1 Selected additional topics

This material is optional, and consists of a few applications and extensions of our standard course material that may be of interest to people in particular ﬁelds. I will use my usual target designations P=physics, C=chemistry, E=engineering, EE=electrical engineering, CS=computer science, and B=biology.

1.1 Semiconductor devices. Diodes^P,CS,E,EE

Consider copper, which is an excellent conductor. A 1.0 cm² cross section bit of length 1.0 cm has resistance R = 1.7 × 10^-6Ω, whereas a typical ceramic will have R ≈ 1.0 × 10⁶Ω.
Germanium and Silicon are semiconductors, the same fragment having R ≈ 6.0Ω. Diodes and transistors are usually made of doped Germanium, meaning Germanium with lattice replacement impurities of Arsenic or Indium.
The Germanium atoms have four valence electrons that form covalent bonds with nearby Germanium atoms, resulting in a stable crystal structure with a fairly high resistance. Pure Germanium is nothing to write home about.

Doping with Arsenic results in radical changes in the physical properties, since the Arsenic has an additional valence electron that can become de-localized and drift through the neutral crystal. These electrons allow the crystal to conduct a current when a voltage is applied Doping Germanium with Arsenic results in an N-type semiconductor, since the charge-carriers that are involved in conduction will be the extra negatively charged electrons.

Doping with Indium, which has three valence electrons, will result in holes in the covalent bond lattice of the crystal, and since nearby atoms will pull electrons away from neighbors to complete their octets, the holes can drift and act like positively charged charge-carriers. Now we have a P-type semiconductor. This is very similar to the electron-hole model of Dirac electron theory, in which the holes in the Dirac sea are anti-electrons (positrons). In this case the holes are just holes in the valence band of the solid.

If a P-type semiconductor is joined to an N-type, you have a pn-junction, and now the electrical properties become quite special.
Consider connecting the battery to the junction shown, with the N-type on the left, and P-type on the right. This is called forward-biased. The battery will push the positive charge carriers to the right, and will draw electrons into itself from the P-type semiconductor.

Electrons will enter the N-type from the ground, and will drift right. The junction will be enriched on both sides with charge-carriers, electrons can cross the junction from the left, ﬁll holes in the P-type, and new holes will be created as electrons are pulled out by the battery. The entire junction will conduct.

If the junction is reverse-biased, the situation changes dramatically, since the charge carriers drift in opposite directions, depleting the junction of carriers altogether.
In this case no current can cross the junction, and we have a one-way conduit for current, depending on how the junction is biased. This is a solid-state diode. The transistor is a very similar conﬁguration.

Working with diodes is very simple; they drop 0.6 V when conducting, the current through them is largely uninhibited by the diode and do not pass any current in the reverse direction unless the reverse-bias voltage exceeds a critical value, the break-down voltage. This is usually around 75 V . There is a non-linear relation between the current I through a diode and the voltage dropped across it, but in practice we simply use the 0.6 V value for the voltage drop for any current through the diode in the forward sense.

1.1.1 The voltage clamp

The clamp is a voltage limiter that will both allow a voltage at a chosen point in a circuit to exceed a particular value. This has obvious applications. The circuit symbol for the diode is the arrow.

The voltage at point b will be held at V _c by the battery, but b is the cathode of the diode, and as long as the diode conducts forward, V (a) - V (b) = V _a - V _c = 0.6 V , which very effectively pins the voltage at point a at V (a) = V _c + 0.6 V .
Notice that the battery is trying to push current the wrong way through the diode, but cannot unless it exceeds 75 V . This means that the battery and diode do nothing unless V (a) > V _c + 0.6 V , as if they were not even present.

Clamps are often used to protect circuits from excessive voltages. A good use would be in the construction of analog-digital interfaces that plug into a computer parallel port. The pins on the port all must be maintained between a certain speciﬁed pair of minimum and maximum values.

1.1.2 Signal rectiﬁcation

AC power source voltages oscillate between positive and negative values. Sometimes we want DC power, but batteries are not the most convenient choice for DC since they are limited in range and will run down. A better choice is to use a rectiﬁed AC power supply which will provide a constant DC voltage.
A diode placed in a simple loop powered by a generator V _in(t) suppling

Vin(t) = V0 sin ωt

will chop off the current when the generator reverse-biases the diode. The diode only conducts forward as long as V ₀ < 75 V . The output voltage (dropped across the resistor) is then half-wave rectiﬁed. Notice that according to our simple rules, the voltage across the resistor will peak at

Vout,max = Vin,max - 0.6 V

The ﬁgure below is for R = 2.0Ω, V _in,max = 4.0 V , ω = 3.141rad s , and I_s = 0.05 A. The half-wave rectiﬁer can be made to produce a nearly DC signal by ﬁltering the output through a low-pass ﬁlter.

The half-wave rectiﬁer can be made to produce a nearly DC signal by ﬁltering the output through a low-pass ﬁlter such as that illustrated below. The rectiﬁed voltage ﬁlls the battery which maintains a nearly constant charge.

The voltage across the capacitor will be slightly out of phase with the generator powering the circuit, but the RC decay of the peak signal can be stretched out to the beginning of the next full generator cycle, producing a stable DC baseline output with a small “ripple”
whose magnitude is

Imax-- Imax--π- ΔV ≈ Δt ≈ C ωC

This particular output is not useful if the ripple is large.

A center-tapped transformer has ﬁve leads, two on the primary and three on the secondary. The third secondary lead is connected to the center of the secondary coil. This splits the secondary into two coils. Together with a pair of heavy duty diodes this can be used to make a very effective full wave rectiﬁer.
If the primary voltage is ℰ_p = ℰ₀ sin ωt, then in an ordinary transformer there will be a secondary emf ℰ_s = ℰ_s sin(ωt - φ). In a center tap transformer, the voltage drop in the secondary between top and bottom leads is ℰ_s sin(ωt - φ), but between center-tap and top lead the voltage drop is ℰ_top - 0 = ℰ_s 2 sin(ωt - φ), and between bottom and center-tap ℰ_bottom - 0 = -ℰ_s 2 sin(ωt - φ). The center-tap transformer is used to create a pair of synchronized (in-phase) equal and opposite emf AC-voltage sources.

Consider standard AC, what comes out of the wall socket is 110V AC, 60 Hz, but this is actually RMS voltage. A digital multimeter set to measure AC voltage will measure the RMS value, deﬁned via the average

2 1 ∫ T 2 V rms = --- V (t) dt T 0

which for wall-socket power is

∫ --1-- (110 V )2 = 60 H z 60 Hz V 2 sin2(2 π ⋅60 H z t) dt 0 peak

2 Vpeak-- = 2

so the actual peak voltage is V _peak = 110 V ⋅ √2-- = 155.56 V . Suppose that we want to build a power supply that outputs 25.2 V (RMS), then we need a turn ratio of

Ns-- 25.2 V = 110 V Np

If we obtain a center-tapped 25.2 V transformer and ground the center tap, we can produce a 12.0 V fully rectiﬁed output voltage by passing the signal through two power diodes (where we lose 0.6 V );
The capacitors used as ﬁlters after this stage in a power supply are usually large electrolytic devices, 4500 μF or larger.
Computation of the ripple depends on the load resistance (actually load current). Recall that for the ﬁlter capacitor

dQ I dt I T dV = ----, dV = ------≈ ----- C C 2C

for a full-wave rectiﬁer. For a 60 Hz signal, connected to a load drawing 1.5 A, a 4500 μF ﬁlter capacitor will create a ripple of

1.5-A---⋅ 0.016--s- dV ≈ - 3 ≈ 2.78 V 2 ⋅ 45 × 10

which is 1.96 V (RMS) of ripple.

How well do you understand this topic? Try the following problems.

1 Let V _in(t) = V ₀ sin ωt, and ﬁnd the voltage difference V _out,1 -V _out,2 for this circuit.

Diodes have become even more ubiquitous in modern electronics than resistors. You see them in power supplies, rectiﬁers, clamps, crowbars (a circuit used to safely operate switched transformers), parallel-port pin-sets, voltage regulators, DC power supplies, and of course as constituents in transistors.

1.2 Signal ampliﬁcation with transistors^P,E,EE,CS

Transistors are active devices that have revolutionized modern electronics. Active means that they are powered. The transistor can act like a true ampliﬁer, not only amplifying voltage, but power as well.

Transistors are three-terminal devices whose function in electronics replaces the vacuum tube, the terminals being base, collector and emitter, which are hold-overs from the terminology of triode-tubes, a device that the transistor has largely replaced. Transistors come in two ﬂavors, npn and pnp, which consist of two semiconductor junctions and are indicated with the device symbols

The device has no moving parts, is extremely inexpensive to mass-produce, and simply consists of a triple layer of three semiconducting materials and a set of three conducting leads. It is a triumph of modern physics (quantum theory of electron energy bands in solids).
Its operation is incredibly simple, and can be understood with only the most rudimentary knowledge of circuits. There are four rules for the npn transistor;

1. V _C must be more positive than V _E.
2. Base-to-emitter and base-to-collector circuits act like diodes with base being the anode. This means that V _BE ≈ 0.6 - 0.8 V , and that under normal correct operation V _B ≈ V _E + 0.6 V . If you connect a voltage exceeding 0, 6V across base-emitter, you will probably ruin the transistor.
3. A given transistor has maximum I_C, I_B and V _CE values. Exceeding them will ruin the device. There are tables of limiting parameters for common transistors, and it is a good idea to know the speciﬁcations of the transistors that you are using in a project.
4. If all three rules are obeyed, then the formula

IC = β IB

is obeyed, β is the current gain, and varies between 50 and 250 even for different transistors of the same type. The exact value of this number should not be relied upon.

During correct operation, basic properties of semiconductor physics dictates that there will be a (roughly) 0.6 V drop between the base and emitter terminals

VB = VE + 0.6 V

The transistor and its predecessor the vacuum tube can amplify the power of a signal. Where does the extra energy come from? These are powered devices, and the extra energy comes from the power source (such as a battery) used to bias the transistor; the term used to describe the application of voltages to the collector and base such that the conditions required for operation according to the three primary rules of transistors are satisﬁed.

The simplest transistor circuit that we can build is the emitter-follower, which will simply DC-offset the input voltage by 0.6 V and output it, as long as the transistor is provided with proper operating conditions. The output voltage is the emitter voltage, and this is 0.6 V below base voltage while the transistor is working. We use this simple circuit to demonstrate that the output impedance

V V Z = R = --out-= --E- out I I out E

will be much higher than the input impedance

V V Z = --in- = --B- in I I in B

which is a simple consequence of the model and rule 4

IE = IC + IB = βIB + IB

and so

VB-- -VB---- VE--+--0.6V---- Zin = = --IE-- = (1 + β) ≈ (1 + β)Zout = (1 + β) R IB (1+ β) IE

The voltage source V _in therefore sees a huge impedance in front of it, even though the load R may be rather small. The follower may not be amplifying the voltage, but look at what it is doing to the power;

2 2 V-in- V-out- ℘in = Vin IB = , ℘out = Zin R

but V _in and V _out are nearly the same, differing by a lousy 0.6 V , so

℘out-- Zin-- ≈ ≈ (1 + β) ℘in R

making the device a power ampliﬁer. The idea here is that a microscopic current into the base can control a huge current down through the collector into the emitter. The circuit should be designed so that the base current I_B is microscopic, it will stimulate the transistor to pull a massive current through the collector, and through the emitter resistor.

1.2.1 Common-emitter ampliﬁer

The device illustrated below is so named because the transistor emitter is pulled to ground (“common”). The output voltage here is V _out = V _C, and the input voltage is provided by the signal generator (which we want to amplify) and the voltage V powering the transistor. V not only powers the ampliﬁer by keeping the collector at a high positive potential, it also provides bias voltage, preventing the input signal V _in from pulling the base voltage low enough to shut off the transistor.

How does it work? The circuit resistances are chosen so that the downward currents through R₁ and R₂ ( I₁ and I₂) are large compared to any current that will be pulled off at x into the base. In this case we ﬁnd that

----R1------ VB = Vx + ℰsig = V + ℰsig R1 + R2

which puts the emitter at

----R1------ VE = R + R V + ℰsig - 0.6 1 2

while the transistor works, this pulls a current

V I = --E-- E R E

out of the emitter into the ground. All conditions for rules 1 - 3 are met, so

---β---- IC = βIB , IE = (1 + β)IB , IC = IE ≈ IE 1 + β

and so

R β R β R V = V - I R = V - ---C---------V = V - --C---------(-------1----V + ℰ - 0.6) C C C R 1 + β E R 1 + β R + R sig E E 1 2

The generator applied to point x represents the voltage signal that we wish to amplify. We can see that our output voltage V _out = V _C will have a DC-baseline of

RC-----β----- ---R1------- Vout,DC = V - ( V - 0.6) RE 1 + β R1 + R2

on top of which there will be an ampliﬁed, inverted version of the input signal

RC------β---- Vout = VC = Vout,DC - ℰsig RE 1 + β

These two very simple applications give you some indication of what the transistor can do. Power and signal voltage ampliﬁcation are just two examples out of many thousands. The transformer can amplify the voltage or the current of a signal, but not both. The transistor can do both simultaneously, along with lots of other things. Very precise switching, without “ringing” can be accomplished with transistors. Logic gates (AND/OR/NAND...) can be designed using a transistor and a resistor, such gates are RTL logic gates and preceded CMOS and TTL gates in computer applications.
Another interesting bit of physics is the fact that when a voltage is applied to certain types of crystals, they mechanically deform or warp (piezo-electric effect). A suitable crystal, with a voltage applied to it provided by an antenna, will oscillate. It can be mechanically coupled to a solenoid to form a speaker or mechanical vibrator, and you have a simple radio-wave receiver.

1.3 Thermoelectric and Peltier effects^P,C,B,CS,E

There are all sorts of currents, current is just the ﬂow of some quantity. There are electrical currents (ﬂow of charged particles), energy currents, heat currents, and even entropy currents. Currents are driven by gradients or ocal variations in potentials or affinities of different sorts. For example consider Newton’s law of cooling, which relates the ﬂow of heat to a temperature gradient with no accompanying ﬂow of matter

dQ--- κA--- dT--- = δT = κA dt ℓ dx

Write this in terms of the heat current (density)

dQ- dt-- dT--- 2-d-- -1- JQ = = κ = - κT ( ) A dx dx T

and now write Ohm’s law, the ﬂow of electrical current in response to a voltage or emf variation within a conductor at constant temperature

d Je = e JN = σEx = - σ ----(ℰ ) dx

where σ = 1 ρ is the conductivity and e is the charge on the particles actually ﬂowing.
These two formulas relate the current densities to gradients in potentials or affinities driving them.
Suppose that we consider the ﬂow of electrical current (charged particles) and the ﬂow of heat simultaneously in a material. It is reasonable to assume that the two currents J_e and J_Q would depend on both of the potentials in some way, and in the simplest approximation that makes both currents zero when both potential gradients are zero we would have

-d-- -d-- 1-- 2 -d-- 1-- d--- Je = - σ ( ℰ ) + β ( ), JQ = - κT ( ) + γ (ℰ ) dx dx T dx T dx

Some rather deep considerations lead to the fact that γ = β, which is called the Onsager reciprocity relation. It is this fact that leads to both the Seebach and Peltier effects; the ability of a thermocouple or junction between two dissimilar metals to generate a voltage difference when the end of the junction are maintained at different temperatures, and the isothermal generation of a heat ﬂow through a metal junction when a current is passed through it. Both of these effects are mainstays of modern computer technology.

This is a thermocouple; two different metals A and B (wires) joined together with the two junctions maintained at different (constant) temperatures T₁ and T₂

We connect a voltmeter between the ends so no electrical current passes from end to end, this makes J_e = 0, and therefore a voltage drop is created between the ends of each piece of metal;

-d-- β---d-- 1-- (ℰ ) = ( ) dx σ dx T

add up the three portions

β β β ℰ - ℰ = -B--(T - T ), ℰ - ℰ = --A-(T - T ), ℰ - ℰ = --B-(T - T ) 1 4 σ 1 3 2 1 σ 2 1 3 2 σ 3 1 B A B

the sum gives us the voltage drop across the whole device

βA-- βB-- ℰ3 - ℰ4 = ( - )(T1 - T2) σA σB

Thermocouples typically generate only a volt or a fraction of a volt, but they can be connected in series and so generate a signiﬁcant voltage. The device essentially uses a heat source to maintain the temperature difference, transforming the heat energy at least partially into a useable source of emf.

Consider a thermocouple junction between two different metals, one of which is a heat-sink (the one on the right), the other a source of heat that we wish to remove heat from. By passing an electrical current through the junction, we can draw heat through it even under isothermal conditions T_A = T_B.

With no temperature gradients we have (using Onsager’s relation)

-d-- -d-- Je = - σ (ℰ ), JQ = β (ℰ ) dx dx

and so for each material

σ Je = - --JQ β

The electrical current into the junction must equal the electrical current ﬂowing out of the junction, and this causes there to be an imbalance in the heat ﬂows into and out of the junction,

βB βA JQ,A - JQ,B = ( ----- ----) Je σB σA

and so the junction can be cooled by running the current in the appropriate direction. This is Peltier cooling.

There really is something for everyone in this section. Chemical diffusion or diffusion of biological populations is a simple phenomenon that can be explained with the same physics that we used here. Lets quickly review the relationship between currents and potentials or affinities;
Electrical current ﬂows in response to an emf difference (differential),
heat ﬂows in response to a temperature differential,
and so we conclude that matter too will ﬂow in response to a matter or chemical potential. We saw in 201 that the work done in moving dN particles from a cell where the chemical potential is μ_f from one where the chemical potential is μ_i is dN ⋅ (μ_f - μ_i), and

dU = dQ + μ dN + qd ℰ = T dS + μ dN + ℰ dq

We see couplings in this formula; divide by dt and the cross-sectional area of the cell of matter at x of width dx and we obtain, with deﬁnitions

1 dS 1 dU 1 dN 1 dq J = -------, J = -------, J = --------, J = ------ S A dt U A dt N A dt e A dt

for all of the currents,

Ju = T JS + μJN + ℰ Je

with each current on the right coupled with its potential or affinity.

Consider how the number of particles dN in the middle cell could change, by a current of matter into the cell from the left, our out into the cell to the right;

d d 1 dN J (x) - J (x + dx) ---(dN ) = A J (x) - A J (x+dx), ---(--------) = --N-----------N------------- dt N N dt A dx dx

-d-- = - JN (x) dx

This is a matter conservation law, call n(x) = 1 A dN dx , which is the concentration or number of particles per unit volume. We have a matter conservation law

dn-- -d-- = - JN dt dx

Now lets ﬁgure out what potential is responsible for our matter ﬂow; from our current equation above we see that it is the chemical potential, and so in the simplest linear relationship between current and potential gradient we would suppose that

d JN = - ν ---μ dx

where μ is some constant. Lets suppose that the matter in the cells, liquids or gases, are ideal, so that we can use our formula from 201 for the chemical potential of a normal system

d-μ- 1- dn-- μ(x) = kT ln n(x) + f(T ), = kT dx n dx

then we obtain Fick’s law

νkT dn J = - ---------- N n dx

and

dn d2n 1 d2n 1 dn ---- = ν ----- = ν kT ( ------- - ----(----)2) dt dx2 n dx2 n2 dx

If we let n(x) = n₀ + c(x) where c(x) is a small variation we get the diffusion equation upon which volumes of chemistry and biology depends

dc(x) ν kT d2c(x) --------≈ --------------- + ⋅ ⋅ ⋅ dt n0 dx2

We see derived what chemists take for granted; matter ﬂow is driven by concentration gradients. Graham’s law of diffusion is buried in here.
I hope this section gives you some insights into what potentials are, their gradients drive ﬂows of charge, matter, heat, and energy. In the simplest approximations these ﬂows are proportional to the potential gradients. These two concepts can be applied to dynamics in any scientiﬁc ﬁeld.