Practical_Antenna_Handbook_0071639586

716 A p p e n d i x A : U s e f u l M a t h
your ride, your angular velocity is constant; that is, you are always moving through an
equal number of degrees of the circle each second. However, both your velocity in the x
direction and your velocity in the y direction are constantly changing (and occasionally
they have the same magnitude). For example: As you reach the top of the circular path
you’re traversing, your y (or vertical) speed slows as you approach the top, goes to zero
exactly at the top, and then starts to increase in the opposite direction as you head back
toward ground level. In other words, your y velocity was positive but is now negative.
Halfway to ground level, when your x position is at its farthest extension in either
direction from the loading gate, your y (or vertical) speed is at its greatest, but your
velocity along the x axis is now going through zero one-quarter rotation after your y
velocity did! So it turns out that while your x position is the sine function of Eq. (A.3.2),
your x velocity is actually a cosine function similar to—but with a different constant or
scale factor in front of—Eq. (A.3.3). In calculus terminology, velocity is the first derivative
of position, so if your position is becoming more negative (i.e., you’re heading back
down toward the ground), your velocity is a negative number.
Similarly, acceleration is the rate of change of velocity. Do you remember the strange
feeling you had the first time you went “over the top” on a Ferris wheel—perhaps even
a momentary sense of weightlessness? That’s because you were reversing your vertical
(or y) velocity at that point. Even though your instantaneous y velocity at the top went
to zero, you were experiencing maximum acceleration in the (negative) Y direction at
that very instant. Through an analysis similar to the one we performed for velocity
versus position, a velocity that can be described as a cosine function is the result of an
acceleration that is a sine function of the same frequency but with a minus sign. In calculus
terminology, acceleration is the second derivative of position. In summary:
( ) ( )
x( t) = k sin 2π ft and y( t) = k cos 2π ft
(A.3.4)
v ( t) = 2πfk cos 2π ft and v ( t)
= −2πfk sin 2π ft (A.3.5)
X
( ) ( )
Y
2 2
a ( t) = − 2πf k sin 2π ft and a ( t)
= − 2πf k cos 2π ft (A.3.6)
X
( ) ( ) ( ) ( )
Y
where k = constant proportional to mechanical distance (or electrical amplitude)
f = frequency of driving force (mechanical or electrical)
x and y express instantaneous position along either axis
v X and v Y express instantaneous velocity along either axis
a X and a Y express instantaneous acceleration along either axis
x(t) means x varies with time t (spoken “x as a function of t” or simply “x of t”)
From Eqs. (A.3.4) through (A.3.6) you can see that if an object’s position can be described
with a sine function of frequency f, its acceleration is also a sine function of
frequency f but with the opposite sign (or polarity). This is of particular interest to us in
Chap. 3, when we examine the behavior of electrical charges on a very short conductor
as they are driven back and forth in sinusoidal motion by an alternating voltage of frequency
f instead of the motor and gears of the Ferris wheel.