Example 2 - Waveform Representation

Example:

a) What is the period of this waveform?
b) Find equations representing the waveform above over two full cycles (or periods)

Solution:
a) The period is the time of one full cycle
T = 20 seconds

b) To represent first period, need equations for line segments (1) 0 ≤ t ≤ 10;
(2) 10 ≤ t ≤ 15; (3) 15 ≤ t ≤ 20; ...

Equation for line (1); (i in Amp., t in seconds)
i(t) = m t + b; m = ∆i / ∆t = (v2 - v1) / (t2 - t1)
i(t) = (15 - (-5)) / (10 - 0) = 20 / 10 = 2
\ i = 2 t + b; b = -5, by inspection of graph
\ i = 2 t - 5; (0 ≤ t ≤ 10)

Equation for line (2) 10 ≤ t ≤ 15; (i in Amp., t in seconds)
i(t) = m t + b; m = 0
\ i = b
\ i = 15; (10 ≤ t ≤ 15)

Equation for line (3) 15 ≤ t ≤ 20; (i in Amp., t in seconds)
i(t) = m t + b; m = ∆i / ∆t = (v2 - v1) / (t2 - t1) = (-5 - 15) / (20 - 15) = -4
\ i = -4 t + b
Calculate b ..., use point on line (20, -5):
-5 = -4 (20) + b; --> b = 75
\ i = -4 t + 75; (15 ≤ t ≤  20)


Equations for second cycle or period (20 ≤ t ≤ 40)... use shifting conception
i = 2 (t - T) - 5; i = 2 (t - 20) - 5 or i = 2 t - 45; (20 ≤ t ≤ 30)
i = 15; (30 ≤ t ≤ 35)
i = -4 (t - 20) + 75 or i = -4 t + 155; (35 ≤ t ≤ 40)