## 2. Distance Determination

An EDM uses the EM signal structure and determines distance using a phase shift. The EM signal has a sinusoidal wave form. Remember from trigonometry that the sine curve looks like, Figure A-3:

Figure A-3 Sine Curve |

This wave form repeats every 360°. The distance between wave form ends is the wavelength, λ, Figure A-4.

Figure A-4 Wavelength Relationship |

Different wavelengths are generated at different modulation frequencies, f. Wavelength, frequency, and the speed of light are related by:

λ = c/f | Eqn (A-1) | |

λ: wavelength c: speed of light f: frequency |

The wavelength is a known quantity since it is generated by the EDM at a specific frequency. The signal leaves the EDM at 0° phase, goes thru *N* number of full phases on its path to then from the reflector, and returns to the EDM at some angle between 0° and 360° creating a partial wavelength, p, Figure A-5:

Figure A-5 Phase Shift |

The EDM can very accurately determine the length of the last partial wavelength from its phase. The total EDM-reflector-EDM distance is (Nλ + p).

*Example*

The wavelength in Figure A-5 is 20.00 ft.There are 10 full wavelengths before the last partial one. What is the EDM to reflector distance?

The last partial wave is: p = (82°30'39.9"/360°00'00") x 20.00 ft = 4.584 ft

Including the full wavelengths the total distance EDM-reflector-EDM is: 10 x 20.00 ft + 4.854 ft = 204.584 ft

The distance between the EDM and reflector is half that: 204.584 ft / 2 = 102.292 ft.

Unfortunately, while the EDM can accurately measure the last partial wavelength, it doesn't know how many full wavelengths occurred before it (This is similar to the *integer ambiguity* in carrier-based GPS measurements).

So how does the EDM resolve this dilemma?

By decreasing the frequency by a factor of 10 and repeating the process. Decreasing the frequency by a factor of 10 increases the wavelength by a like amount. The partial wavelength at this level will give the next higher distance digit. This is repeated a number of times until the distance is resolved.

Figure A-6 illustrates three frequencies each folded out to show a continuous EDM-reflector-EDM path:

Figure A-6 Multiple Frequencies |

*Example*

The follwing table shows the last partial length for each of 4 different wavelengths. What is the total distance EDM-reflector?

λ, m |
p, m |
dist, m |

10.00 | 3.68 | |

100.0 | 53.7 | |

1,000 | 454 | |

10,000 | 8450 |

The digits in bold represent the digits added to the distance as a result of each partial wavelength.

λ, m |
p, m |
dist, m |

10.00 | 3.68 |
3.68 |

100.0 | 53.7 |
53.68 |

1,000 | 454 |
453.68 |

10,000 | 8450 |
8453.68 |

The *total distance* is 8456.68 m; the EDM-reflector distance is 8453.68/2 = 4226.84 m