Miscellaneous notes.

Introduction.

This is a collection of notes on different subjects.
They are in no particular order.

Electrolytic capacitor impedance.

Fig.1: Capacitor equivalent circuit and impedances.

 Type A B Capacitance 1000 µF 1000 µF Voltage 50 V 50 V Diameter 12.5 mm 16 mm Length 25 mm 25 mm Maximum ripple current @ 120 Hz 1.05 A @ 85 °C 1.8 A @ 105 °C Dissipation factor (=DF=tan φ) @ 120 Hz 0.12 0.1 Impedance @ 100 kHz No data 21 mΩ ESR (calculated) 1.6 mΩ 1.3 mΩ Self-inductance (calculated) No data 36 nH Self resonant frequency (calculated) No data 26.6 kHz

Xc = 1 / ( 2 * π * f * C ), where:
Xc is the capacitor reactance.
f is the frequency.
C is the capacitance.

Xl = 2 * π * f * L, where:
Xl is the inductor reactance.
f is the frequency.
L is the self-inductance.

ESR = ( DF / 100 ) / ( 2 * π * fDF * C ), where:
ESR is Equivalent Series Resistance
DF is dissipation factor. Note that this value is in % even if this is not stated in the data-sheet.
fDF is the frequency where DF is specified.
C is the capacitance.

Z = √ ESR² + ( Xl - Xc )², where:
Z is the total impedance
ESR is Equivalent Series Resistance
Xl is the inductor reactance.
Xc is the capacitor reactance.

Fr = 1 / ( 2 * π * √ L * C ), where:
Fr is self-resonant frequency
C is the capacitance.
L is the self-inductance.

L = ( Xc + √ Z² - ESR² ) / ( 2 * π * f ), where:
L is the self-inductance.
Xc is the capacitor reactance at the frequency where Z is specified.
Z is the total impedance at some high frequency.
ESR is Equivalent Series Resistance
f is the frequency where Z is specified.

Example ESL calculation (Type B in table 1):
ESR = ( DF / 100 ) / ( 2 * π * fDF * C ) = ( 0.1 / 100 ) / ( 2 * π * 120 * 1.00E-3 ) = 1.33E-3 = 1.33 mΩ

Example self-inductance calculation (Type B in table 1):
Z = 21 mΩ @ 100 kHz.
Xc = 1 / ( 2 * π * f * C ) = 1 / ( 2 * π * 1.00E5 * 1.00E-3 ) = 1.59E-3 = 1.59 mΩ
L = ( Xc + √ Z² - ESR² ) / ( 2 * π * f ) = ( 1.59e-3 + √ 2.10E-2² - 1.33E-3² ) / ( 2 * π * 1.00E5 ) = 3.59E-8 = 35.9 nH

Example self-resonant frequency calculation (Type B in table 1):
Fr = 1 / ( 2 * π * √ L * C ) = 1 / ( 2 * π * √ 3.59E-8 * 1.00E-3 ) = 2.66E4 = 26.6 kHz

If you need to measure a capacitor yourself, the easiest is to measure the self-resonant frequency.
The impedance at Fr is ESR.
L = 1 / ( 4 * π² * C * Fr² )

More detailed info can be found in [1] and [2].

Resistor color codes.

This table shows the standard color codes:
Color: Guess what.
Code: IEC60757 standardized abbreviation for the color.
Value: Numerical value.
Multiplier: Multiplier value.
Tolerance: Tolerance value. The letter in () is the standard letter abbreviation for the tolerance. This is brown for a 1% resistor.
Temperature coefficient: Temperature coefficient value. The letter in () is the standard letter abbreviation for the tolerance. This will typically be red for a 1% resistor.

 Color Code Value Multiplier Tolerance Temperature coefficient Black BK 0 1 (=100) 250ppm/K (U) Brown BN 1 10 (=101) 1% (F) 100ppm/K (S) Red RD 2 100 (=102) 2% (G) 50ppm/K (R) Orange OG 3 1000 (=103) 15ppm/K (P) Yellow YE 4 10000 (=104) 25ppm/K (Q) Green GN 5 100000 (=105) 0.5% (D) 20ppm/K (Z) Blue BU 6 1000000 (=106) 0.25% (C) 10ppm/K (Z) Violet VT 7 10000000 (=107) 0.1% (B) 5ppm/K (M) Gray GY 8 100000000 (=108) 0.05% (A) 1ppm/K (K) White WH 9 1000000000 (=109) Gold GD 0.1 (=10-1) 5% (J) Silver SR 0.01 (=10-2) 10% (K) None 20% (M)

Resistor E series.

 E3 E6 E12 E24 E3 E6 E12 E24 E3 E6 E12 E24 E3 E6 E12 E24 10 10 10 10 18 18 33 33 33 56 56 11 20 36 62 12 12 22 22 22 22 39 39 68 68 68 13 24 43 75 15 15 15 27 27 47 47 47 47 82 82 16 30 51 91

 E48 E96 E192 E48 E96 E192 E48 E96 E192 E48 E96 E192 E48 E96 E192 E48 E96 E192 E48 E96 E192 E48 E96 E192 10.0 10.0 10.0 13.3 13.3 13.3 17.8 17.8 17.8 23.7 23.7 23.7 31.6 31.6 31.6 42.2 42.2 42.2 56.2 56.2 56.2 75.0 75.0 75.0 10.1 13.5 18.0 24.0 32.0 42.7 56.9 75.9 10.2 10.2 13.7 13.7 18.2 18.2 24.3 24.3 32.4 32.4 43.2 43.2 57.6 57.6 76.8 76.8 10.4 13.8 18.4 24.6 32.8 43.7 58.3 77.7 10.5 10.5 10.5 14.0 14.0 14.0 18.7 18.7 18.7 24.9 24.9 24.9 33.2 33.2 33.2 44.2 44.2 44.2 59.0 59.0 59.0 78.7 78.7 78.7 10.6 14.2 18.9 25.2 33.6 44.8 59.7 79.6 10.7 10.7 14.3 14.3 19.1 19.1 25.5 25.5 34.0 34.0 45.3 45.3 60.4 60.4 80.6 80.6 10.9 14.5 19.3 25.8 34.4 45.9 61.2 81.6 11.0 11.0 11.0 14.7 14.7 14.7 19.6 19.6 19.6 26.1 26.1 26.1 34.8 34.8 34.8 46.4 46.4 46.4 61.9 61.9 61.9 82.5 82.5 82.5 11.1 14.9 19.8 26.4 35.2 47.0 62.6 83.5 11.3 11.3 15.0 15.0 20.0 20.0 26.7 26.7 35.7 35.7 47.5 47.5 63.4 63.4 84.5 84.5 11.4 15.2 20.3 27.1 36.1 48.1 64.2 85.6 11.5 11.5 11.5 15.4 15.4 15.4 20.5 20.5 20.5 27.4 27.4 27.4 36.5 36.5 36.5 48.7 48.7 48.7 64.9 64.9 64.9 86.6 86.6 86.6 11.7 15.6 20.8 27.7 37.0 49.3 65.7 87.6 11.8 11.8 15.8 15.8 21.0 21.0 28.0 28.0 37.4 37.4 49.9 49.9 66.5 66.5 88.7 88.7 12.0 16.0 21.3 28.4 37.9 50.5 67.3 89.8 12.1 12.1 12.1 16.2 16.2 16.2 21.5 21.5 21.5 28.7 28.7 28.7 38.3 38.3 38.3 51.1 51.1 51.1 68.1 68.1 68.1 90.9 90.9 90.9 12.3 16.4 21.8 29.1 38.8 51.7 69.0 92.0 12.4 12.4 16.5 16.5 22.1 22.1 29.4 29.4 39.2 39.2 52.3 52.3 69.8 69.8 93.1 93.1 12.6 16.7 22.3 29.8 39.7 53.0 70.6 94.2 12.7 12.7 12.7 16.9 16.9 16.9 22.6 22.6 22.6 30.1 30.1 30.1 40.2 40.2 40.2 53.6 53.6 53.6 71.5 71.5 71.5 95.3 95.3 95.3 12.9 17.2 22.9 30.5 40.7 54.2 72.3 96.5 13.0 13.0 17.4 17.4 23.2 23.2 30.9 30.9 41.2 41.2 54.9 54.9 73.2 73.2 97.6 97.6 13.2 17.6 23.4 31.2 41.7 55.6 74.1 98.8

The values for the E48..E192 series can be calculated:
Value = 10 ^ ( n / e ), where:
n is the number in the series.
e is the series (48, 96 or 192).

Thermally coupling two TO-92 transistors.

In some cases you need to get 2 TO-92 transistors to track thermally.
This is easily done by inserting the 2 cases into one end of a piece of 6.0 mm heat-shrink sleeving and heating it slowly.
I normally hold it around 10 mm above a hot soldering iron for 5..10 minutes.
Do not use more heat than that (like a paint-removal tool) as it is very likely the transistors will be destroyed.
For optimal thermal contact, use a little non-conductive compound.
If it is 2 different transistor types, cut one lead on one of them a little shorter so you know the pin-out of the assembly.
If you use more than one type of assembly for a project, use different colored tubing.

Fig.2: Transistors after shrinking the tubing.

Fig.3: Transistors after cutting away excessive tubing.

Headphone sensitivity is specified as dBSPL @ 1 mW or as dBSPL @ 1 V.
Translating between the two:
dBSPL(V) = 10 * LOG * ( 1000 / R ) + dBSPL(mW) or
dBSPL(mW) = dBSPL(V) - 10 * LOG * ( 1000 / R ), where:
dBSPL(V) is the sensitivity in dBSPL @ 1 V
dBSPL(mW) is the sensitivity in dBSPL @ 1 mW
R is the nominal headphone impedance.

Example:
A 32 Ω headphone with a sensitivity of 100 dBSPL @ 1 mW has a sensitivity of:
10 * LOG ( 1000 / 32 ) + 100 = 115 dBSPL @ 1 V

This is a very simplified calculation for the cross-talk in 3-wire headphone connections.

Fig.4: Equivalent circuit for 3-wire headphone connection.

Ro: Amplifier output impedance. See amplifier data sheet.
Rg: Amplifier ground output impedance. This should be a very low value.
Rc: Connector resistance. This is normally only specified as a maximum value ( typically 50 mΩ ).
Rw1: Common cable resistance for the 2 channels.
Rw2: Cable resistance from output to each channel.
Rw3: Cable resistance from each transducer to common ground point A.
Rh: Nominal transducer impedance.

Assume one channel has signal and the other is muted and find the voltage in point A:
Rchannel = Ro + Rc + Rw2 + Rh + Rw3
Rground = 1 / ( 1 / ( Rg + Rc + Rw1 ) + 1 / Rchannel )
VA = Rground / ( Rground + Rchannel )
Find the voltage in the muted transducer:
V = VA * Rh / Rchannel

 Ro Rg Rc Rh Rw1 Rw2 Rw3 Cross-talk Comment 1 Ω 10 mΩ 50 mΩ 32 Ω 0.33 Ω (1) 0.33 Ω (1) 17 mΩ (2) 40 dB 1 Ω 10 mΩ 50 mΩ 32 Ω 0 0.33 Ω (1) 0.33 Ω (1) 65 dB A 4-wire connection with the 2 GNDwires connected in the 3-way connector. 1 Ω 1 Ω 50 mΩ 32 Ω 0.33 Ω (1) 0.33 Ω (1) 17 mΩ (2) 25 dB A "3-channel" amplifier.

Note 1: 2 m 0.1 mm² wire.
Note 2: 10 cm 0.1 mm² wire.

Fig.5: Example resistance vs. temperature plot for an NTC resistor.

Reading a value from a logarithmic plot with reasonable accuracy can be difficult ( assuming the plot is accurate in the first place ).
In the example above, we are interested in the resistance value at 37.5 °C.
This value is called yv. The values at the bottom and the top of the plot are called yb and yt.
You can use any value for yb and yt that are easy to read on the plot.
Using a ruler ( in this case an on-screen ruler ), you can measure the positions of yb, yv and yt. These are called pb, pv and pt.
In this example, pb = 466 pixel, pv = 217 pixel and pt = 0 pixel ( after resizing the plot ).
Then:
yv = yb * 10 ^ ( ( pv - pb ) / ( pt - pb ) * log ( yt / yb ) )
yv = 0.01 Ω * 10 ^ ( ( 217 px - 466 px ) / ( 0 px - 466 px ) * log ( 100 Ω / 0.01 Ω ) ) = 1.37 Ω

A logarithmic on-screen ruler would be very useful, but I have not found one?

References.

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