-
it may go 1000 V positive before the switch contact “blows
signed for transient voltage protection, are variously avail-
over.” This shortens the life of the switch and also gener-
able at voltage ratings from 10 to 1000 volts, and can
ates impulsive interference that may affect other circuits
handle transient currents up to thousands o f amperes (see
nearby. If the switch happens to be a transistor, it would be
Chapter 9x). Including a transient suppressor (with ap-
an understatement to say that its life is shortened; its life is
propriate fusing) across the ac powerline terminals makes
ended.
good sense in a piece of electronic equipment, not only to
The best solution usually is to put a diode across the
prevent inductive spike interference to other nearby instru-
inductor, as in Figure 1.84. When the switch is on, the
ments but also to prevent occasional large powerline spikes
diode is back-biased (from the dc drop across the induc-
from damaging the instrument itself.
tor’s winding resistance). At turn-off the diode goes into
conduction, putting the switch terminal a diode drop above
1.6.8 Interlude: inductors as frie nds
the positive supply voltage. The diode must be able to han-
dle the initial diode current, which equals the steady current Lest we leave the impression that inductance and induc-
that had been flowing through the inductor; something like tors are things only to be feared, let’s look at the circuit in
a 1N4004 is fine for nearly all cases.
The only disadvantage of this simple protection circuit 33 As explained in §9.5.1, you shou ld choo se a cap acito r rated for
is that it lengthens the decay of current through the indue - "across-the-lin e" service.
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40 1.7. Impedance and reactance Art o f Electronics Third Edition
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Figure 1.86. The goal is to charge up the capacitor from a

source of dc voltage Vm. In the top circuit (Figure 1.86A)
we've done it the conventional way, with a series resis-
tor to limit the peak current demanded from the voltage
source. OK, it does work - but it has a drawback that can
be serious, namely that half the energy is lost as heat in the
resistor. By contrast, in the circuit with the inductor (Fig-
ure 1.86B) no energy is lost (assuming ideal components);
and. as a bonus, the capacitor gets charged to twice the
input voltage. The output-voltage waveform is a sinusoidal
half-cycle at the resonant frequency / = 1/ \j7nI.C, a topic
we’ll see soon (§ 1.7.14).34 35 audible frequencies -
in fra so n ic ultrasonic
frequency (log scale)

Figure 1.87. Example of frequency analysis: “boom box” loud-

speaker equalization. The lowest and highest piano notes, called
AO and C8, are at 27.5 Hz and 4.2 kHz; they are four octaves below
A440 and four octaves above middle C, respectively.

1.7 Im pedance a n d reactance

Warning: this section is somewhat mathematical; you may
wish to skip over the mathematics, but be sure to pay atten-
tion to the results and graphs.
Circuits with capacitors and inductors are more com-
plicated than the resistive circuits we talked about ear-
lier, in that their behavior depends on frequency: a “volt-
Figure 1.86. Resonant charging is lossless (with ideal compo- age divider” containing a capacitor or inductor will have
nents) compared with the 50% efficiency of resistive charging. a frequency-dependent division ratio. In addition, circuits
Charging is complete after tf, equal to a half-cycle of the resonant containing these components (known collectively as re-
frequency. The series diode terminates the cycle, which would oth- active components) “corrupt” input waveforms such as
erwise continue to oscillate between 0 and 2V,.. square waves, as we saw earlier.
However, both capacitors and inductors are linear de-
vices, meaning that the amplitude of the output waveform,
whatever its shape, increases exactly in proportion to the
A mechanical analogy may be helpful here. Imagine dropping pack-
input waveform's amplitude. This linearity has many con-
ages onto a conveyor belt that is moving at speed v , the packages are sequences, the most important of which is probably the
accelerated to that speed by friction, with 50% efficiency, finally reach- following: the output of a linear circuit, driven with a
ing the belt speed V, at which speed they ride into the sunset. That’s re- sinewave at some frequency f , is itself a sinewave at the
sistive charging. Now we try something completely different, namely, same frequency (with, at most, changed amplitude and
we rig up a conveyor belt with little catchers attached by springs to the phase).
belt; and alongside it we have a second belt, running at twice the speed Because of this remarkable property of circuits contain-
(2l>). Now when we drop a package onto the first conveyor it com- ing resistors, capacitors, and inductors (and, later, linear
presses a spring, then rebounds at 2 v ; and it makes a soft landing onto
amplifiers), it is particularly convenient to analyze any such
the second conveyor. No energy is lost (ideal springs), and the package
circuit by asking how the output voltage (amplitude and
rides off into the sunset at 2v. That’s reactive charging.
35 Resonant charging is used for the high-voltage supply in flashlamps
phase) depends on the input voltage for sinewave input at a
and stroboscopes, with the advantages of (a) full charge between single frequency, even though this may not be the intended
flashes (spaced no closer than tf), and (b) no current immediately after use. A graph of the resulting frequency response, in which
discharge (see waveforms), thus permitting the flashlamp to “quench” the ratio of output to input is plotted for each sinewave
after each flash. frequency, is useful for thinking about many kinds of