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Passive Components for Active Crossovers  439

A closer approach to the desired value while keeping the values near equal is possible by using combinations of three resistors rather than two. The cost is still low because resistors are cheap, but seeking out the best combination is an unwieldy process. By the time this book appears I hope to have a Javascript solution on my website. [6]

All of the statistical features described here apply to capacitors as well but are harder to apply because capacitors come in sparse value series, and it is also much more expensive to use multiple parts to obtain an arbitrary value. It is worth going to a good deal of trouble to come up with a circuit design that uses only standard capacitor values; the resistors will then almost certainly be all non-standard values, but this is easier and cheaper to deal with. A good example of the use of multiple parallel capacitors, both to improve accuracy and to make up values larger than those available, is the Signal

Transfer RIAApreamplifier; [7] [8] see also the end of this book. This design is noted for giving very accurate RIAAequalisation at a reasonable cost. There are two capacitances required in the RIAA network, one being made up of four polystyrene capacitors in parallel and the other of five polystyrene capacitors in parallel. This also allows non-standard capacitance values to be used.

Other Resistor Combinations

So far we have looked at serial and parallel combinations of components to make up one value,

as in Figure 15.3a and 15.3b. Other important combinations are the resistive divider at Figure 15.5c, which is frequently used as the negative feedback network for non-inverting amplifiers.Atwooutput divider is shown at Figure 15.3d and a three-output divider at Figure 15.3e.Another important configuration is the inverting amplifier at Figure 15.5f, where the gain is set by the ratio R2/R1.All resistors are assumed to have the same tolerance about an exact mean value.

I suggest it is not obvious whether the divider ratio of Figure 15.5c, which is R2/(R1 + R2), will be more or less accurate than the resistor tolerance, even in the simple case with R1 = R2 giving a divider

Figure 15.3: Resistor combinations: (a) series; (b) parallel; (c) one-tap divider;

(d) two-tap divider; (e) three-tap divider; (f ) inverting amplifier.

440  Passive Components for Active Crossovers

ratio of 0.5. However, the Monte Carlo method shows that in this case partial cancellation of errors still occurs, and the division ratio is more accurate by a factor of √2.

This factor depends on the divider ratio, as a simple physical argument shows:

If the top resistor R1 is zero, then the divider ratio is obviously one with complete accuracy, the bottom resistor value is irrelevant, and the output voltage tolerance is zero.

If the bottom resistor R2 is zero, there is no output and accuracy is meaningless, but if instead R2 is very small compared with R1, then R1 completely determines the current through R2, and R2 turns this into the output voltage. Therefore the tolerances of R1 and R2 act independently, and so the combined output voltage tolerance is worse by a factor of √2.

Some more Monte Carlo work, with 8000 trials per data point, revealed that there is a linear relationship between accuracy and the “tap position” of the output between R1 and R2, as shown in Figure 15.4. Plotting against division ratio would not give a straight line. With R1 = R2 the tap is at 50%, and accuracy improved by a factor of √2, as noted earlier. With a tap at about 30% (R1 = 7 kΩ,

Figure 15.4: The accuracy of the output of a resistive divider made up with components of the same tolerance varies with the divider ratio.

Passive Components for Active Crossovers  441

R2 = 3 kΩ) the accuracy is the same as the resistors used. This assessment is not applicable to potentiometers, as the two sections of the pot are not uncorrelated in value—they are highly correlated.

The two-tap divider (Figure 15.3d) and three-tap divider(Figure 15.3e) were also given a Monte Carlo work-out, though only for equal resistors. The two-tap divider has an accuracy factor of 0.404 at OUT

1 and 0.809 at OUT 2. These numbers are very close to √2/(2√3) and √2/(√3) respectively. The threetap divider has an accuracy factor of 0.289 at OUT 1, of 0.500 at OUT 2, and of 0.864 at OUT 3. The middle figure is clearly 1/2 (twice as many resistors as a one-tap divider, so √2 times more accurate), while the first and last numbers are very close to √3/6 and √3/2 respectively. It would be helpful if someone could prove analytically that the factors proposed are actually correct.

For the inverting amplifier of Figure 15.3f, the values of R1 and R2 are completely uncorrelated, and so the accuracy of the gain is always √2 worse than the tolerance of each resistor, assuming the tolerances are equal. The nominal resistor values have no effect on this. We therefore have the interesting situation that a non-inverting amplifier will always be equally or more accurate in its gain than an inverting amplifier. So far as I know this is a new result. Maybe it will be useful.

Resistor Noise: Johnson and Excess Noise

All resistors, no matter what their method of construction, generate Johnson noise. This is white noise, which has equal power in equal absolute bandwidth, i.e. with the bandwidth measured in Hz, not octaves. There is the same noise power between 100 and 200 Hz as there is between 1100 and 1200

Hz. The level of Johnson noise that a resistor generates is determined solely by its resistance value, the absolute temperature (in degrees Kelvin), and the bandwidth over which the noise is being measured. For our purposes the temperature is 25 °C and the bandwidth is 22 kHz, so the resistance is really the only variable. The level of Johnson noise is based on fundamental physics and is not subject to modification, negotiation, or any sort of rule-bending . . . however, you might want to look up “load synthesis”, in which a high-value resistor is made to act like a low-value resistor but still having the low Johnson current noise of the high-value resistor. [9] Sometimes Johnson noise from resistors places the limit on how quiet a circuit can be, though more often the noise from the active devices is dominant. It is a constant refrain in this book that resistor values should be kept as low as possible, without introducing distortion by overloading the circuitry, in order to minimise the Johnson noise contribution and the effects of opamp current noise.

The rms amplitude of Johnson noise is calculated from the classic equation:

 

vn = 4kTRB

15.4

Where:

 

 

vn is the rms noise voltage

T is absolute temperature in °K

B is the bandwidth in Hz

k is Boltzmann’s constant

R is the resistance in Ohms

 

The thing to be careful with here is to use Boltzmann’s constant (1.380662 × 10−23), and NOT the Stefan-Boltzmann constant (5.67 10−08), which relates to black-body radiation, has nothing to do with

442  Passive Components for Active Crossovers

resistors, and will give some impressively wrong answers. The voltage noise is often left in its squared form for ease of RMS-summing with other noise sources.

The noise voltage is inseparable from the resistance, so the equivalent circuit is of a voltage source in series with the resistance present. Johnson noise is usually represented as a voltage, but it can also be treated as a Johnson noise current, by means of the Thevenin-Norton transformation, [10] which gives the alternative equivalent circuit of a current source in shunt with the resistance. The equation for the noise current is simply the Johnson voltage divided by the value of the resistor it comes from:

in = vn /R.

15.5

Excess resistor noise refers to the fact that some resistors, with a constant voltage drop across them, generate extra noise in addition to their inherent Johnson noise. This is a very variable quantity, but is essentially proportional to the DC voltage across the component; the specification is therefore in the form of a “noise index” such as “1 uV/V”. The uV/V parameter increases with increasing resistor value and decreases with increasing resistor size or power dissipation capacity. Excess noise has a 1/f frequency distribution. It is usually only of interest if you are using carbon or thick-film resistors— metal film and wirewound types should have little or no excess noise.Arough guide to the likely range for excess noise specs is given in Table 15.13

One of the great benefits of opamp circuitry is that it allows simple dual-rail operation, and so it is noticeably free of resistors with large DC voltages across them; the offset voltages and bias currents involved are much too low to cause measurable excess noise. If you are designing an active crossover on this basis, then you can probably forget about the issue. If, however, you are using discrete transistor circuitry, it might possibly arise; specifying metal film resistors throughout, as you no doubt would anyway, should ensure you have no problems.

To get a feel for the magnitude of excess resistor noise, consider a 100 kΩ 1/4 W carbon film resistor with a steady 10 V across it. The manufacturer’s data gives a noise parameter of about 0.7 uV/V, and so the excess noise will be of the order of 7 uV, which is −101 dBu. That could definitely be a problem in a low-noise preamplifier stage.

Table 15.13: Resistor excess noise.

Type

Noise Index uV/V

 

 

Metal film TH

0

Carbon film TH

0.2–3

Metal oxide TH

0.1–1

Thin film SM

0.05–0.4

Bulk metal foil TH

0.01

Wirewound TH

0

 

 

(Wirewound resistors are normally considered to be completely free of excess noise.)