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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

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 3: 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 4: 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.


References.

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

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