Problem 5

Pulse width modulation analysis of flywheel system. This problem extends the flywheel

analysis of Problem 4. Pulse width modulation (PWM) is a common technique for specifying the

moment applied by an electric motor. In PWM, the motor current is either "on" or "off" with the

switching occurring with regularity (often called the repetition rate). This is done to maximize the

efficiency of the amplifier. It is assumed that the moment created by the motor also follows the

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current switching. The rapid switching of the moment is filtered by the inertia of the motor load

(the flywheels in this problem) so that the angular velocities of the flywheels are a "smoothed"

version of the moment. This problem explores these ideas in a quantitative manner.

1. The average value of a periodic signal 9 is defined

(d, T- signal period

Prove the following general result: if an asymptotically stable system is subjected to a periodic

input, then

Have (0) Ma

(2)

where (0) is the frequency response evaluated at -0, u is the periodic input and y is the

periodic output. Refer to the graphs of fly, fly and fly from Problem 4 to see that they agree

with (2). Also refer to Problem 3 to show its agreement with (2).

Hist: Apply the "averaging operator" to the Fourier series representations of the input and

output and note that t=0 for all 0 (recall -20/T).

2. Although the average value of f, can be specified according to (2) there can be considerable

variation in the angular velocity about the mean value. Some applications require that the

angular velocity not deviate too much from the mean value. A conservative upper limit on

the period I will be derived that guarantees that a specified deviation from the mean value

will not be exceeded. Consider these steps:

(a) Consider the rectangular wave with period 7>0 and duty cycle a € [0, 1]. One period

for t€ 0,7] is defined below:

(1 teBar)

(t)= 10 t€aTT)

The duty cycle determines the duration that the input is "on" or "off" in one period:

a-0 means - 0 for all t; a-1 means -1 for all t; a-0.5 means -1 for half

the period and zero for the remaining half. Find the Fourier series coefficients, denoted

(b) Determine an upper bound for ea, &0, where are the Fourier series coefficients

of . This upper bound will have the index k.

(e) Let Hy be the frequency response function associated with f. Find an approximation

for H₁ for sufficiently large . "Sufficiently large" means the dominant terms in the

numerator and denominator are the are the highest powers of ./nnumerator and denominator are the are the highest powers of us.

(d) The Fourier series for fly can be written as

- - Σκουλαρίκια

12(1) -

+2ΣRe [8 (kuva)

- +2

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What is the relation between fave and a? Thus, conclude that

over a certain range simply by selecting a.

can be specified

(e) Despite the relation between a and Shaw, the "ripple" in the angular velocity cannot

be too large. Thus, consider the following sequence of bounds

||$21 (1) — £21,000| — |

|-|(2Σe [ (int)

Re

52 Rei

ام و لا (س) 2 >

(س) -

Substitute the upper bound for a/T and the "high frequency" approximation of |H₂|-

Then, do the sum to find an upper bound for (1) -- The period I should be

a parameter. Hint:

1²

(f) Based on the bound, determine the largest value of T so that f(t) — £₁,| ≤0.01. In

other words, fly(t) is very close to its mean value.

(g) Graph four periods of fly with the selected value of T and a(0.25, 0.5, 0.75). Use

-1000,0,1000) in the partial sum of the Fourier series. The time grid can

need to be customized. Select the time spacing in the grid, t, so that there are 1000

points per period. Show that the deviation of f(t) from he doe not exceed C

all cases of a.