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Miscellaneous notes.

Contents


Introduction.

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


Electrolytic capacitor impedance.

Capacitor equivalent circuit and impedances.
Fig.1: Capacitor equivalent circuit and impedances.

Table 1: Data for 2 different capacitors. A is a low-cost general-purpose type and B is a low-impedance type designed for switching applications.
TypeAB
Capacitance1000 µF1000 µF
Voltage50 V50 V
Diameter12.5 mm16 mm
Length25 mm25 mm
Maximum ripple current @ 120 Hz1.05 A @ 85 °C1.8 A @ 105 °C
Dissipation factor (=DF=tan φ) @ 120 Hz0.120.1
Impedance @ 100 kHzNo data21 mΩ
ESR (calculated)1.6 mΩ1.3 mΩ
Self-inductance (calculated)No data36 nH
Self resonant frequency (calculated)No data26.6 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

You can download a spread-sheet with these calculations here.

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.

Table 2: Resistor color codes.
ColorCodeValueMultiplierToleranceTemperature coefficient
BlackBK01 (=100)250ppm/K (U)
BrownBN110 (=101)1% (F)100ppm/K (S)
RedRD2100 (=102)2% (G)50ppm/K (R)
OrangeOG31000 (=103)15ppm/K (P)
YellowYE410000 (=104)25ppm/K (Q)
GreenGN5100000 (=105)0.5% (D)20ppm/K (Z)
BlueBU61000000 (=106)0.25% (C)10ppm/K (Z)
VioletVT710000000 (=107)0.1% (B)5ppm/K (M)
GrayGY8100000000 (=108)0.05% (A)1ppm/K (K)
WhiteWH91000000000 (=109)
GoldGD0.1 (=10-1)5% (J)
SilverSR0.01 (=10-2)10% (K)
None20% (M)

Resistor E series.

Table 3: E3, E6, E12 and E24 values.
E3E6E12E24E3E6E12E24E3E6E12E24E3E6E12E24
1010101018183333335656
11203662
1212222222223939686868
13244375
1515152727474747478282
16305191

Table 4: E48, E96 and E192 values.
E48E96E192E48E96E192E48E96E192E48E96E192E48E96E192E48E96E192E48E96E192E48E96E192
10.010.010.013.313.313.317.817.817.823.723.723.731.631.631.642.242.242.256.256.256.275.075.075.0
10.113.518.024.032.042.756.975.9
10.210.213.713.718.218.224.324.332.432.443.243.257.657.676.876.8
10.413.818.424.632.843.758.377.7
10.510.510.514.014.014.018.718.718.724.924.924.933.233.233.244.244.244.259.059.059.078.778.778.7
10.614.218.925.233.644.859.779.6
10.710.714.314.319.119.125.525.534.034.045.345.360.460.480.680.6
10.914.519.325.834.445.961.281.6
11.011.011.014.714.714.719.619.619.626.126.126.134.834.834.846.446.446.461.961.961.982.582.582.5
11.114.919.826.435.247.062.683.5
11.311.315.015.020.020.026.726.735.735.747.547.563.463.484.584.5
11.415.220.327.136.148.164.285.6
11.511.511.515.415.415.420.520.520.527.427.427.436.536.536.548.748.748.764.964.964.986.686.686.6
11.715.620.827.737.049.365.787.6
11.811.815.815.821.021.028.028.037.437.449.949.966.566.588.788.7
12.016.021.328.437.950.567.389.8
12.112.112.116.216.216.221.521.521.528.728.728.738.338.338.351.151.151.168.168.168.190.990.990.9
12.316.421.829.138.851.769.092.0
12.412.416.516.522.122.129.429.439.239.252.352.369.869.893.193.1
12.616.722.329.839.753.070.694.2
12.712.712.716.916.916.922.622.622.630.130.130.140.240.240.253.653.653.671.571.571.595.395.395.3
12.917.222.930.540.754.272.396.5
13.013.017.417.423.223.230.930.941.241.254.954.973.273.297.697.6
13.217.623.431.241.755.674.198.8

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).

You can download a spread-sheet with the values here.


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.

Transistors after shrinking.
Fig.2: Transistors after shrinking the tubing.

Transistors after shrinking and cutting.
Fig.3: Transistors after cutting away excessive tubing.


Headphone sensitivity.

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


Headphone cross-talk.

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

Headphone equivalent circuit.
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

You can download a spread-sheet with the calculations here.

Table 5: Some calculated values.
RoRgRcRhRw1Rw2Rw3Cross-talkComment
1 Ω10 mΩ50 mΩ32 Ω0.33 Ω (1)0.33 Ω (1)17 mΩ (2)40 dB
1 Ω10 mΩ50 mΩ32 Ω00.33 Ω (1)0.33 Ω (1)65 dBA 4-wire connection with the 2 GND
wires connected in the 3-way connector.
1 Ω1 Ω50 mΩ32 Ω0.33 Ω (1)0.33 Ω (1)17 mΩ (2)25 dBA "3-channel" amplifier.

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


Reading logarithmic plots.

Example resistance vs. temperature plot for an NTC resistor.
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

You can download a spread-sheet with the calculations here.

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


References.

[1] Illinois Capacitor "Impedance, Dissipation Factor and ESR"
[2] Cornell Dubilier "Aluminum Electrolytic Capacitor Application Guide"

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