Distance by phase comparison method

In the phase-comparison method of EDM, a continuous carrier wave is modulated with a known frequency, sent to a reflector and compared with the returned wave. The phase difference gives the distance.

Principle

Let the modulation frequency be $f$ and velocity $v$, so wavelength $\lambda = v/f$. The wave travels from the instrument A to the reflector B and back, a total of $2D$. Hence

$$2D = N\,\lambda + \Delta\lambda, \qquad \Delta\lambda = \frac{\phi}{2\pi}\,\lambda$$ $$D = \frac{1}{2}\left(N + \frac{\phi}{2\pi}\right)\lambda$$

where $N$ is the integer number of complete wavelengths in the double path and $\phi$ the phase difference (in radians) measured by the instrument.

 Sent wave      /\  /\  /\  /\ ...
 Received wave   /\  /\  /\ ...
                  |<-->| phase shift (phi)

Steps

1. The transmitted wave and the received wave are compared in a phase detector, and $\phi$ (the fractional part $\Delta\lambda$) is obtained.
2. The unknown integer $N$ cannot be found from one frequency. Hence several modulation frequencies are used, e.g. a low frequency (long wavelength, e.g. 1500 m) for the coarse distance and a higher frequency (short wavelength, e.g. 10 m) for fine reading. The readings of different wavelengths remove the ambiguity.
3. The microprocessor combines the results, applies atmospheric corrections and displays the distance.

Example

For $\lambda = 20$ m, $N = 10$ and $\phi = 90^\circ$ ($\tfrac{1}{4}$ cycle):

$$D = \frac{1}{2}(10 + 0.25)\times 20 = 102.5\ \text{m}$$