RESISTOR.HTM --- Part of Manual for Driver Parameter Calculator --- by Claus Futtrup.
Created 19. October 1997, last revised 23. August 2003. Ported to XHTML 1.0 on 2. October 2004. Last modified 25. October 2004.

Using the compare resistor method to calculate complex impedance, you will need a good, low-inductive power resistor, preferably around 10 ohm if you intend to measure drivers with nominal impedance in the 4-8 ohm range. The resistor may be of high tolerance, but then you must measure it yourself, for higher accuracy (preferably less than 1%). A good quality 0.1% resistor which is non-inductive and temperature-constant can usually be had within a reasonable price range.

When using the compare resistor method, most of the variation from the signal generator etc. is applied to both the resistor and the speaker driver, so these variations are virtually removed by this method, making it quite accurate. For this reason it is also obvious that any inaccuracy in the resistor value will affect the precision of your measurements. Here follows a discussion on measuring your resistor for higher accuracy, and various equipment.

From Dansk Audio Teknik (DAT) I bought a 15 W, 10 ohm non-inductive power-resistor (from Danotherm), class J. Counting letters I realize that J means the precision is only 10%, with a value between 9 and 11 ohm a better knowledge was required. The resistor is of the type which consist of an aluminium case, and supposed to be mounted on a heat sink.

The resistor is a non-inductive resistor because the wires are wound in a bifilar manner, which means the wires are wound as two opposite twisting coils, which cancels out most of the inductance found in low-quality wire-wound power-resistors. Most power-resistors are made this way to be resistive in a wide frequency range.

Preferably your resistor is such a non-inductive type, but alternatively you can use 0.5 W metalfilm (they're inductive, but not much) or carbonfilm (they are non-inductive but make noise) resistors, you could take like 10 resistors each of 100 ohm and connect them in parallel to get a 10 ohm, 5 W power-resistor.

The question arises, how many Watt is a measuring resistor supposed to withstand. First I recommend that you do not dimension it too small. Second, I recommend that you measure with a decent current, say an average of 50-100 mA. With a 10 ohm resistor this gives you up to 1 W power which the resistor must dissipate. A 2 W resistor would be minimum, I think. More is better, especially if you intend to experiment with higher power.

Some examination that I have made of power-resistors has shown, that the aluminium case resistor type is not particular non-inductive and not at all temperature-independent without a heat sink. Instead the usual white ceramic type is very linear and temperature-independent. I can highly recommend this standard type. Bigger is better, but they might be slightly more inductive, ie. frequency dependent, which is bad of course. Watch out. Try to measure the inductance of your resistor.

I have personally measured some MODULOHM resistors, which shows no inductance or capacitance tendency within the 20k Hz range of interest. They have a slightly negative temperature coefficient, but are more linear than other power resistors, and I consider them to be a very reliable brand.

### Various measurements

The following survey is provided for inspiration.

At Dansk Audio Teknik/DAT they measured the resistor to 10.082 ohm with a good-quality LCR-meter.

At Aalborg University the resistor was measured to 9.974 ohm using a Philips PM 6303 RCL meter with approx 0.25% precision. The instrument had recently been calibrated. This instrument can show when there is a capacitance or inductance, it has two wires on each side of the resistor. I waited until the display was stable, and everything was freely suspended in air (no contact to the resistor house). I consider this measurement to be extremely accurate.

At home I have a Lutron DM9090 digital multimeter, a handheld with a 200 ohm range I measured 10.1 ohm. Checking for resistance in the wires I got 0.3 ohm, but an instrument of this quality might be calibrated with lead wires included so this is not taken into account. I assume this instrument is measuring 10.1 ohm.

A friend has a dirt-cheap (10 bucks handheld) multimeter, and in the 200 ohm range it measured the resistor to 10.8 ohm, and with short circuited wires 0.8 ohm. Since such a cheap instrument is (probably) not calibrated, I will assume the measurement to be 10.0 ohm. This precision (when compared to the most expensive equipment above) is a coincidense---you cannot rely on cheap instruments for anything but an approximate value unless supported by other instruments of different type.

At Aalborg University the resistor was measured with a simple multimeter, 2 kohm range, value 9.4 ohm. This is obviously a very inaccurate measurement. The instrument was of average quality for digital multimeters, but with a 2 kohm range as the lowest option this instrument is not intended for measuring low-value resistors.

I also tried to use an expensive Fluke 45 Dual Display Multimeter, a table-instrument, but it showed up not to be calibrated and we got 10.89 ohm. Measuring with short circuited wires we got 0.89. This should not be necessary with such instruments, but I will assume 10.00 ohm.

At a local electronics store they measured the resistor with a Fluke 87 instrument, giving me 10.3 ohm (probably 10.25 according to the guy who did the measuring). This is an expensive instrument and at the store they said the precision was quite high, though measuring in a 200 ohm range. Though this is an expensive instrument, I do not believe them any longer. Most handheld instruments, even the more expensive ones, do not have the proper range to measure 10 ohm resistors, and when pulling current they are too weak. They are simply not meanth to operate in this range.

At the store they also had a second, cheaper handheld instrument, which showed that the resistor should be 9.8 ohm, on a 200 ohm scale. The guy at the store said this instrument was less accurate than the Fluke 87.

Measuring with my earlier measuring-resistor, which I had assumed was around 6.9 ohm (nominal 6.8 ohm) showed up to be inaccurate. I used the compare-resistances measurement method as described above. First applying a 9 Volt battery the results were inaccurate (because the battery was drained). With a 1.5 Volt battery using 6.9 ohm the new resistor showed up to be around 10.6 ohm. Voltage drops were 0.594 and 0.914 using a very accurate HP table-instrument voltmeter with 0.1% precision. Though very old the HP instrument is digital and samples the voltage-drop so that the internal resistance is above 20 Mohm in the entire audio frequency range (I only used DC, where the precision is twice as good as with AC). Perhaps my former and not so well examined measuring resistor is only about 6.6 ohm.

Again measuring with my Lutron DM9090 multimeter, 200 ohm range, I got 10.5 ohm and with short circuited wires 0.5 ohm(?). Perhaps the difference from earlier is because I had the resistor loaded with 9 Volt before measuring, giving 0.9 Ampere (by applying ohms law) the power consumption must have been around 9 * 0.9 = 8 Watt and therefore the resistor was warm. I will use the previously mentioned figure, 10.1 ohm.

At Aalborg University (again) the resistor was measured to 9.962, 9.960 and 9.943 ohm using a Wayne Kerr Automatic LCR Meter 4225, using 100 Hz, 1 kHz and 10 kHz frequency range with 0.25% precision (the precision is not so good at 10 kHz). The instrument had not been calibrated for a long time, but the measurement is considered valid, and much more accurate than what's provided by handheld instruments (even a Hewlett Packard that was floating around). The guy who helped me claimed that the Phillips PM 6303 RCL meter was probably more accurate, apparently this instrument was bugged by some (internal) inductance.

I also asked another place at the University. They had an ABB digital multimeter, named MegaWatt Mega 16 S. This instrument showed 10.25 ohm. The helper wisely told me that the precision of the instrument would only be like 10% for the range (though higher precision than the usual 3.5 number displays), and further said that the resistance as well as the instrument would change with temperature.

### List of results

First we can list the results, without comments:

• DAT, 10.082 ohm
• Aab. Univ. Phillips PM 6303, 9.974 ohm
• Lutron DM9090, 10.1 ohm (200 range)
• Dirt-cheap instrument, 10.0 ohm (200 range)
• Aab. Univ. average instrument, 9.4 ohm (2k range)
• Fluke 45, 10.00 ohm
• Fluke 87, 10.3 ohm (200 range)
• Handheld, 9.8 ohm (200 range)
• Earlier resistor (comparison), 10.6 ohm
• Wayne Kerr 4225, 9.962 ohm (best precision at 100 Hz)
• ABB MegaWatt Mega 16 S, 10.25 ohm

I think assuming the resistor to be exactly 10.00 ohm is very accurate (I was lucky).

### Finding mean and standard deviation

If you do not have super-good instruments available, like some of those above, I would say you could measure with various handhend instruments, then calculate the mean value + the standard deviation (assuming a normal probability distribution).

For this method to be good, you will need enough data, at least 10 measurements are required. It assumes that all instruments are equally good.

You simply calculate the mean value by adding all results togeter, and dividing by the number of data. I would get approx. 10.04 with the above data.

Calculating the standard deviation (the squareroot of the variance) you first calculate the variance by subtracting the mean value from all data, then square each of these values. Add the values together and divide by the number of data MINUS one. In my case I divide by 8 and get approx. 0.1, take the square root and find that the standard deviation is approx. 0.30 ohm. In my case I get that the resistance is 10.04 ohm +- 0.30 ohm.

In my case I gain a good increase in precision from such a calculation, but still has approx. 3% inaccuracy. Instead I must evaluate each instrument and its quality, as I did above, and judge which instruments to trust, and which instruments may not be so accurate.

### Taking instrument precisions into account

First we can list the results, with an assumed precision (both percentages and lower/upper). Given such an interval this method requires that it is very likely that the result is inside the range, something like 95% confidence. This means it is better to assume a bit too large a tolerance instead of specifying too small tolerances.

• 10.082 ohm +- 0.5% (lower=10.032, upper=10.132)
• 9.974 ohm +- 0.5% (lower=9.924, upper=10.024)
• 10.1 ohm +- 5% (lower=9.612, upper=10.605)
• 10.0 ohm +- 10% (lower=9.091, upper=11)
• 9.4 ohm +- 15% (lower=8.174, upper=10.81)
• 10.00 ohm +- 5% (lower=9.524, upper=10.5)
• 10.3 ohm +- 5% (lower=9.810, upper=10.815)
• 9.8 ohm +- 5% (lower=9.333, upper=10.29)
• 10.6 ohm +- 10% (lower=9.636, upper=11.66)
• 9.962 ohm +- 0.5% (lower=9.912, upper=10.012)
• 10.25 ohm +- 5% (lower=9.762, upper=10.763)

I have chosen to give the range-200 ohm handheld instruments 5%, while the expensive LCR-meters get precision/tolerance of only 0.5% (I was told that the Wayne Kerr would be within 0.25%, but that is probably only with 68% confidence). The range-2k ohm get 15% tolerance. The less accurate ones (my earlier measurement resistor, and the dirt-cheap instrument) gets 10% precision.

Working with 95% confidence means that all instruments should (more or less) overlap in range (lower to upper). If they do not, then you must use higher tolerances - or the result will be a rugged curve, not really useful for anything, but perhaps cubistic art.

Locating the highest value in LOWER (10.032) and the lowest value in UPPER (10.012) it's clear that not all instruments fit into a range, probably some instruments have been specified too precise (the LCR-meters). Anyway, something like 10.02 seems to be a good bet. An alternative would be to give the DAT LCR-meter 0.75% precision interval instead, and perhaps also the Wayne Kerr LCR-meter. In that case LOWER = 10.007 and UPPER = 10.024 (now the Phillips LCR-meter is limiting upwards) giving us a valid range with 10.02 (or 10.0155 to be exact) as a middle value.

You can clarify the picture by drawing each instrument as a triangle, where the area is unity (ie. 1, representing one measurement) and with a height = 2A/b, where A = area and b = baseline = upper - lower. This is just a method, which is empirical and I have described it here to make a simple graph and get a broad view on the collected data.

Plot these triangles (an approx. of a gaussian probability curve), and add the curves together. The summary-curve shows the probability density - higher density means that the instruments all together indicate there is a good chance that the value is around that value. Then divide each corner on the curve with the number of data and on the x-axis you have ohm, and on the y-axis you have probability.

Finally try to cumulate the data, which means that for each corner from the left you add the y-value to the previous found value - this will give you an S-shaped curve, running from 0% to 100% (ie. from 0 to 1). What you have now is a very valuable probability plot, at 50% (ie. 0.5) you have the median - read this value - and that's what the resistor value most likely is. Also try to read 0.25 and 0.75 values, they're good indicators of the precision.

In my case you will see that the professional LCR-meters completely dominates the picture because their precision is many times better. This is true, though, eg. the Wayne Kerr LCR-meter does not use regular cables - they would be infected by parasitic capacitance and inductance - instead it uses two solid bars that can be pushed togeter to connect to the component you want to measure.

### Summary

As we can see, determining a resistor with accurate results is quite demanding. Using handheld instruments, even of good quality probably is not going to cut it (at least expect the instrument to use a 20 ohm scale, not 200 ohm scale). Using a handcalculator with statistical functions may increase the speed dramatically.

Carefully clean all contacts in order to remove any dirt, corrosion, etc. As we are measuring a very low resistance such factors can play a major role. Short circuit the test leads of the multimeter and make sure that the Ohms reading of the multimeter does not change when you move the connectors a bit or pull the wires a bit. If the reading does change, then clean the contacts again: use cleansing sprays, grind the contacts, or treat them with sandpaper.

When you have one good reference resistor, you may get a useful reference for measuring resistors, capacitors and inductors, but you need that first resistor to be accurate. If you have no other options, buy one or more 0.1% precision resistors. Remember to get a decent Watt specification, at least a 2 W resistor is required (8 pieces of 0.25 W resistors will do).

### The ultimate choice

The ultimate choice for a measuring resistor would be something like the VISHAY VHP-4. According to VISHAY the range of 10 ohm and high power handling (10W) is a very difficult range to manufacture, but the VHP-4 has the following data anyway:

```Tolerance from 10 ohm: +/- 0.01 %
Temperature Coefficient of Resistance (TCR): 5 ppm / deg C
Lifetime stability: Load life 2000 hours, max R change = +/- 0.005%
Current Noise: 0.010 microVolt (RMS) per Volt applied
Voltage coefficient: < 0.1 ppm / V (not measurable)
Maximum working voltage: 600 V
Thermal EMF: max lead effect 0.1 microVolt / degC
Thermal EMF: max power effect 2.5 microVolt / Watt
Operating Temperature Range: -55 degC to (max) 150 degC
Inductance: 0.1 microHenry (due mainly to the leads)
Capacitance: max 1.0 pF, typical 0.5 pF
```

Reading this many data about a resistor is amazing. I can guarantee you that the data are amazing too. It is impossible to manufacture such good data resistors with traditional ceramic power resistors. Not many manufacturers are available for products this kind of high quality resistors.

Besides the above you could also be lucky to find data like:

```Power rating, eg. by 70 degC, eg. without a heatsink
Voltage rating, eg. 250 V
Max. overload voltage, eg. 400 V
Overload capacity, eg. 4 x Power-rating for 5 seconds
Insulation resistance, eg. > 10^6 Mohm (for housed resistors)
Dielectric strength, eg. 5000 V
Shelf lifetime stability, eg. +/- 25 ppm/year
Rise time, eg. 1 nanosecond, given certain circumstances
```

The VISHAY bulk metal foil resistors, like the VHP-4, all have low inductance, low capacitance, low noise, low thermal EMF (electromotive forces, eg. from material incompatibility), low voltage-coefficient, low temperature coefficient, long life and long load-life stability. The inductance and capacitance issue will be determined by where on the legs are you measuring! With low reactances comes high speed (fast rise time).

Noise from resistors is either thermal noise or current noise. Thermal noise is only dependent on the resistor value and therefore independent on type or manufacturer. Current noise is inversely proportional to the frequency (also called 1/f noise) and is material dependent. Worst is semiconductors and pure carbon resistors whereas it almost does not exist in metallic conductors like metal film resistors. It is not (directly) a problem of interest in passive loudspeakers, but in amplifiers (measuring equipment) it is of course highly interesting because the noise level will be amplified by the amplifier gain.

VHP-4 is equipped with 4 legs. The 2 extra legs are for measuring. The legs are attached with a technique, which gives very good measurements, but the sensible attachment cannot handle high currents. If you by accident switches between the two measuring legs and the resistor-legs, then the measuring legs will "blow" like a fuse, and the feature is gone. Always take care when using VHP-4 that the legs are not interchanged.

VHP-4 is mounted in a heat sinking hat/house of socalled TO3 design. Mounted on a heat sink the resistor will handle higher power, alternatively not get as hot and therefore not change its resistive value as much.

VISHAY resistors are as far as I know unique by their bulk metal foil design, which is what gives them their excellent properties. Most measuring equipment (at least if an expensive type) is filled with VISHAY resistors.

The only problem is that such a resistor costs around 100 USD. You can save 20% of the price if you only need 0.1% accuracy.