Venikov, V. A. Electric Power Systems: Automatic Power Systems Control /translated from Russian by Y. M. Matskovsky. – Moscow: MIR Publishers, 1982. – 445 p. (Pp. 362-364).
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Trigonometry. Units of Measurement
by Bob Connell
Trigonometry is a branch of mathematics concerned with functions that describe angles. Although knowledge of trigonometry is valuable in surveying and navigation, in control systems engineering its virtue lies in the fact that trigonometric functions can be used to describe the status of objects that exhibit repeatable behavior. This includes the motion of the planets, pendulums, a mass suspended on a spring, and perhaps most relevant here, the oscillation of process variables under control.
The most common unit of measurement for angles is the degree, which is 1/360 of a whole circle
A lesser used unit is the radian. Although the radian is not ordinarily used in angular measurement, it should be understood because when differential equations, which occur in control systems engineering, are solved, the angles emerge in radians.
On the circumference of a circle, if an arc equal in length to the radius of the circle is marked off, then the arc will subtend, at the center of the circle, an angle of 1 radian. The angle θ (or POB) in Figure 1, illustrates this.
Figure 1 - A radian defined
In line with this definition of a radian, the relationship between radians and degrees can be worked out. The full circumference of the circle (length 2π r) subtends an angle of 360° at the center of the circle. An arc of length r will subtend an angle of
Therefore 1 radian = 180/π deg, or π radians = 180°.
The actual value of a radian is 57°17'45", although this value is hardly ever required in control systems analysis.
If the base line OB in Figure remains fixed and the radius OP is allowed to rotate counterclockwise around the center O, then the angle 6 (or POB) increases. If the starting point for OP is coincident with OB, and OP rotates one complete rotation (or cycle) until it is again coincident with OB, then the angle 6 will be 360°. From this it is evident that 1 cycle = 360° = 2n radians.
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