RELAYS 101

Horizontal Characteristic

This routine allows you to see and predict the distance (in ohms) to a fault plot that falls on the border of the Horizontal Characteristic. For any given reach (to border of characteristic) at 90 degrees then there is a prediction one can accomplish to prove that the characteristic is indeed a Horizontal Characteristic (straight, flat line that is parallel to the zero reference line). With modern protective relays the characteristic is driven by mathematics within the relays' programming. With older electromechanical or solid-state relays the characteristic was created in other manners. After one proves the reach at 90 degrees then, a reach test is done at angular points on either side of the 90 degree point. A horizontal characteristic can thus be proven. The math to prove a horizontal characteristic is: Reach at Test Angle (TA) = Reach at 90 degrees/sin(Test Angle). 
Note that there is no "MTA" test as the Angle of Maximum Torque is defined as the angle at which the least amount of current can still cause the trip contacts to close. 
The horizontal characteristic is inherently non-directional (omni-directional). The 90 degree mark is, realistically, the closest reach to the point of origin and therefore requires the most amount of current. 90 degrees is the point to set the reach on your relay because that is the setpoint. Other points on the line will be tested to prove the characteristic and (typically) to check for pickup at a loadpoint. For example: with a set point reach of 5 ohms and a test fault angle of 30 degrees then the test reach that proves the characteristic is horizontal would be 10 ohms. 10 = 5/sin(30). Or another example might be a setpoint reach of 6 ohms and a test fault angle of 45 degrees then the reach would be (reach at TA = reach at 90/sin(TA)) 6/sin45 = 8.48 ohms.

Reach at Test Angle = (Reach at 90 degrees) / sin(Test Angle)

Look for a horizontal line from fault point at test angle to set point reach at 90 degrees.