The US Coast Guard closed the LORAN-C system on February 8, 2010. This problem is left as an example of how hyperbolas can be used.

## Introduction

LORAN-C is a low frequency hyperbolic radio-navigation system. The U.S. Department of Transportation has designated LORAN-C as the government-provided radio-navigation system for the Costal Confluence Zone (CCZ). To obtain a line of position (LOP) the navigator measures the difference between the time arrival of a pulse from the master transmitter and a secondary transmitter of a particular chain. The measurement is plotted on a LORAN-C chart. The crossing point of two or more LOP's, where each LOP is derived using the same master transmitter but a different secondary transmitter, fixes the receiver in latitude and longitude.

The LORAN-C stations for the 8970 Great Lakes Chain with master transmitter (M) at Dana, IN, are shown on the map. Secondary transmitters for the 8970 chain are denoted W, X, Y, and Z.

Transmitter Location Emission Delay (μs) Coding Delay (μs) Power (kW)
8970-M Dana, IN     400
8970-W Malone, FL 14,355.11 11,000 800
8970-X Seneca, NY 31,162.06 28,000 800
8970-Y Baudette, MN 47,753.74 44,000 800
8970-Z Boise City, OK 63,669.46 59,000 900

The radio waves used in Loran-C travel at the speed of light or 0.186 miles per microsecond (μs). The emission delay is the coding delay plus the time it takes for the secondary signal to reach the master transmitter. As an example, the coding delay for Baudette, MN, is 44,000.00 μs and the emission delay is 47,753.74 μs. This means that the Baudette secondary transmitter signal sends out its pulse 44,000.00 μs after the Dana transmitter sends its signal and the signal from Baudette arrives in Dana 47,753.74 μs after the Dana signal is sent. The difference between the emission delay and coding delay (3,753.74 μs) is the time it takes for the signal to go from Baudette to Dana. Since the signal travels at the speed of light, which is 0.186 miles/μs, the distance between Baudette and Dana is 3,753.74 ( 0.186 ) = 698.19564 miles.

Since the transmitters are located at the foci of a hyperbola, the distance between the foci for the Dana / Baudette hyperbola would be 698.19564 miles. The focal length c, the distance from the focus (transmitter) to the center, is half of that distance or 349.09782 miles.

The dial reading is used to determine the distance between the vertices of the hyperbola and which branch of the hyperbola to use. The craft (plane or ship) is closer to the secondary transmitter if the dial reading is less than the emission delay and closer to the master transmitter if the dial reading is more than the emission delay. The navigator uses this information to select the proper branch of the hyperbola.

The difference between the dial reading and the emission delay tells you how far apart in time the signals between the two transmitters arrive at the craft. After converting that time difference into miles, you arrive at the difference in distances between the foci, which is 2a in hyperbola jargon.

## Example

Assume that a plane has a dial reading of 49,143.57 μs from the Yankee transmitter. Since the dial reading is more than the emission delay of 47,753.74 μs, the plane is closer to the master transmitter in Dana, IN, than it is to the secondary transmitter in Baudette, MN. The time difference in the arrival of the signals is 1,389.83 μs. Since radio waves travel at 0.186 miles/μs, the difference in distances from the plane to the transmitters is 1,389.83 ( 0.186 ) = 258.50838 miles. That distance is how far it is between the vertices of the hyperbola, so 2a = 258.50838. Taking half of that value tells us that the distance from the center of the hyperbola to the vertex is a = 129.25419 miles.

Once you know a and c, you can use the Pythagorean relationship for hyperbolas, a2 + b2 = c2, to find b. Since a = 129.25419 and c = 349.09782, then b = 324.28790.

Plotting the branch of the hyperbola that is closer to Dana on a map would give the graph to the right. Do this for both dial readings and where the two hyperbolas intersect determines your location.