11.4.1.1 Kilogram

The kilogram is the mass of a particular object, the International Prototype Kilogram kept in France. All other masses are related to the kilogram by comparison.

11.4.1.2 Metre

The definition of the metre incorporates the agreed value for the speed of light (c = 299 792 458 m/s). This speed does not have to be measured. The metre is not in practice established by the time-of-flight experiment implied in the definition, but by using given wavelengths of suitable laser radiation stated in the small print of the definition.

11.4.1.3 Second

The second is established by setting up a caesium clock apparatus. This apparatus enables a man-made oscillator to be locked onto a frequency inherent in the nature of the caesium atom, thus eliminating the inperfections of the man-made oscillator, such as temperature sensitivity and ageing.

Using this apparatus, the second can be established extremely close in size to that implied in the definition. Time and frequency can be propagated by radio signals and so calibration of, for example, a frequency counter consists in displaying the frequency of a stable frequency transmission such as BBC Radio 4 on 198 kHz. Commercial off-air frequency standards are available to facilitate such calibration.

11.4.1.4 Voltage

Work by Kibble² and others has led to the value of 483 597.9 GHz/V being ascribed to the height of the voltage step which is seen when a Josephson junction is irradiated with electromagnetic energy. A Josephson junction is a device formed by the separation of two superconductors by a very small gap. The voltage produced by such a junction is independent of temperature, age, materials used and apparently any other influencing quantity. It is therefore an excellent, if somewhat inconvenient, reference voltage. It is possible to cascade thousands of devices on one substrate to obtain volt level outputs at convenient frequencies. Industrial voltage measurements can be related to the Josephson voltage by the traceability hierachy.

11.4.1.5 Impedance

Since 1990 measurements of electrical resistance have been traceable to the quantised Hall effect at NPL.³ Capacitance and inductance standards are realised from DC standards using a chain of AC and DC bridges. Impedance measurements in industry can be related through the traceability hierarchy to these maintained measurement standards. Research is in progress at NPL and elsewhere aimed at using the AC quantum hall effect for the direct realisation of impedance standards.

11.4.1.6 Kelvin

The fundamental fixed point is the triple point of water, to which is assigned the thermodynamic temperature of 273.16 K exactly; on this scale the temperature of the ice point is 273.15 K. (or 0°C). Platinum resistance and thermocouple thermometers are the primary instruments up to 1336K (gold point), above which the standard optical pyrometer is used to about 4000 K.

(1) Electromagnetic torque: moving-coil, moving-iron, induction and electrodynamic (dynamometer).

(2) Electric torque: electrostatic voltmeter.

(3) Electrothermal torque: maximum demand.

11.5.1.1 Torque effects

Instrument dynamics is discussed later. The relevant instantaneous torques are: driving torque, produced by means of energy drawn from the network being monitored; acceleration torque; damping and friction torque, by air dashpot or eddy current reaction; restoring torque, due usually to spring action, but occasionally to gravity or opposing magnetic field.

The higher the driving torque the better, in general, are the design and sensitivity; but high driving torque is usually associated with movements having large mass and inertia. The torque/mass ratio is one indication of relative performance, if associated with low power demand. For small instruments the torque is 0.001–0.005 mN-m for full-scale deflection; for 10–20 cm scales it is 0.05–0.1 mN-m, the higher figures being for induction and the lower for electrostatic instruments.

Friction torque, always a source of error, is due to imperfections in pivots and jewel bearings. Increasing use is now made of taut-band suspensions (Figure 11.2); this eliminates pivot friction and also replaces control springs. High-sensitivity moving-coil movements require only 0.005 mN-m for a 15 cm scale length: for moving-iron movements the torques are similar to those for the conventional pivoted instruments.

11.5.1.2 Scale shapes

The moving-coil instrument has a linear scale owing to the constant energy of the permanent magnet providing one of the two ‘force’ elements. All other classes of indicators are inherently of the double-energy type, giving a square-law scale of the linear property (voltage or current) being measured; for a wattmeter, the scale is linear for the average scalar product of voltage and current. The rectifier instrument, although an a.c. instrument, has a scale which is usually linear, as it depends on the moving-coil characteristic and the rectifier effect is usually negligible. (In low-range voltmeters, the rectifier has some effect and the scale shape is between linear and square law.) Thermal ammeters and electrodynamic voltmeters and ammeters usually have a true square-law scale, as any scale shaping requires extra torque and it is already low in these types. Moving-iron instruments are easily designed with high torque, and scale shaping is almost always carried out in order to approach a linear scale. In some cases the scale is actually contracted at the top in order to give an indication of overloads that would otherwise be off-scale. The best moving-iron scale shape is contracted only for about the initial 10% and is then nearly linear. Logarithmic scales have the advantage of equal percentage accuracy over all the scale, but they are difficult to read, owing to sudden changes in values of adjacent scale divisions. Logarithmic scales are unusual in switchboard instruments, but they are sometimes found in portable instruments, such as self-contained ohmmeters.

11.5.2.1 Moving-coil indicator

This instrument comprises a coil, usually wound on a conducting former to provide eddy-current damping, with taut-band or pivot/control-spring suspension. In each case the coil rotates in the short air gap of a permanent magnet. The direction of the deflection depends on the polarity, so that unmodified indicators are usable only on d.c., and may have a centre zero if required. The error may be as low as ±0.1% of full-scale deflection; the range, from a few microamperes or millivolts up to 600 A and 750 V, which makes possible multirange d.c. test sets. The scale is generally linear (equispaced) and easily read. Non-linear scales can be obtained by shaping the magnet poles or the core to give a non-uniform air gap.

11.5.2.2 Corrections

The total power taken by a normal-range voltmeter can be 50 μW/V or more. For an ammeter the total series loss is 1–50 mW/A. Such powers may be a significant fraction of the total delivered to some electronic networks: in such cases electronic and digital voltmeters with trivial power loss should be used instead.

11.5.2.3 Induced moving-magnet indicator

The instrument is polarised by a fixed permanent magnet and so is only suitable for d.c. A small pivoted iron vane is arranged to lie across the magnet poles, the field of which applies the restoring torque. One or more turns of wire carry the current to be measured around the vane, producing a magnetic field at right angles to that of the magnet, and this drive torque deflects the movement to the equilibrium position. It is a cheap, low-accuracy instrument which can have a nearly linear centre-zero scale, well suited to monitoring battery charge conditions.

11.5.3.1 Moving-iron indicator

This is the most common instrument for a.c. at industrial frequency. Some (but not all) instruments may also be used on d.c. The current to be measured passes through a magnetising coil enclosing the movement. The latter is formed either from a nickel–iron alloy vane that is attracted into the field, or from two magnetic alloy members which, becoming similarly polarised, mutually repel. The latter is more usual but its deflection is limited to about 90°; however, when combined with attraction vanes, a 240° deflection is obtainable but can be used only on a.c. on account of the use of material of high hysteresis.

The inherent torque deflection is square law and so, in principle, the instrument measures true r.m.s. values. By careful shaping of vanes when made from modern magnetic alloys, a good linear scale has been achieved except for the bottom 10%, which readily distinguishes the instrument from the moving-coil linear scale. Damping torque is produced by eddy-current reaction in an aluminium disc moving through a permanent-magnet field.

11.5.3.2 Precautions and corrections

These instruments work at low flux densities (e.g. 7 mT), and care is necessary when they are used near strong ambient magnetic fields despite the magnetic shield often provided. As a result of the shape of the B/H curve, a.c./d.c. indicators used on a.c. read high at the lower end and low at the higher end of the scale. The effect of hysteresis with d.c. measurements is to make the ‘descending’ reading higher than the ‘ascending’.

Moving-iron instruments are available with class index values of 0.3–5.0, and many have similar accuracies for a.c. or d.c. applications. They tend to be single-frequency power instruments with a higher frequency behaviour that enables good r.m.s. readings to be obtained from supplies with distorted current waveforms, although excessive distortion will cause errors. Nevertheless, instruments can be designed for use in the lower audio range, provided that the power loss (a few watts) can be accepted. Ranges are from a few milliamperes and several volts upward. The power loss is usually significant and may have to be allowed for.

11.5.3.3 Electrodynamic indicator

In this the permanent magnet of the moving-coil instrument is replaced by an electromagnet (field coil). For ammeters the moving coil is connected in series with two fixed field coils; for voltmeters the moving and the fixed coils each have series-connected resistors to give the same time-constant, and the combination is connected in parallel across the voltage to be measured. With ammeters the current is limited by the suspension to a fraction of 1 A, and for higher currents it is necessary to employ non-reactive shunts or current transformers.

11.5.3.4 Precautions and corrections

The flux density is of the same order as that in moving-iron instruments, and similar precautions are necessary. The torque has a square-law characteristic and the scale is cramped at the lower end. By using the mean of reversed readings these instruments can be d.c. calibrated and used as adequate d.c./a.c. transfer devices for calibrating other a.c. instruments such as moving-iron indicators. The power loss (a few watts) may have to be allowed for in calculations derived from the readings.

11.5.3.5 Induction indicator

This single-frequency instrument is robust, but limited to class index (CI) number from 1.0 to 5.0. The current or voltage produces a proportional alternating magnetic field normal to an aluminium disc arranged to rotate. A second, phase-displaced field is necessary to develop torque. Originally it was obtained by pole ‘shading’, but this method is obsolete; modern methods include separate magnets one of which is shunted (Ockenden), a cylindrical aluminium movement (Lipman) or two electromagnets coupled by loops (Banner). The induction principle is most generally employed for the measurement of energy.

11.5.3.6 Moving-coil rectifier indicator

The place of the former copper oxide and selenium rectifiers has been taken by silicon diodes. Used with taut-band suspension, full-wave instrument rectifier units provide improved versions of the useful rectifier indicator.

11.5.3.7 Precautions and corrections

Owing to its inertia, the polarised moving-coil instrument gives a mean deflection proportional to the mean rectified current. The scale is marked in r.m.s. values on the assumption that the waveform of the current to be measured is sinusoidal with a form factor of 1.11. On non-sinusoidal waveforms the true r.m.s. value cannot be inferred from the reading: only the true average can be known (from the r.m.s. scale reading divided by 1.11). When true r.m.s. values are required, it is necessary to use a thermal, electrodynamic or square-law electronic instrument.

11.5.3.8 Multirange indicator

The rectifier diodes have capacitance, limiting the upper frequency of multirange instruments to about 20 kHz. The lower limit is 20–30 Hz, depending on the inertia of the movement. Indicators are available with CI numbers from 1.0 to 5.0, the multirange versions catering for a wide range of direct and alternating voltages and currents: a.c. 2.5V–2.5kV and 200 mA–10 A; d.c. 100 μV–2.5kV and 50 μA–10A. Non-linear resistance scales are normally included; values are derived from internal battery-driven out-of-balance bridge networks.

11.5.3.9 Moving-coil thermocouple indicator

The current to be measured (or a known proportion of it) is used as a heater current for the thermocouple, and the equivalent r.m.s. output voltage is steady because of thermal inertia–-except for very low frequency (5 Hz and below). With heater current of about 5 mA in a 100Ω resistor as typical, the possible voltage and current ranges are dictated by the usual series and shunt resistance, under the restriction that only the upper two-thirds of the square-law scale is within the effective range. The instrument has a frequency range from zero to 100 MHz. Normal radiofrequency (r.f.) low-range self-contained ammeters can be used up to 5 MHz with CI from 1.0 to 5.0. Instruments of this kind could be used in the secondary circuit of a r.f. current transformer for measurement of aerial current. The minimum measurable voltage is about 1 V.

The thermocouple, giving a true r.m.s. indication, is a primary a.c./d.c. transfer device. Recent evidence⁴ indicates that the transfer uncertainty is only a few parts in 10⁶. Such devices provide an essential link in the traceability chain between microwave power, r.f. voltage and the primary standards of direct voltage and resistance.

11.5.3.10 Precautions and corrections

The response is slow, e.g. 5–10 s from zero to full-scale deflection, and the overload capacity is negligible: the heater may be destroyed by a switching surge. Thermocouple voltmeters are low-impedance devices (taking, e.g., 500 mW at 100 V) and may be unsuitable for electronic circuit measurements.

11.5.4.1 Electrostatic voltmeter

This is in effect a variable capacitor with fixed and moving vanes. The power taken (theoretically zero) is in fact sufficient to provide the small dielectric loss. The basically square-law characteristic can be modified by shaping the vanes to give a reasonably linear upper scale. The minimum useful range is 50–150 V in a small instrument, up to some hundreds of kilovolts for large fixed instruments employing capacitor multipliers. Medium-voltage direct-connected voltmeters for ranges up to 15 kV have CI ratings from 1.0 to 5.0. The electrostatic indicator is a true primary alternating/direct voltage (a.v./d.v.) transfer device, but has now been superseded by the thermocouple instrument. The effective isolation property of the electrostatic voltmeter is attractive on grounds of safety for high-voltage measurements.

11.5.5.1 Electrodynamic indicator

The instrument is essentially similar to the electrodynamic voltmeter and ammeter. It is usually ‘air-cored’, but ‘iron-cored’ wattmeters with high-permeability material to give a better torque/mass ratio are made with little sacrifice in accuracy. The current (fixed) coils are connected in series with the load, and the voltage (moving) coil with its series range resistor is connected either (a) across the load side of the current coil, or (b) across the supply side. In (a) the instrument reading must be reduced by the power loss in the volt coil circuit (typically 5 W), and in (b) by the current coil loss (typically 1 W).

The volt coil circuit power corrections are avoided by use of a compensated wattmeter, in which an additional winding in series with the volt circuit is wound, turn-for-turn, with the current coils, and the connection is (a) above. The m.m.f. due to the volt-coil circuit current in the compensating coil will cancel the m.m.f. due to the volt circuit current in the current coils.

The volt circuit terminal marked ± is immediately adjacent to the voltage coil; it should be connected to the current terminal similarly marked, to ensure that the wattmeter reads positive power and that negligible p.d. exists between fixed and moving coils, so safeguarding insulation and eliminating error due to electric torque.

When a wattmeter is used on d.c., the power should be taken from the mean of reversed readings; wattmeters read also the active (average) a.c. power. For the measurement of power at very low power factor, special wattmeters are made with weak restoring torque to give f.s.d. at, e.g., a power factor of 0.4. Range extension for all wattmeters on a.c. can be obtained by internal or external current transformers, internal resistive volt-range selectors or external voltage transformers. Typical self-contained ranges are between 0.5 and 20 A, 75 and 300 V. The usable range of frequency is about 30–150 Hz, with a best CI of 0.05.

Three-phase power can be measured by single dual-element instruments.

11.5.5.2 Thermocouple indicator

The modern versions of this instrument are more correctly termed electronic wattmeters. The outputs from current-and voltage-sensing thermocouples, multiplied together and amplified, can be displayed as average power. Typical d.c. and a.c. ranges are 300 mV–300V and 10 mA–10 A, from pulsed and other non-sinusoidal sources at frequencies up to a few hundred kilohertz. The interaction with the network is low; e.g. the voltage network has typically an impedance of 10 k Ω/V.

11.5.5.3 Induction indicator

The power-reading instrument includes both current and voltage coils. The energy meter is referred to later. The instruments are frequency sensitive and are used only for fixed-frequency switchboard applications. The CI number is 1.0–5.0, usually the latter.

11.5.5.4 Electrodynamic reactive-power indicator

The active-power instrument can read reactive power if the phase of the volt circuit supply is changed by 90°, to give Q = VI sin φ instead of P = VI cos φ (for sinusoidal conditions).

11.5.6.1 Maximum-demand instrument

The use of an auxiliary pointer carried forward by the main pointer and remaining in position makes possible the indication of maximum current values, but the method is not satisfactory, because it demands large torques, a condition that reduces effective damping and gives rise to overswing. A truer maximum demand indication, and one that is insensitive to momentary peaks, is obtainable with the aid of a thermal bimetal. Passing the current to be indicated through the bimetal gives a thermal lag of a few minutes. For long-period indication (1 h or more) a separate heater is required.

Such instruments have been adapted for three-phase operation using separate heaters. A recent development permits a similar device to be used for two phases of a three-phase supply. The currents are summed linearly, and the maximum of the combined unbalanced currents maintained for 45 min is recorded on a kV-A scale marked in terms of nominal voltage.

11.6.1.1 Performance of single-phase meters

The limits of permissible error are defined in BS 37. In general, the error should not exceed ± 2% over most of the working range for credit meters. For prepayment meters, +2 and −3% for loads above 1/30 full load at unity power factor, at any price setting to be used, are the specified limits. Tests for compliance with this requirement at and below 1/10 marked current must be made with the coin condition that not less than one nor more than three coins of the highest denomination acceptable by the meter are prepaid.

It is usual for manufacturers to adjust their meters to less than one-half of the permissible error over much of the working range. The mean error of an individual meter is, of course, less than the maximum observable error, and the mean collective error of a large number of meters is likely to be much less than ± 1% at rated voltage and frequency. If no attempt is made during production testing to bias the error in one direction, the weighted mean error of many meters taken collectively is probably within 0.5%, and is likely to be positive (see Figure 11.5(a)).

11.6.1.2 Temperature

The temperature error of a.c. meters with no temperature compensation is negligible under the conditions normally existing in the UK. In North America it is common practice to fix meters on the outside of buildings, where they are subjected to wide temperature ranges (e.g. 80°C), which makes compensation desirable. However, even in the UK it is common to provide compensation (when required) in the form of a strip of nickel-rich alloy encircling the stator magnet. Its use is advantageous for short-duration testing, as there is an observable difference in error of a meter when cold and after a 30 min run.

11.6.1.3 Voltage and frequency

Within the usual limits of ± 4% in voltage and ± 0.2% in frequency, the consequent errors in a meter are negligible. Load-shedding, however, may mean substantial reductions of voltage and frequency. A significant reduction in voltage usually causes the meter to read high on all loads, as shown in Figure 11.5(b). Reduction of the supply frequency makes the meter run faster on high-power-factor loads and slower on lagging reactive loads: an increase of frequency has the opposite effect (Figure 11.5)(c). The errors are cumulative if both voltage and frequency variations occur together.

11.7.1.1 Linear ramp

This is a voltage/time conversion in which a linear time-base is used to determine the time taken for the internally generated voltage v_s to change by the value of the unknown voltage V. The block diagram (Figure 11.6(b)) shows the use of comparison networks to compare V with the rising (or falling) v_s; these networks open and close the path between the continuously running oscillator, which provides the counting pulses at a fixed clock rate, and the counter. Counting is performed by one of the binary-coded sequences; the translation networks give the visual decimal output. In addition, a binary-coded decimal output may be provided for monitoring or control purposes.

Limitations are imposed by small non-linearities in the ramp, the instability of the ramp and oscillator, imprecision of the coincidence networks at instants y and z, and the inherent lack of noise rejection. The overall uncertainty is about ±0.05% and the measurement cycle would be repeated every 200 ms for a typical four-digit display.

Linear ‘staircase’ ramp instruments are available in which V is measured by counting the number of equal voltage ‘steps’ required to reach it. The staircase is generated by a solid state diode pump network, and linearities and accuracies achievable are similar to those with the linear ramp.

11.7.1.2 Charge balancing

This principle⁵ employs a pair of differential-input transistors used to charge a capacitor by a current proportional to the unknown d.c. voltage, and the capacitor is then discharged by a large number of small +δq and -δq quantities, the elemental discharges being sensed and directed by fast comparator/flip-flop circuits and the total numbers being stored. Zero is in the middle of the measurement range and the displayed result is proportional to the difference between the number of +δq, -δq events.

The technique which, unlike dual-ramp methods, is inherently bipolar, claims some other advantages, such as improved linearity, more rapid recovery from overloads, higher sensitivity and reduced noise due to the averaging of thousands of zero crossings during the measurement, as well as a true digital auto-zero subtraction from the next measurement, as compared with the more usual capacitive-stored analogue offset p.d. being subtracted from the measured p.d.

Applications of this digital voltmeter system (which is available on a monolithic integrated circuit, Ferranti ZN 450) apart from d.c. and a.c. multimeter uses, include interfacing it directly with a wide range of conventional transducers such as thermocouples, strain gauges, resistance thermometers, etc.

11.7.1.3 Successive approximation

As it is based on the potentiometer principle, this class produces very high accuracy. The arrows in the block diagram of Figure 11.7 show the signal-flow path for one version; the resistors are selected in sequence so that, with a constant current supply, the test voltage is created within the voltmeter.

Each decade of the unknown voltage is assessed in terms of a sequence of accurate stable voltages, graded in descending magnitude in accordance with a binary (or similar) counting scale. After each voltage approximation of the final result has been made and stored, the residual voltage is then automatically re-assessed against smaller standard voltages, and so on to the smallest voltage discrimination required in the result. Probably four logic decisions are needed to select the major decade value of the unknown voltage, and this process will be repeated for each lower decade in decimal sequence until, after a few milliseconds, the required voltage is stored in a coded form. This voltage is then translated for decimal display. A binary-coded sequence could be as shown in Table 11.2, where the numerals in bold type represent a logical rejection of that number and a progress to the next lower value. The actual sequence of logical decisions is more complicated than is suggested by the example. It is necessary to sense the initial polarity of the unknown signal, and to select the range and decimal marker for the read-out; the time for the logic networks to settle must be longer for the earlier (higher voltage) choices than for the later ones, because they must be of the highest possible accuracy; offset voltages may be added to the earlier logic choices, to be withdrawn later in the sequence; and so forth.

The total measurement and display takes about 5 ms. When noise is present in the input, the necessary insertion of filters may extend the time to about 1 s. As noise is more troublesome for the smaller residuals in the process, it is sometimes convenient to use some different techniques for the latter part. One such is the voltage–frequency principle (see below), which has a high noise rejection ratio. The reduced accuracy of this technique can be tolerated as it applies only to the least significant figures.

11.7.1.4 Voltage–frequency

The converter (Figure 11.8) provides output pulses at a frequency proportional to the instantaneous unknown input voltage, and the non-uniform pulse spacing represents the variable frequency output. The decade counter accumulates the pulses for a predetermined time T and in effect measures the average frequency during this period. When T is selected to coincide with the longest period time of interfering noise (e.g. mains supply frequency), such noise averages out to zero.

Instruments must operate at high conversion frequencies if adequate discrimination is required in the final result. If a six-digit display were required within 5 ms for a range from zero to 1.000 00 V to ± 0.01%, then 10⁵ counts during 5 ms are called for, i.e. a 20 MHz change from zero frequency with a 0.01% voltage–frequency linearity. To reduce the frequency range, the measuring time is increased to 200 ms or higher. Even at the more practical frequency of 0.5 MHz that results, the inaccuracy of the instrument is still determined largely by the non-linearity of the voltage–frequency conversion process.

In many instruments the input network consists of an integrating operational amplifier in which the average input voltage is ‘accumulated’ in terms of charge on a capacitor in a given time.

11.7.1.5 Dual-slope

This instrument uses a composite technique consisting of an integration (as mentioned above) and an accurate measuring network based on the ramp technique. During the integration (Figure 11.9) the unknown voltage V_x is switched at time zero to the integration amplifier, and the initially uncharged capacitor C is charged to V_x in a known time T₁, which may be chosen so as to reduce noise interference. The ramp part of the process consists in replacing V_x by a reversed biased direct reference voltage V_r which produces a constant-current discharge of C; hence, a known linear voltage/time change occurs across C. The total voltage change to zero can be measured by the method used in the linear-ramp instrument, except that the slope is negative and that counting begins at maximum voltage. From the diagram in Figure 11.9(a) it follows that tan θ_x/tan θ_r = (T₂ − T₁)/T₁ = V_x/V_r so that V_x is directly proportional to T₂−T₁. The dual-slope method is seen to depend ultimately on time-base linearity and on the measurement of time difference, and is subject to the same limitations as the linear-ramp method, but with the important and fundamental quality of inherent noise rejection capability.

Noise interference (of series-mode type) is principally due to e.m.f.s at power supply frequency being induced into the d.c. source of V_x. When T₁ equals the periodic time of the interference (20 ms for 50 Hz), the charging p.d. v_c across C would follow the dotted path (Figure 11.9(a)) without changing the final required value of v_c, thus eliminating this interference.

11.7.1.6 Multislope

At time T₁ (above) the maximum dielectric absorption in C will coincide with v_c maximum; this effect degrades (a) the linearity of the run-down slope during T₂ − T₁ and (b) the identification of zero p.d. One improved ‘multislope’ technique reduces dielectric absorption in C by inserting various reference voltages during the run-up period and, after subtraction, leaves a lower p.d., v′_c, prior to run-down while storing the most significant digit of V_x during the run-up period. During run-down, C is discharged rapidly to measure the next smaller digit; the residue of v′_c, including overshoot, is assessed to give the remaining three digits in sequence using three v_r/t functions of slope +1/10, −1/100, + 1/1000 compared with the initial rapid discharge. Measurement time is reduced, owing to the successive digits being accumulated during the measurement process.⁶

11.7.1.7 Mixed techniques

Several techniques can be combined in the one instrument in order to exploit to advantage the best features of each. One accurate, precise, digital voltmeter is based upon precision inductive potentiometers, successive approximation, and the dual-slope technique for the least significant figures. An uncertainty of 10 parts in 10⁶ for a 3-month period is claimed, with short-term stability and a precision of about 2 parts in 10⁶.

11.7.1.8 Digital multimeters

Any digital voltmeter can be scaled to read d.c. or a.c. voltage, current, immittance or any other physical property, provided that an appropriate transducer is inserted. Instruments scaled for alternating voltage and current normally incorporate one of the a.c./d.c. converter units listed in a previous paragraph, and the quality of the result is limited by the characteristics inherent in such converters. The digital part of the measurement is more accurate and precise than the analogue counterpart, but may be more expensive.

For systems application, programmed signals can be inserted into, and binary-coded or analogue measurements received from, the instrument through multiway socket connections, enabling the instrument to form an active element in a control system (e.g. IEC bus, IEEE-488 bus).

Resistance, capacitance and inductance measurements depend to some extent on the adaptability of the basic voltage-measuring process. The dual-slope technique can be easily adapted for two-, three- or four-terminal ratio measurements of resistance by using the positive and negative ramps in sequence; with other techniques separate impedance units are necessary. (See Table 11.3)

11.7.1.9 Input and dynamic impedance

The high precision and small uncertainty of digital voltmeters make it essential that they have a high input impedance if these qualities are to be exploited. Low test voltages are often associated with source impedances of several hundred kilo-ohms; for example, to measure a voltage with source resistance 50 k Ω to an uncertainty of ±0.005% demands an instrument of input resistance 1 G Ω, and for a practical instrument this must be 10 G Ω if the loading error is limited to one-tenth of the total uncertainty.

The dynamic impedance will vary considerably during the measuring period, and it will always be lower than the quoted null, passive, input impedance. These changes in dynamic impedance are coincident with voltage ‘spikes’ which appear at the terminals owing to normal logic functions; this noise can adversely affect components connected to the terminals, e.g. Weston standard cells.

Input resistances of the order of 1–10 G Ω represent the conventional range of good-quality insulators. To these must be added the stray parallel reactance paths through unwanted capacitive coupling to various conducting and earth planes, frames, chassis, common rails, etc.

11.7.1.10 Noise limitation

The information signal exists as the p.d. between the two input leads; but each lead can have unique voltage and impedance conditions superimposed on it with respect to the basic reference or ground potential of the system, as well as another and different set of values with respect to a local earth reference plane.

An elementary electronic instrumentation system will have at least one ground potential and several earth connections–-possibly through the (earthed) neutral of the main supply, the signal source, a read-out recorder or a cathode ray oscilloscope. Most true earth connections are at different electrical potentials with respect to each other, owing to circulation of currents (d.c. to u.h.f.) from other apparatus, through a finite earth resistance path. When multiple earth connections are made to various parts of a high-gain amplifier system, it is possible that a significant frequency spectrum of these signals will be introduced as electrical noise. It is this interference which has to be rejected by the input networks of the instrumentation, quite apart from the concomitant removal of any electrostatic/electromagnetic noise introduced by direct coupling into the signal paths. The total contamination voltage can be many times larger (say 100) than the useful information voltage level.

Electrostatic interference in input cables can be greatly reduced by ‘screened’ cables (which may be 80% effective as screens), and electromagnetic effects minimised by transposition of the input wires and reduction in the ‘aerial loop’ area of the various conductors. Any residual effects, together with the introduction of ‘ground and earth-loop’ currents into the system, are collectively referred to as series and/or common-mode signals.

11.7.1.11 Series and common-mode signals

Series-mode (normal) interference signals, V_sm, occur in series with the required information signal. Common-mode interference signals, V_cm, are present in both input leads with respect to the reference potential plane: the required information signal is the difference voltage between these leads. The results are expressed as rejection ratio (in decibels) with respect to the effective input error signal, V_e, that the interference signals produce, i.e.

where K is the rejection ratio. The series-mode rejection networks are within the amplifier, so V_c is measured with zero normal input signal as V_e = output interference voltage ÷ gain of the amplifier appropriate to the bandwidth of V_e.

Consider the elementary case in Figure 11.10, where the input-lead resistances are unequal, as would occur with a transducer input. Let r be the difference resistance, C the cable capacitance, with common-mode signal and error voltages V_cm and V_e, respectively. Then the common-mode numerical ratio (c.m.r.) is

assuming the cable insulation to be ideal, and X_C r. Clearly, for a common-mode statement to be complete, it must have a stated frequency range and include the resistive imbalance of the source. (It is often assumed in c.m.r. statements that r = 1 k Ω.)

The c.m.r. for a digital voltmeter could be typically 140dB (corresponding to a ratio of 10⁷/1) at 50 Hz with a 1 k Ω line imbalance, and leading consequently to C = 0.3 pF. As the normal input cable capacitance is of the order of 100 pF/m, the situation is not feasible. The solution is to inhibit the return path of the current i by the introduction of a guard network. Typical guard and shield parameters are shown in Figure 11.11 for a six-figure digital display on a voltmeter with ± 0.005% uncertainty. Consider the magnitude of the common-mode error signal due to a 5 V, 50 Hz common-mode voltage between the shield earth E₁ and the signal earth E₂:

11.7.1.12 Floating-voltage measurement

If the d.c. voltage difference to be measured has a p.d. to E₂ of 100 V, as shown, then with N–G open the change in p.d. across r will be 50 μV, as a series-mode error of 0.005% for a 1 V measurement. With N–G connected the change will be 1 μV, which is negligible.

The interconnection of electronic apparatus must be carefully made to avoid systematic measurement errors (and short circuits) arising from incorrect screen, ground or earth potentials.

In general, it is preferable, wherever possible, to use a single common reference node, which should be at zero signal reference potential to avoid leakage current through r. Indiscriminate interconnection of the shields and screens of adjacent components can increase noise currents by short-circuiting the high-impedance stray path between the screens.

11.7.1.13 Instrument selection

A precise seven-digit voltmeter, when used for a 10 V measurement, has a discrimination of ±1 part in 10⁶ (i.e. ±10μV), but has an uncertainty (‘accuracy’) of about ±10 parts in 10⁶ (i.e. ±100 μV). The distinction is important with digital read-out devices, lest a higher quality be accorded to the number indicated than is in fact justified. The quality of any reading must be based upon the time stability of the total instrument since it was last calibrated against external standards, and the cumulative evidence of previous calibrations of a like kind.

Selection of a digital voltmeter from the list of types given in Table 11.3 is based on the following considerations:

11.7.1.14 Calibration

It will be seen from Table 11.3 that digital voltmeters should be recalibrated at intervals between 3 and 12 months. Built-in self-checking facilities are normally provided to confirm satisfactory operational behaviour, but the ‘accuracy’ check cannot be better than that of the included standard cell or Zener diode reference voltage and will apply only to one range. If the user has available some low-noise stable voltage supplies (preferably batteries) and some high-resistance helical voltage dividers, it is easy to check logic sequences, resolution, and the approximate ratio between ranges. In order to demonstrate traceability to national standards, a voltage source whose voltage is traceable to national standards should be used. This can either be done by the user or, more commonly, by sending the instrument to an accredited calibration laboratory.

11.7.5.1 Network analysers

Used in the design and synthesis of complex cascaded networks and systems. The instrument normally contains a swept-frequency sinusoidal source which energises the test system and, using suitable connectors or transducers to the input and output ports of the system, determines the frequency-domain characteristic in ‘lumped’ parameters and the transfer function equations in both magnitude and phase: high-frequency analysers often characterise networks in ‘distributed’ s-parameters for measurements of insertion loss, complex reflection coefficients, group delay, etc. Three or four separate instruments may be required for tests up to 40 GHz. Each instrument would have a wide dynamic range (e.g. 100 dB) for frequency and amplitude, coupled with high resolution (e.g. 0.1 Hz and 0.01 dB) and good accuracy, often derived from comparative readings or ratios made from a built-in reference measurement channel.

Such instruments have wide applications in research and development laboratories, quality control and production testing–-and they are particularly versatile as an automatic measurement testing facility, particularly when provided with built-in programmable information and a statistical storage capability for assessment of the product. Data display is by X – Y recorder and/or cathode-ray oscilloscope (c.r.o.) and the time-base is locked to the swept frequency.

11.7.5.2 Spectrum analysers

Spectrum analysers normally do not act as sources; they employ a swept-frequency technique which could include a very selective, narrow-bandwidth filter with intermediate frequency (i.f.) signals compared, in sequence, with those selected from the swept frequency; hence, close control of frequency stability is needed to permit a few per cent resolution within, say, a 10 Hz bandwidth filter. Instruments can be narrow range, 0.1 Hz–30 kHz, up to wide range, 1 MHz–40 GHz, according to the precision required in the results; typically, an 80 dB dynamic range can have a few hertz bandwidth and 0.5 dB amplitude accuracy. Phase information is not directly available from this technique. The instrument is useful for tests involving the output signals from networks used for frequency mixing, amplitude and frequency modulation or pulsed power generation. A steady c.r.o. display is obtained from the sequential measurements owing to digital storage of the separate results when coupled with a conventional variable-persistence cathode-ray tube (a special cathode-ray tube is not required for a digital oscilloscope): the digitised results can, in some instruments, be processed by internal programs for (a) time averaging of the input parameters, (b) probability density and cumulative distribution, and (c) a comprehensive range of statistical functions such as average power or r.m.s. spectrum, coherence, autocorrelation and cross-correlation functions, etc.

11.7.5.3 Fourier analysers

Fourier analysers use digital signalling techniques to provide facilities similar to the spectrum analysers but with more flexibility. These ‘Fourier’ techniques are based upon the calculation of the discrete Fourier transform using the fast Fourier transform algorithm (which calculates the magnitude and phase of each frequency component using a group of time-domain signals from the input signal variation of the sampling rate); it enables the long measurement time needed for very-low-frequency ( 1 Hz) assessments to be completed in a shorter time than that for a conventional swept measurement—together with good resolution (by digital translation) at high frequencies. Such specialised instruments are usable only up to 100 kHz but they are particularly suitable for examining low-frequency phenomena such as vibration, noise transmission through media and the precise measurement of random signals obscured by noise, etc.

11.8.1.1 Electron gun and beam deflection

The section of the tube from which the beam emerges includes the heater, cathode, grid and the first accelerating anode; these collectively form the electron gun. A simplified diagram of the connections is given in Figure 11.13. The electric field of the relatively positive anode partly penetrates through the aperture of the grid electrode and determines on the cathode a disc-shaped area (bounded by the −15 kV equipotential) within which electrons are drawn away from the cathode: outside this area they are driven back. With a sufficiently negative grid potential the area shrinks to a point and the beam current is zero. With rising grid potential, both the emitting area and the current density within it grow rapidly.

The diagram also shows the electron paths that originate at the emitting area of the cathode. It can be seen that those which leave the cathode at right angles cross the axis at a point not far from the cathode. This is a consequence of the powerful lens action of the strongly curved equipotential surfaces in the grid aperture. But, as electrons are emitted with random thermal velocities in all directions to the normal, the ‘cross-over’ is not a point but a small patch, the image of which appears on the screen when the spot is adjusted to maximum sharpness.

The conically shaped beam of divergent electrons from the electron gun have been accelerated as they rise up the steep potential gradient. The electrical potentials of the remaining nickel anodes, together with the post-deflection accelerating anode formed by the graphite coating, are selected to provide additional acceleration and, by field shaping, to refocus the beam to a spot on the screen. To achieve sharply defined traces it is essential that the focused spot should be as small as possible; the area of this image is, in part, dictated by the location and size of the ‘point source’ formed in front of the cathode. The rate at which electrons emerge from the cathode region is most directly controlled by adjustment of the negative potential of the grid; this is effected by the external brightness control. A secondary result of changes in the brightness control may be to modify, slightly, the shape of the point source. The slight de-focusing effect is corrected by the focus control, which adjusts an anode potential in the electron lens assembly.

11.8.1.2 Performance (electrostatic deflection)

If an electron with zero initial velocity rises through a potential difference V, acquiring a speed u, the change in its kinetic energy is , so that the speed is

using accepted values of electron charge, e, and rest-mass, m. The performance is dependent to a considerable extent on the beam velocity, u.

The deflection sensitivity of the tube is increased by shaping the deflection plates so that, at maximum deflection, the beam grazes their surface (Y₁Y₂; Figure 11.12). The sensitivity and performance limits may be assessed from the idealised diagrams (and notation) in Figure 11.14. Experience shows that a good cathode-ray tube produces a beam-current density J_s at the centre of the luminous spot in rough agreement with that obtained on theoretical grounds by Langmuir, i.e.

(11.1)

where J_c is the current density at the cathode, V is the total accelerating voltage, T is the absolute temperature of the cathode emitting surface, θ is one-half of the beam convergence angle, and k is a constant. If i_b is the beam current, then the apparent diameter of the spot is given approximately by .

The voltage V_d applied to the deflection plates produces an angular deflection δ given by

(11.2)

Deflection is possible, however, only up to a maximum angle α, such that

(11.3)

From these three expressions, certain quantities fundamental to the rating of a cathode-ray tube with electric field deflection can be derived.

11.8.1.3 Definition

This is the maximum sweep of the spot on the screen, D = 2αL, divided by the spot diameter, i.e. N = D/s. In television N is prescribed by the number of lines in the picture transmission. In oscillography N is between 200 and 300. Better definition can be obtained only at the expense of deflection sensitivity.

11.8.1.4 Specific deflection sensitivity

This is the inverse of the voltage that deflects the spot by a distance equal to its own diameter, i.e. the voltage which produces the smallest perceptible detail. The corresponding deflection angle is δ₀ = s/L. Use of equations (11.1)–(11.3) gives for the specific deflection sensitivity

(11.4)

where k₁ is a constant. Thus, the larger the tube dimensions the better the sensitivity at the smallest beam power consistent with adequate brightness of trace.

11.8.1.5 Maximum recording speed

The trace remains visible so long as its brightness exceeds a certain minimum. The brightness is proportional to the beam power density VJ, to the luminous efficiency σ of the screen, and to the time s/U during which the beam sweeps at speed U over one element of the screen. Hence,

(11.5)

Recording speeds up to 40 000km/s have been achieved with sealed-off tubes, permitting single-stroke photography of the trace.

11.8.1.6 Maximum specified recording speed

Unless compared with the spot size s, the value of U does not itself give adequate information. The specific speed U_s= U/s is more useful: it is the inverse measure of the shortest time-interval in which detail can be recorded. Combining equations (11.5) and (11.1)

(11.6)

The product ζ(Vθ)² is the essential figure in the speed rating of a cathode-ray tube.

11.8.1.7 Limiting frequency

A high recording speed is useful only if the test phenomena are recorded faithfully. The cathode-ray tube fails in this respect at frequencies such that the time of passage of an electron through the deflection field is more than about one-quarter of the period of an alternating quantity. This gives the condition of maximum frequency f_m:

(11.7)

which means that the smallest tube at the highest acceleration voltage has the highest frequency limit. This conflicts with the condition for deflection sensitivity. Combining the two expressions gives the product of the two:

(11.8)

Compromise must be made according to the specific requirement. It may be observed that the optimum current i_b is not zero as might appear, because for such a condition the convergence angle θ is also zero. In general, the most advantageous current will not differ from that for which the electron beam just fills the aperture.

Limitation of cathode-ray-tube performance by space charge is an effect important only at low accelerating voltages and relatively large beam currents. For these conditions the expressions above do not give practical ratings.

11.8.1.8 Fluorescent screen

The kinetic energy of each incident electron is converted into light owing to the exchange of energy which occurs within the phosphors coated on the inside of the screen. The result, luminescence, is usually described as either fluorescence or phosphorescence. Fluorescence is the almost instantaneous energy conversion into light, whereas phosphorescence is restricted to the phosphors which ‘store’ the converted energy and release it, as light, after a short delay. The afterglow can last many seconds. The respective terms are applied to the phosphors in accordance with their dominant light-emitting behaviour; however, each property is always present in a phosphor. In most cathode-ray tubes fluorescent phosphors are used with a very short afterglow (40 μs); a longer afterglow would blur a quickly changing display.

The most common fluorescent substances are silicates and sulphides, with calcium tungstate for photographic work. An excellent zinc silicate occurs naturally in the mineral willemite, but all other substances are prepared artificially. After purification, the substances are activated by the addition of very small quantities of suitable metals, and then ground to a powder of grain size 5–10 μm (for sulphides 10–50 μm).

In applying the powder to the glass several methods are used. Very finely ground silicates can be applied without a binder, by setting the powder from water, or an electrolyte. Sulphides are usually compounded with the glass wall by means of a binder such as sodium silicate; good results have been obtained also by baking them directly into the glass. In choosing the method two conflicting considerations must be taken into account. Baking or compounding gives good thermal contact, and prevents overheating of the powder particles by the electron beam. But at the same time good optical contact is established with the glass, with the result that light from the powder enters the glass also at angles exceeding the total reflection angle. This part of the light cannot escape at the viewing side, but is reflected and forms a luminous ‘halo’ which blurs the picture.

While silicates fluoresce usually green or blue-green, white fluorescence can be obtained by mixing cadmium sulphide and zinc cadmium sulphide in suitable proportions. The luminous efficiency of modern fluorescent powders is very high. Zinc silicate emits up to 2.5cd/W of beam power, and some sulphides are even more effective. The luminous efficacy increases with voltage up to about 10 kV, beyond which it decreases, partly because fast electrons are not stopped by the crystal grains.

The range of phosphors is classified by ‘P’ numbers, as in Table 11.4.

The light persistence of the most widely used phosphor (P31) is 40 μs; however, there are tubes available, using this phosphor, which have both variable persistence and ‘mesh storage’ facilities. Adjustable persistence enables very-low-frequency signals, lasting for a minute, to be viewed as a complete trace while retaining the normal ‘writing’ properties of the tube. Storage, at reduced brightness, is possible for up to 8 h.

Phosphor materials are insulators. The current circuit for all the incident beam electrons is completed by an equal number of low-energy electrons being attracted along various paths through the vacuum to the positive conducting-graphite coating. The result of secondary emission of electrons is to leave the screen with a positive charge only a little lower in potential than that of the graphite coating; thus, the surplus secondary emission electrons return to the phosphor-coated screen surface. This mechanism works perfectly at the voltages usually employed in oscillography and television (3–20 kV), also in small projection-tube sealed-off high-speed oscillographs (10–25 kV). Above 25 kV the mechanism is unsatisfactory and the dispersion of charge has to be supplemented by other means.

The power concentration in the luminous spot of a cathode-ray tube is very high. In direct-vision television tubes the momentary luminance may exceed 10 cd/mm². A 3 mA 80 kV beam of a large-screen projection tube may be concentrated in an area of 0.1 mm², representing a power density of 2kW/mm². (By comparison, melting tungsten radiates about 10 W/mm².) The width of the trace in a conventional 10 kV cathode-ray tube with a beam current of 10 μA is only 0.35 mm.

11.8.2.1 Ramp voltage (saw-tooth) generator

If a constant current I is fed to a capacitor C, its terminal voltage increases linearly with time. An amplified version of the voltage is applied to the X plates. Various sweep times are obtained by selecting different capacitors or other constant current values.

The method shown in Figure 11.15 employs a d.c. amplifier of high gain −G with a very large negative feedback coupled through an ideal capacitor C_f. The amplifier behaves in a manner dictated by the feedback signal to node B rather than by the small direct input voltage E_s. The resultant signal vi is small enough for node B to be considered as virtually at earth potential. Analysing the network based on the ‘virtual earth’ principle, then v₁ = 0 and i_i = 0, the input resistance R_i being typically 1 MΩ. Then i_s + i_f = 0, whence

The output voltage v_o increases linearly with time to provide the ramp function input to the X amplifier.

11.8.2.2 Variable delay

When two similar X amplifiers, A and B, are provided, B can be triggered by A; if B is set for, say, five times the frequency of A and this had a variable delay facility, then any one-fifth of the complete A waveform display can be extended to fill the screen. Both complete and expanded waveforms can be viewed simultaneously by automatic switching of A and B to the X plates during alternate sweeps.

11.8.2.3 Signal amplifiers

The X- and Y-plate voltages for maximum deflection in a modern 10 cm cathode-ray tube may lie between 30 and 200 V. As many voltage phenomena of interest have amplitudes ranging between 500 V and a few millivolts, both impedance matched attenuators and voltage amplifiers are called for. The amplifiers must be linear and should not introduce phase distortion in the working frequency range; the pulse response must be fast (e.g. 2 ns); and the drift and noise must be small (e.g. 3 μV).

Amplifiers have d.c. and a.c. inputs. The d.c. coupling is conventional, to minimise the use of capacitors (which tend to produce low-frequency phase distortion). The d.c. inputs also permit the use of a reference line. The a.c. input switch introduces a decoupling capacitor.

Two separate Y amplifiers are often provided; in conjunction either with one switched or with two separate X amplifiers, they permit the examination of two voltages simultaneously. It is important to distinguish the dual-trace oscilloscope (in which a single beam is shared by separate signal amplifiers) and the true double-beam device (with a gun producing two distinct beams).

11.8.3.1 Other features

With all c.r.o.s, the X–Y cartesian coordinate display can be augmented by Z-modulation. This superimposes a beam-intensity control on the X–Y signals to produce the illusion of three dimensions. The beam control may brighten one spot only, or gradually ‘shape’ the picture.

Oscilloscopes are supplemented by such accessories as special probes for matching and/or attenuating the signal to the signal/cable interface. Anti-microphony cables are available which minimise cable-generated noise and capacitance change. Many types of high-speed camera can be used, including single-shot cameras that can employ ultraviolet techniques to ‘brighten’ a record from a trace of low visual intensity.

11.8.4.1 Impedance matching

Voltage signals are often presented to a c.r.o. through wires or coaxial cables terminated on crocodile clips. Although this may be adequate at low frequencies, it can behave like a badly matched transmission line, causing attenuation distortion and phase distortion. The earth screen of a coaxial cable is not fully effective, and in consequence the cable can pick up local radiated interference. In a normal cable the central conductor is not bonded to the insulation, and flexure of the cable may distort the signal waveform because of changes in the capacitance distribution, or generated noise. The cable capacitance of about 50 pF/m is additive to the c.r.o. input capacitance, and will distort the signal, particularly if the frequency and source impedance are high.

These adverse effects can be mitigated by use of fully screened probes and cables of low and constant capacitance associated with low loss. The probes may be active (including an impedance matching unity-gain amplifier) or passive (including resistance with adjustable capacitance compensation); attenuators are incorporated in each to reduce both the signal level and the capacitive loading of the source, the compensation being most effective for high-impedance sources. The active probe can be used to match the source to the standard 50Ω input impedance of a high-frequency c.r.o. (100 MHz and above), an important matter in that a quarter wavelength at 100 MHz is about 75 cm, and severe standing-wave effects may otherwise occur to invalidate the measurements.

11.8.4.2 Earths and grounds

Ground planes (e.g. extended screens) and earth planes (e.g. trunking) constitute low-impedance paths through which interfering signals can circulate, of a wide variety of frequencies and waveforms. The arbitrary connection of screens to earth and ground terminals can have the unwelcome result of increasing noise, due to earth-loop currents. The latter must not enter the input signal lines of either the system or the c.r.o.; and the interference can be minimised by using a single ground/earth connection to serve the whole equipment, if this is possible. The common connection should be made at the zero-potential input signal lead.

(1) they allow long-term display of a transient quantity;

(2) it is easy to produce a hard copy of the display;

(3) they allow computation and signal processing within the d.s.o.;

(4) they allow easy transfer of data to a computer; and

(5) they use cheap cathode-ray-tube construction because the deflection is not at signal frequency but much lower than the maximum signal frequency.

11.8.8.1 Sampling rate

Clearly the sampling rate must be high enough to give a faithful representation of the applied signal. Nyquist’s theorem states that a periodic signal must be sampled at more than twice the highest frequency component of the signal. In practice, because of the finite time available, a sample rate somewhat higher than this is necessary. A sample rate of 4 per cycle at oscilloscope bandwidth would be typical. Therefore, for single shot, a digital oscilloscope specification will give a bandwidth of 100 MHz or a sample rate of 400 megasamples/s. If the display is a dot display, then something like 20 samples per cycle is necessary if the eye is to interpret the display easily. Usually some form of interpolation is provided to aid interpretation. The simplest method is to draw straight lines between successive sample parts. Other interpolation methods are used such as sine interpolation, designed for fairly pure sine waves. This interpolation will give erroneous displays for non-sinusoidal inputs.

All this means that great care must be taken in interpreting a display when the input approaches the oscilloscope bandwidth, particularly if it has significant high frequency components such as a square wave. This is also true of course for an analogue c.r.o.

The d.s.o. in repetitive mode can build up the display over a number of sections of the displayed waveform, and so the sampling-rate problems are not so severe. Specifications will give bandwidth, single-shot bandwidth and sampling rate. If single-shot mode is to be used, then care must be taken to look for the adequacy of the single-shot bandwidth or the sampling rate.

For an analogue c.r.o., a rule of thumb relates the rise time of the c.r.o. to its bandwidth rise time (s) × bandwidth (Hz) = 0.35. So, a 100 MHz oscilloscope has a rise time of 3.5 ns. A signal rise time approaching this cannot be faithfully displayed, as the fastest displayed rise time is that of the instrument itself. Usually the fastest rise time display-able is 8 times the c.r.o. rise time. So the fastest signal rise time = 2.8 ÷ bandwidth. For a 100 MHz c.r.o. this would be 28 ns.

A similar restriction occurs for the d.s.o. A fast signal pulse on an expanded time scale is shown in Figure 11.20. The worst case giving the slowest rise time is when one of the samples is in the middle of the edge. The time from 10% to 90% of the rise (the rise time) is given by (80/100) × 2 sample periods (= 1.6 sample periods).

So for a 100 megasample/s d.s.o., the sampling interval is 10 ns and the displayed edge, regardless of the rise time of the applied signal, is at worst 16 ns. Again, in order to faithfully display the signal, and not the characteristics of the d.s.o., the fastest signal rise time should be about 8 times this limit, say 10 sample intervals. So, as a rule of thumb:

e.g. for 100 megasamples/s d.s.o. the fastest signal rise time is 100 ns. For a signal rise time faster than this it is likely that the performance of the d.s.o. will show itself in the display of the signal.

11.8.8.2 Vertical resolution

Vertical resolution is another important parameter of the d.s.o. An important contribution to this is the number of bits of the A/D converter. The more bits, the greater the resolution. The minimum discernible level is

where n is the number of bits of the A/D.

For example, for an input voltage of 1 V and an 8-bit A/D, the minimum discernible level is

This value is sometimes given as a percentage and in this case would be ±0.2%.

So on a 1-V signal, noise of less than 4 mV will not be displayed. Increasing the number of bits in the A/D converter will help, but internally generated noise limits the advantage achieved. This leads to the concept of effective bits of the d.s.o., which is performance of a hypothetical idealised A/D converter which gives the same vertical resolution of the actual A/D converter with its noise limitation.