A
419 say of it all, as Fourier said originally-" I t is derived from primary notions
. . . left us without proof."
Now it is on just such notions which underly explicitly or otherwise the development of physical science that those interested in the philosophy of science must concentrate critical attention.
T h e treatment developed in this paper is the result of critical consideration of the acts of measurement whereby numbers are introduced into nature, and follows the application of the same method to the problems of electric and mag- netic induction (Brown,
Kelvin
1940 a and b). It appears that there are only two fundamental measurements, those of length and time, and this has led to the view that all physical quantities must be capable of expression in terms of length and time alone. It this possibility, although he does not seem to have favoured its adoption. Lord
(1889) was interesting, therefore, to find that Maxwell also considered the possibility, saying :
"
(1881) mentions
There is something exceedingly interesting in seeing that we can practically found a metrical system on a unit of length and a unit of time. There is nothing new in it, since it has been known from the time of Newton, but it is still a subject full of fresh interest.
The very thought of such a thing is full of many lessons in science that have scarcely yet been realised, especially as to the ultimate properties of matter." by
T h e treatment adopted in this paper had as its original stimulus two definitions
Sir Arthur Eddington. The first was his emphasis on the fact that the whole subject matter of exact science consists of pointer readings and similar indica- tions '' (Eddington, 1928), which is really a definition of exact
science, and the second was his definition of the electric charge e, viz., that e is only manifested in the presence of other charges, and in particular, for simplicity, in the presence of one other charge, and it is a measure of the mutual interaction.
Consideration of the first led to the view that the only measurements that
physicists actually require to make are either of space-like or time-like intervals,
and the second (since Eddington's definition applies equally to M,
L and T, the mass) follows is more in line with the present philosophy of physics than the classical
" intuitive
" presentation.
§ 2. P O I N T E R R E A D I N G S
I n exact science, numbers are introduced into nature by certain conventional and historic processes, If we exclude the process of pure counting, by which, for instance, we find the number of Jupiter's moons, these conventional processes are found to consist of visual observations of the alignment of two marks, together with counting. I n exact
G.
and this has led Eddington to describe a physicist as a man who requires no sense organs beyond one colour-blind eye, and to call the objects of his emasculated attention pointer
T h e physicist claims that he measures many things-mass, force, potentials and so on-but what he actually observes are pointer readings, and this, together with the faculty of counting, is all that is required for that part, and it is important to remember that it is only a part, of science called exact science. A hint in this direction might have been obtained by recalling that mass is a coefficient invented by Newton, force is the hypothetical cause of change, and potential is a pure
Western-European myth. None of them cauld, therefore, be measured directly.
Now consideration will show that there are only two direct ways of making use of pointer readings in the process of introducing numbers into nature: in one case the alignment is of two different pairs of marks ; in the other case one of the marks at least, and usually both marks, must be the same in each pair of alignments. T h e first type of observation leads to the mea'surement of a length or space interval and the second to a measurement of a time interval. I n the case of measurement of mass by a balance, or resistance by a bridge method, although a pointer reading is observed, it is not this reading which introduces a number, i.e. it is not by counting marks on this scale that the number is obtained. The number is got by pure counting elsewhere.
Consider first the simple measurement of length. This involves the use of a material scale which is placed in contact with both marks on the object to be measured. T h e zero mark on the scale is aligned with one of the marks on the object, and then the mark on the scale which is in contact with the second mark on the object is noted. By a process of counting marks on the scale between the two marks with which alignment has been made, a number is arrived at which is defined to be the length between the two marks on the object. Usually the ob- server moves from one point to the other to observe the two alignments, so that a time interval is necessarily involved in the measurement of a length. T o avoid movement the observer may remain at rest at the point of zero alignment and upon the zero position, so that both may be observed with the same glance. But even in this case, as we know, a time interval is involved. This is usually expressed
by saying that
" light takes time to travel portant to avoid this popular form of speech. We shall emphasize that, in the
".
I n this paper it will be most im- not, therefore, make a hypothesis for which there is no evidence whatever ; we shall restrict ourselves to the discussion of interaction, which is what we observe.
Now the question which arises in connection with the above discussion of measurement is this : is interaction instantaneous ? I n other words, if a disturb- ance is produced at the point occupied by the observer, does its interaction with a distant object occur at the same time (on the observer's clock) as that at which the disturbance was caused ? I t is, of course, well known that this is not the case.
T h e experiment that shows this is one of the Fizeau-wheel type. A disturbance

Lecture 2: Electrical Measurements 2-2
Back when physicists were emasculated colour-blind individuals they made use of a pointer based technique for measuring and comparing voltage, current and impedance. We will have a quick look at this for historical reasons before moving on to more interesting things. A diagram of a typical device is given in Figure 1.
Figure 1: Moving coil meters: D’Arsonval style
As current is passed through the coils a magnetic force is developed between the central coil and the static magnetic field of the permanent magnet. The entire central coil is mounted on jewelled bearings.
The springs attached to the moving coil also bring the current into the coil. The meter therefore measures current. How do you measure voltage and resistance with such a device. Simple,
• Voltage: Use a high quality resistor to convert the voltage to be measured into a current. Then pass this current through the moving coil.
• Resistance: Use a high quality voltage source and a series resistor to set-up a current through the resistor of interest. We set the output current to be equal to the full scale value when the output terminals are shorted together. We then include the resistor to be measured between the terminals and measure the decrease in the current flowing.
In order to minimize the load placed on the circuit (and therefore minimize the effect of the measurement itself) one wants a current meter with the lowest possible impedance, while for a voltage measurement you want the highest possible impedance (i.e. the high quality resistor should be of high value). For the resistance measurement it is probable that other things are important (i.e. probably don’t want to pass too much current through the resistor because you may heat it and therefore change its impedance
- termed self-heating).
2.2.1.1
Changing range on the multimeter
The coil will have a fixed full-scale current range determined by the design of the coil, springs and magnet. To allow for varying the range of current to be measured one uses a current shunt. This is merely another resistor in parallel with the coil that provides another current path. The current that passes through the meter,
R m i m is now reduced to i m i tot
=
R sh
R sh
+ R m where R sh is the shunt resistance and is the meter coil resistance. The shunt resistor is chosen by some external switch. For the highest current values the shunt resistor becomes impractically small - in this case one inserts a series resistor with the meter to raise its effective resistance.

Lecture 2: Electrical Measurements 2-3
In order to switch the sensitivity in the voltmeter mode we switch the value of the series resistor. For the impedance measurements we change the value of the series resistor.
If we do this we can compare an unknown voltage against a known voltage. Most techniques to do this require a sensitive “null” detector i.e. a device that has the ability to detect small non-zero voltages or currents (see Figure 2). i.e. How do we make these adjustable voltage sources and the null detector?
Detector
Unknown
Voltage
Vx Vr
Adjustable
Known
Voltage
Figure 2: The “null” detector detects either the small current flowing or the small voltage difference
For the variable source we can use a precision variable divider along with a voltage standard. In the
19th century this voltage source was based on a chemical energy gap (i.e. a battery), whereas today they are more typically based on a semiconductor energy gap (i.e. a Zener diode). For the utmost precision experiments (in Standards Laboratories) they make use of the superconducting energy gap in a Josephson Junction - this voltage can be related to fundamental properties of the Universe. When
Vx r
Null
Detector
R 2
R 1 r s
Vs
Figure 3: The “null” detector detects the small current flowing or the small voltage difference between the variable source and unknown voltage balanced V x
= V
R
=
R
1
R
1
+ R
2
V
S if R
S
R
1
+ R
2
. This voltage measurement has essentially come down to a measurement of an accurate resistance ratio, ρ = R
1
/ ( R
1
+ R
2
) The good thing about this design is that at the measurement point no current is drawn from the unknown source (so the value of r is irrelevant). What are the bad things:
1. How do we make an accurate measurement of ρ ?
2. Does current from V
S cause irreversible changes? (e.g. is V
S comes from a battery
3. Is r
S small enough?, or at least constant?
To make sure that problems (2) and (3) are avoided one typically uses an intermediate voltage standard that can supply lots of current (i.e.
r
S is small). One then regularly calibrates this new standard against the fixed standard to eliminate drifts. By using the intermediate standard in the right hand position of
Figure 3 on both occasions one avoids ever needing to draw substantial current from either the standard or the voltage source under test.
In order to achieve challenge (1) one can take a number of routes: one can use a simple analog rotary or linear potentiometer and measure its position or angle as a measure of its actual impedance. This is easy but is going to be low precision. A much better approach is to to use a sequence of switched
“digital” potentiometers in series. The internals of one such “digital pot” is shown in Figure 4 - a switch chooses how many resistors are included in the circuit. By ganging several of this devices in series (as shown in Figure 5) one obtains a large range with potentially high precision and accuracy (as long as the constituent resistors have good stability and accuracy). This works very well but can be difficult
(expensive) to purchase a large number of different valued resistors with accurately set values.

Lecture 2: Electrical Measurements 2-4
ISO-9001
Registered
This difficulty can be overcome by using an R-2R network (see Figure 6) in which the output voltage is set by opening and closing the various switches.
The output voltage of the network is equal to
V out
= V r
N/ (2 n −
R
1) where n
= R R thevenin
= R R thevenin
= R R thevenin
= R R thevenin
= R
R R
V o u t
2 R
(T e rm .)
B it N
R
2 R
R
B it N -1
R
2 R
R
B it 2
2 R
R R R R
2 R
R R
2 R
Figure 3. Thevenin Resistance
Digit a l in for m a t ion is p r es en t ed t o t h e la d d er a s in d ivid u a l b it s of a d igit a l wor d s wit ch ed b et ween a r efer en ce volt a ge (Vr ) a n d gr ou n d (Figu r e 4 ).
Dep en d in g on t h e n u m b er a n d loca t ion of t h e b it s s wit ch ed t o Vr or gr ou n d , Vou t will i t
M S B
V o u
V r /
S in ce a n R/ 2 R la d d er is a lin ea r cir cu it , we va r y b et ween 0 volt s a n d Vr . If a ll in p u t s
1
B # t
2 ca n a p p ly t h e p r in cip le of s u p er p os it ion t o
Figure 4: An example of a digital potentiometer a t t h e ou t p u t , if a ll in p u t s a r e con n ect ed t o
Vr , t h e ou t p u t volt a ge a p p r oa ch es Vr , a n d if s om e in p u t s a r e con n ect ed t o gr ou n d a n d s om e t o Vr t h en a n ou t p u t volt a ge b et ween
0 volt s a n d Vr occu r s . Th es e in p u t s (a ls o ca lled b it s in t h e d igit a l lin go) r a n ge fr om t h e Mos t S ign ifica n t Bit t o t h e Lea s t S ign ifica n t Bit . As t h e n a m es in d ica t e, t h e
2
3
4
5
6
7 e ffe c t of in d ivid u a l b it loc a t ion s t o t h e N t h b it .
Not ice t h a t s in ce b it 1 h a s t h e gr ea t es t effect on t h e ou t p u t volt a ge it is d es ign a t ed t h e Mos t S ign ifica n t Bit .
V r / 4
V r / 8
V r / 1 6
V r / 3 2
V r / 6 4
V r / 1 2 8 a ge is ca lcu la t ed b y s u m m in g t h e effect of a ll b its con n ected to Vr . For exa m p le, if b its
1 a n d 3 a r e con n ect ed t o Vr wit h a ll ot h er in p u t s gr ou n d ed , t h e ou t p u t volt a ge is ca lcu la t ed b y:
Vou t = (Vr / 2 )+(Vr / 8 )
10,000 Ω
9 Ω wh en a ct iva t ed , will ca u s e t h e s m a lles t
8
9
0 to
V r / 2 5 6
90 Ω
V r / 5 1 2
900 Ω
1 0 V r / 1 0 2 4
0 to
9000 Ω
Vou t = 5 Vr / 8 .
t h e b it s (or in p u t s ) b it 1 t o b it N t h e ou t p u t
1 1 V r / 2 0 4 8
Th e R/ 2 R la d d er is a b in a r y cir cu it . Th e volt a ge ca u s ed b y con n ect in g a p a r t icu la r effect of ea ch s u cces s ive b it a p p r oa ch in g th e
N
1 2
L S B
V r /
V r
4
/
0 9
2 N
6 b it t o Vr wit h a ll ot h er b it s gr ou n d ed is : LS B is 1 / 2 of t h e p r eviou s b it . If t h is s e-
Figure 5: A series of digital potentiometers
Vou t = Vr / 2 N wh er e N is t h e b it n u m b er . For b it 1 , Vou t =Vr / 2 , for b it 2 , Vou t = Vr / 4 et c. Th e t a b le s h ows t h e b it s , t h e e ffe c t of t h e LS B on Vou t a p p r oa ch es 0 . Con ver s ely, t h e fu ll-s ca le ou t p u t of t h e n et wor k (wit h a ll b it s con n ect ed t o Vr ) a p p r oa ch es Vr a s s h own in equ a t ion (1 ).
T e rm .
2 R
R R
B it N
LS B
2 R
B it N -1
2 R
R R
B it 3
2 R
B it 2
2 R
B it 1
M S B
2 R
V ou t
V r
Figure 4. R/2R Ladder of N Bits
Figure 6: A R-2R resistor divider network
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Digital multimeters are based on either a digital to analog converter, or a dual-slope converter (both explained later). Figure 7 shows a typical simple input divider This relies on all these resistors being accurate selected, as well as the resistance being stable. A different type of resistor network as shown in
Figure 6 can effect the same operation but in this case one only needs to manufacture two different types of resistor accurately - one sets the fraction of source voltage required by setting the switches in the binary representation of the fraction desired. These resistor networks are sold with very high accuracy all mounted within a chip.
2.2.3.1
Dual-slope conversion
One popular technique for analog to digital conversion relies on the use of charge integrators to convert a measurement of voltage into a measurement of time. The input voltage to the unit is converted to a current (e.g. most simply by using a resistor) and is then connected to a capacitor for a precise and fixed amount of time. A charge proportional to both the input voltage and the time period is stored on the capacitor. A clock is then started and the capacitor connected to a constant current source. When the

Lecture 2: Electrical Measurements 2-5
Vin
9 M Ω
900k Ω
90k Ω
9k Ω
1k Ω
DVM
Figure 7: A resistor divider network to give multiple ranges on a DVM voltage on the capacitor drops below some threshold value (e.g. 0 V), the clock is stopped and its value read. The time for this discharge to occur is proportional to the original charge on the capacitor and hence to the original voltage to be measured. DVMs from the cheap end up to an Agilent 34401 make use of this process.
2.2.3.2
Voltage-Frequency Converters
Another technique for converting a voltage to a digital value is to, as above, connect the external voltage to a constant current source and then to a capacitor to generate a charge that is increasing linearly with time. The capacitor is also connected to a device that can deposit quanta of charge into the capacitor.
A feedback loop controls the rate of charge pulses in order that the voltage/charge on the capacitor is kept constant with time. Clearly, the rate of charge pulses will be proportional to input voltage and a simple counting of the pulse rate yields a measure of the input voltage.
2.2.3.3
Analog to Digital Conversion: ADC
There are many other ways to convert voltages into a number - we may look at some of these others later in this course, although the upper two techniques are the most widely used approaches in convention
DVMs. I note that the R-2R ladder I described above is an immediate Digital to Analog converter (DAC)
- if I put a digital word onto the switches then the output voltage is an analog voltage proportional to the value of the binary number. We can build an Analog to Digital converter (ADC) from one of these
DACs by comparing the voltage to be measured with the output of the DAC using a null detector. The
DAC word setting is then adjusted until its output is equal to the unknown voltage. We stop at this point and the DAC word then represents the binary equivalent of the unknown voltage (see Chapter 9 in Horowitz and Hill for further details).
AC is loosely intended to mean some voltage that is fluctuating during the time it takes to make a measurement. Usually this would be taken to intend an intentional and periodic modulation of the voltage. The obvious question is what we should measure in order to characterise the magnitude of such a signal. Is it most sensible to measure the amplitude (peak-to-peak, peak, root mean square?), or the power?
All of these measures are obviously different. If we consider a sine-wave V ( t ) = A sin( ωt + φ ), then the measures are:
• peak-to-peak amplitude (p-p): 2 A
• peak amplitude (p) or sometimes (confusingly) just amplitude: A
• root mean square (rms): A/
√
2

Lecture 2: Electrical Measurements 2-6
• power: A 2 R/ 2 = R R t
1
+1 /ω t
1
V 2 ( t ) dt = RV ( t ) 2
However, for other types of signal (e.g. square or triangle waves) or for noise these relationships ar different i.e. for “white noise” the peak-to-peak value (99/
Ways to measure AC signals:
1. If one can sample an AC signal fast enough then it is possible to sample the waveform and therefore obtain all of these measurements by calculation, or even just give the waveform and therefore avoid the difficulty. One could, for example, just look at the waveform on a CRO (up to perhaps 100 GHz these days).
2. Traditionally one used an “true rms voltmeter” to deliver the rms value for a wave. The voltmeter obtains this by essentially determining the average power in the signal and taking the square root of it i.e. the input waveform is squared (rectified) and then averaged, and then the square root is taken of the final value. The AC voltage signal produces the same heating in a resistor as DC signal with a voltage equal to the root mean-square value of the AC signal.
3. Spectrum Analyzer: measures the spectrum of the signal from which is it possible to recover the full time domain behaviour (i.e. give the waveform), or use the spectrum analyzer to deliver the amplitude (in whatever form) of the most important sinusoidal components.
4. Lock-in Amplifier: a tool that allows you to directly measure noise at various frequencies -discussed later in course
To measure resistance accurately one really needs to compare them with precision resistance standards.
The classic accurate method of doing this is the Wheatstone Bridge (actually invented by Hunter Christie see Figure 8). The resistors labelled R
1
, R
2
, and R s are all standard resistors used to measure the
R1 R2
V Null
Detector
Rx
Rs
Figure 8: The Classic Wheatstone Bridge unknown resistance R x
.
R s is a variable resistor and is adjusted until the null detector declares that the bridge is balanced, i.e. that there is no current flowing along the central branch. We can solve to determine what this means in terms of the values of the resistors and we find that the balanced condition corresponds to R x
= R s
R
1
/R
2
. The ratio R
1
/R
2 is referred to as a multiplier, and the ratio of these two resistors could well be switch selectable for values between say 10 − 3 and 10 3 so that a broad range

Lecture 2: Electrical Measurements 2-7 of resistances can be measured. In general the sensitivity is optimal when all impedances are about the same. One can use this bridge for both AC and DC measurements but you need to take into account any reactances in the circuit. We often use such bridges for the sensor in sensitive thermometry in our lab.
The usual method for measuring the resistance is to pass current through it and measure the voltage drop. Unfortunately, this very simple approach can cause some issues. If the resistor is based on many turns of resistive wire then there may be some associated inductance, and there is always a little bit of self-capacitance as well. The resulting impedance is dominated by resistance and may be represented as the network shown here on Figure 9. One can calculate the impedance of this arrangement as:
Figure 9: Equivalent circuit of real resistor
R + jω − CR 2 − CL 2 ω 2 + L
Z =
C 2 L 2 ω 4 + C 2 R 2 ω 2 − 2 CLω 2 + 1
(1)
If R has been constructed so that L and C are small (obviously the desirable situation), then we can assume that ω
2
LC 1 and ω
4
L
2
C
2 being the square of a small number will be real small. The previous expression collapses to:
Z =
R + ıω L (1 − ω
2
LC ) − CR
2
1 + ω 2 C ( CR 2 − 2 L )
(2)
From this we find that the real part is:
R eff
=
R
C ( CR 2 − 2 L ) ω 2 + 1 and the phase angle is approximately (assuming ω 2 LC 1):
(3) tan φ =
L − CR 2
R
ω
(4)
So we can choose to have a frequency independent resistance by setting 2 L/C = R 2 when we design the resistor, or we can choose to have a resistor that has no phase shift (or reactance) by choosing L/C = R 2 .
Of course we can’t do both at the same time. The usual choice is to do the later which means that most resistors have a small frequency dependence.
A resistor needs to be connected to the circuit with some wires and they of course possess resistance as well. This lead resistance adds to the measurement of the resistance, and this is especially a big problem if the resistance to be measured is small. It also becomes a big problem if we are trying to make measurements in the presence of changing lead resistances (perhaps because they are in a large and changing temperature environment), or if the leads are necessarily long. We can overcome this issue by using a very sneaky technique called the 4-wire approach. In this case we connect four leads to the resistor: two leads apply current through the resistor while the other two measure the voltage drop across the resistor. If virtually no current flows in the sense leads then the resistance of the sense leads does not enter into the measurement.

Lecture 2: Electrical Measurements 2-8
The resistance can be measured in a bridge configuration (called a Kelvin bridge - a 4 wire variant of the
Wheatstone bridge) or one can just make use of modern DVMs with input impedances of 10 M-10 GΩ in the sense leads. The unit will pass ∼ 1mA through the current leads (depends on the resistor under study but is about this level for low resistances) while passing a few nA through the sense leads. This leads to a reduction in the effect of lead resistance of about a factor of 10 6 .
Remember from kindergarten that it is always possible to express any network of resistors and batteries that only one of these will work - both are equally valid for any situation, however often one is more appropriate).
R
V I R
One finds the equivalent circuit elements from a real network by a couple of measurements. One measures one then also measures the short-circuit current then one obtains a series resistance. For the Norton equivalent one measures the short-circuit current and that determines the value of the current source. If one also measures the open circuit voltage then one can determine the value of the shunt resistor.
R
R V Ideal
Voltmeter
I Ideal
Ammeter
Figure 11: Here we show equivalent circuits for real meters
Lets model the meter when one is making voltage and current measurements. In Figure 11 we show circuit models for real meters. They are shown as ideal meters in parallel or series with resistors. The ideal voltmeter has an infinite impedance while the ideal ammeter has a zero resistance. If we combine
The intention of course is to measure the output voltage, V of the Th´ I of the Norton equivalent. Any error in this measurement comes about from the non-ideality of the meter and the non-ideality of the source under study.
Lets look first at the voltmeter measurement in Figure 12. The ideal voltmeter is just making a simple resistor divider measurement and so we obtain:
V m
V
=
V + δV
V
δV
V
R
1
=
R
1
+ R
R
= −
R + R
1
(5)
(6)

Lecture 2: Electrical Measurements
R
V
I R
R1
R2
V Ideal
Voltmeter
I Ideal
Ammeter
2-9
Figure 12: Here we show real meters connected to real sources
So the fractional error is given in the final expression. If we now turn to the current measurement one can show that it is equal to :
δI
I
= −
R
2
R + R
2
(7)
It is often the case that one wants to transfer the maximum power from a source into a measurement device (e.g. a meter) - this is especially so when one is trying to minimize noise.
P =
V 2 m
R
=
R
1
V 2
( R + R
1
) 2
(8) where V m when: is the voltage over the internal resistor in the meter. This is a maximum as a function of R
1
∂P
∂R
1
= 0 → R
1
= R and V m
=
1
2
V (9)
This is potentially a major difficulty as it heavily loads the source output voltage i.e.
δV ∼ V / 2 but if
R and R
1 are known (and stable) the error can be calculated and corrected. We will see that this power matched position is often also the point at which the signal-to-noise ratio is maximized.
If one is trying to make low level voltage measurements it is frequently the case that thermoelectric voltages will cause errors. These temperature dependent voltages are generated whenever one connects two dissimilar metals together. The table below, Table 1, shows some typical levels:
Copper to
Copper
Silver
Brass
Aluminium
Tin-Lead Solder
Copper Oxide
Cadmium-Tin Solder
Approx microV/degree
< 0 .
3
0.5
3
5
5
1000
0.2
Table 1: Thermal EMFs

Lecture 2: Electrical Measurements 2-10
It is almost always the case that however careful one is when building electronics that there will always be some electrical pick-up from the mains. In analog systems one can attempt to average these issues away but making lengthy measurements but if you are required to make quick measurements there is no real solution. If on the other hand one uses digital measurement techniques then it is possible to suppress sensitivity by making measurements with an averaging time exactly equal to a multiple of the power line cycle time.
If you are attempting measurements in the presence of large varying magnetic fields then you need to avoid induced currents in the circuit. It is sensible to use twisted pair wires to bring the signal to the meter. Loose or vibrating leads can also induce error voltages so you should make sure the leads are well secured. Whenever possible use physical separation or shielding materials to reduce the magnitude of the magnetic fields.
Since real currents flow through resistors that are being measured it can heat the resistor and therefore change its impedance. This often happens for resistors in cryogenic environments in which the specific heat of materials has become very small, or in vacuum systems in which devices can have very high degree of thermal isolation.
Almost all of the techniques discussed in this section require signal levels of the order of a volt, and currents of the order of 10mA in order to obtain good results. If the signal available is not of this level then one needs to amplify the signal. We looked at some issues associated with noise in amplification etc in the last lecture but will suspend discussion of the details of this until next time.