## Miscellaneous notes. | |

- Introduction
- Electrolytic capacitor impedance
- Resistor color codes
- Resistor E series
- Thermally coupling two TO-92 transistors
- Headphone sensitivity
- Headphone cross-talk
- References

This is a collection of notes on different subjects.

They are in no particular order.

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 |

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 * f_{DF} * 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.

f_{DF} 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 * f_{DF} * 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].

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 (=10^{0}) | 250ppm/K (U) | |

Brown | BN | 1 | 10 (=10^{1}) | 1% (F) | 100ppm/K (S) |

Red | RD | 2 | 100 (=10^{2}) | 2% (G) | 50ppm/K (R) |

Orange | OG | 3 | 1000 (=10^{3}) | 15ppm/K (P) | |

Yellow | YE | 4 | 10000 (=10^{4}) | 25ppm/K (Q) | |

Green | GN | 5 | 100000 (=10^{5}) | 0.5% (D) | 20ppm/K (Z) |

Blue | BU | 6 | 1000000 (=10^{6}) | 0.25% (C) | 10ppm/K (Z) |

Violet | VT | 7 | 10000000 (=10^{7}) | 0.1% (B) | 5ppm/K (M) |

Gray | GY | 8 | 100000000 (=10^{8}) | 0.05% (A) | 1ppm/K (K) |

White | WH | 9 | 1000000000 (=10^{9}) | ||

Gold | GD | 0.1 (=10^{-1}) | 5% (J) | ||

Silver | SR | 0.01 (=10^{-2}) | 10% (K) | ||

None | 20% (M) |

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).

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

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

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

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 GND wires 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.

[1] | Illinois Capacitor "Impedance, Dissipation Factor and ESR" |

[2] | Cornell Dubilier "Aluminum Electrolytic Capacitor Application Guide" |

Poul Petersen, C/Faya 14, 35120 Arguineguín, Las Palmas, Spain.

Poul Petersen home, Poul Petersen DIY index, E-mail: diy@poulpetersen.dk

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