For general formulas, see

Formulas & Concepts Relevant to Diesel-Electric and Electric Locomotives

on

This was originally on my railroad pages, but it had no obvious application in railroading, so I moved it here. Additional information may be found there on D.C. circuits (ohm's law), A.C. circuits (capacitance, inductance, impedance, & resonant circuits), and power & other electrical concepts (power & A.C. motors and electric power transmission).

Applications for SWR are usually in broadcast and other radio frequency applications with antennas and feed lines (so any railfans and others with scanners who may be "listening-in" are hearing something broadcast over an antenna and feed line with some SWR!). For amateurs, SWR is a critical element of good antenna installation.

SWR equals:

- E
_{MAX}/E_{MIN} - or
- I
_{MAX}/I_{MIN} - where
- E = voltage (
**E**lectromotive force) - I = current (in amps)

as measured on a line (maximums and minimums); by formula, this can be computed by

- 1+p/1-p

where p = rho, the voltage reflection coefficient.

Rho is computed in either of the following ways:

- a.) p = (P
_{R}/P_{F})^{1/2} - where
- P
_{R}= reflected power - P
_{F}= forward power

Under balanced conditions, there should be no reflected power; thus, there would be no be no high voltage or low voltage (or current) points on the line, thus no standing waves.

- b.) p = ([(R
_{a}-Z_{o})²+X_{a}²]/[(R_{a}+Z_{o})²+X_{a})^{1/2} - where
- R
_{a}= output resistance (the load - X
_{a}= output - Z
_{o}= the line power

The output resistance and the line impedance should be equal (thus balancing each other out of the numerator) and the output reactance should be as small as possible (thus disappearing from the numerator, making the numerator very small in relation to the denominator). With Ra and Zo equal, the formula

- ([X
_{a}²]/[2Z_{o}²+X_{a}²])^{1/2} - and
- Lim (X
_{a}²)/(2Z_{o}²+X_{a}²), X_{a}-> 0 = 0

The decibel (dB) represents a ratio, expressed logrithmicaly, between two quantities, a reference quantity and the quantity to be compared. The generally used measure of the deciBel is one tenth of the Bel (thus the odd capitalization). It is used to describe the level of something with respect to something else, such as the amount of noise created by idling locomotives or departing aircraft over the ambient (background) noise. For the radio amateurs and other electronic types out there, deciBels is usually used in broadcast and other radio frequency applications to compare power levels (sound, signal strength, noise, etc.).

The Bel or deciBel is unitless: it exists as a ratio with respect to however the event being measured is itself measured. If the ratio is refered to in terms of a specific unit of measurement that measure is indicated by a suffix (e.g., dBm is referenced against 1 milliWatt; dBV is referenced against 1 volt).

Apparently, the measure of the Bel was at first ratio for power, with the basic formula being

B = | log_{10} P_{1}/P_{2} |

Where

- B is the ratio expressed in Bels
- log
_{10}is the base ten logarithm - P
_{1}is the measured level of power - P
_{2}is the reference level of power

The deciBel simply increases the units by ten to make measurements of small changes more readable; thus,

dB = | 10 log_{10} P_{1}/P_{2} |

Where

- dB is the ratio expressed in deciBels (tenths of Bels)
- 10 log
_{10}is ten times the base ten logarithm

As power is proportional to voltage or current squared, the ratio of voltages or currents across a constant impedence is given by

20 log_{10} V_{1}/V_{2}
=or=
20 log_{10} I_{1}/I_{2} |

Where

- 20 log
_{10}is twenty times the base ten logarithm - V
_{1}is the measured level of voltage - V
_{2}is the reference level of voltage

OR - I
_{1}is the measured level of current - I
_{2}is the reference level of current

A gain of 100 volts per volt (e.g., in an amplifier, with 1 volt in and 100 volts out), 100/1, equals

- 20 log
_{10}100, which equals - 20 * 2, which equals
- 40 dB

A gain of 1000 volts per volt, 1000/1, equals

- 20 log
_{10}1000, which equals - 20 * 3, which equals
- 60 dB

A gain of 100 volts per volt (40 dB) through a constant load produces a power gain of 10,000 watts per watt (also 40 dB). Under constant load conditions, e.g., a 50 ohm impedance dipole antenna, a gain of n dB, corresponding to a voltage or current gain of y, is equivalent to a power gain of y²: at any given dB, the voltage or current gain will correspond to the power gain under constant resistance or impedance values; thus, the following voltage or current gains--through a constant load-- correspond to the power gains:

When working with power, 3 dB is twice, 10 dB is 10 times; When working with voltage or current, 6 dB is twice, 20 dB is 10 times:

1 dB = | a power gain of | 1.256 (~26%) | |

3 dB = | a power gain of | ~2.0 (-3 dB = power loss of ~50%) | |

6 dB = | a power gain of | ~4 | |

10 dB = | a power gain of | 10 | |

20 dB = | a power gain of | 100 |

db | Voltage or Current |
Power | ||||
---|---|---|---|---|---|---|

1.00 | = | 1.122 | 1.256 | |||

2.00 | = | 1.259 | 1.585 | |||

3.00 | = | 1.413 | 1.997 | |||

3.01 | = | 1.414 | 2.000 | |||

4.00 | = | 1.585 | 2.240 | |||

4.77 | = | 1.732 | 3.000 | |||

5.00 | = | 1.778 | 3.161 | |||

6.00 | = | 1.995 | 3.980 | |||

6.02 | = | 2.000 | 4.000 | |||

6.99 | = | 2.236 | 5.000 | |||

10.00 | = | 3.162 | 10.000 | |||

13.98 | = | 5.000 | 25.000 | |||

16.99 | = | 7.071 | 50.000 | |||

18.06 | = | 8.000 | 64.000 | |||

20.00 | = | 10.000 | 100.000 | |||

30.00 | = | 31.623 | 1,000.000 | |||

40.00 | = | 100.000 | 10,000.000 | |||

60.00 | = | 10,000.0 | 1,000,000.0 | |||

## Note:These figures for gain are also used to express loss (e.g., -3db is a power loss of nearly 1/2, reference power divided by 1.997; -6dB is a power loss of nearly three quarters, reference power divided by 3.980). |
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At a given dB, square the voltage (or current) gain [or loss] to obtain the power gain [or loss]. This is because power is proportional to I² or V² (P=E²/R or P=I²R, and under constant load conditions, the proportion, dropping the constant, R, is P:E² or P:I²). |