Back to Nautical Basic Principles LORAN C Sat. Nav. Sys. GPS

Electronic System of Position Fixing

Hyperbolic Lines

Principle

Loran stations transmit radio signals of very short duration called pulses. However the frequency of the signals are quite low about 90 kHz to 110 kHz

Duration of pulse varies and is about 200 μs.

If we assume that a pulse duration lasts for 200 μs, then:

1 second = 1000000 μs

If the pulse recurring frequency (PRF) is 25pps then-

25 pulses are sent in 1000000 μs

Therefore the interval between each pulse is 1000000/25 = 40000 μs

Or approximately the transmitter transmits a pulse of 200 μs and rests for 40000 μs.

However the power transmitted during these short bursts is extremely high.

Again we know that radio waves travel at the speed of light, that is at 300000 km per second.

Or in 1000000 μs the radio waves travel 300000000 metres.

Or in 1 μs the radio waves travel 300000000/1000000 = 300 metres.

Applying the above we get 1 NM (1852 metres) is traversed in 6.173 μs

Now let us assume that there are two transmitters (A and B) separated by a distance of 324 NM.

Radio waves will take about 2000 μs (324 x 6.173) to traverse from A to B.

Now let both the transmitters transmit at the same time. Also let a ship be placed at a position where both the signals are received at the same time. Now let the ship move along a course adjusting her course so that the two signals are always received at the same time (no time difference). It is seen that the course line is not a straight line but is a hyperbola.

Now it is seen that if there is a time difference in the arrival of the signals from the two transmitters, then too the track of equal time difference is a hyperbola. And a ship navigating with a receiver where the time difference can be recorded need only to keep the time difference constant to traverse the hyperbola.

However with multiple hyperbola’s drawn for the same two transmitters we would have two identical hyperbolas on either side of the hyperbola of no time difference, as seen in the figure.

The hyperbola of no time difference is known as the CENTRE LINE. And the line extending between the two transmitters is known as the base line extension.

Thus in order to resolve the problem as to on which hyperbola the ship is traversing, we require to create a time difference between the transmission of the two stations.

So to overcome the above ambiguity, a transmitter (MASTER) transmits first and after the other station (SLAVE) receives this signal, it transmits its signal. So the ship will first receive the MASTER signal and then after a time delay will receive the signal from the SLAVE.

Under this system the time difference between the two signals will be maximum on the hyperbola near the MASTER station and minimum on the hyperbola near the SLAVE station.

Since the MASTER signal has to go from the MASTER station to the SLAVE station and activate it, thus a ship near the MASTER station will have to wait after getting a signal from the MASTER, for the signal from MASTER station to go and activate the SLAVE station and then get the SLAVE station signal.

There remains another problem. What happens if at a point the signal from the MASTER arrives at a time that the signal from the SLAVE also arrives (after being triggered by the MASTER), and there is no time difference.

In the above figure, the time difference purposely created between the transmission of the MASTER station and the SLAVE station is indicated as 1000 μs on the base line extension. This time difference is known as the CODING DELAY.

NO coding delay – a ship close to the Slave station Y, receives signal from M and Y at the same time.

Station Master:

Latitude: 46° 48.’455 North

Longitude: 067° 55.’62 West

Station Yankee:

Latitude: 46° 46.54 North

Longitude: 053° 10.’46 West

A SHIP at location:

Latitude: 46° 44.’54 North

Longitude: 052° 50.’46 West

Distance GC: M to S: 618.97 NM (The signal travels from M towards S)

M to Y: 605.139 NM (The signal travels from M towards Y and triggers Y)

Y to S: 13.85 NM (Signal from Y travels towards S)

618.989 NM