� 16 4. Sinusoidal solutions Many things in nature are periodic, even sinusoidal. We will begin by reviewing terms surrounding periodic functions. If an LTI system is fed a periodic input signal, we have a right to hope for a periodic solution. Usually there is exactly one periodic solution, and often all other solutions differ from it by a “transient,” a function that dies off exponentially. This section begins by setting out terms and facts about periodic and sinusoidal functions, and then studies the response of a first order LTI system to a sinusoidal signal. This is a special case of a general theory described in Sections 10 and 14. 4.1. Periodic and sinusoidal functions. A function f(t) is peri-odic if there is a number a > 0 such that f(t + a) = f(t) for all t. It repeats itself over and over, and has done since the world began. The number a is a period. Notice that if a is a period then so is 2a, and 3a, and so in fact is any positive integral multiple of a. If f(t) is continuous and not constant, there is a smallest period, called the minimal period or simply the period, and is often denoted by P . If the independent variable t is a distance rather than a time, the period is also called the wavelength, and denoted in physics by the Greek letter “lambda,” �. A periodic function of time has a frequency, too, often written using the Greek letter “nu,” ω. The frequency is the reciprocal of the minimal period: ω = 1/P. This is the number of cycles per unit time, and its units are, for exam-ple, (sec)−1 . Since many periodic functions are closely related to sine and cosines, it is common to use the angular or circular frequency, denoted by the Greek letter “omega,” �. This is 2ν times the frequency: � = 2νω. If ω is the number of cycles per second, then � is the number of radians per second. In terms of the angular frequency, the period is 2ν P = . The sinusoidal functions make up a particular class of periodic functions, namely, those which can be expressed as a cosine function17 which as been amplified, shifted and compressed: (1) f(t) = A cos(�t − π) The function (1) is periodic of period 2ν/� and frequency �/2ν, and circular frequency �. The parameter A (or, better, A ) is the amplitude of (1). By| |replacing π by π + ν if necessary, we may always assume A ∗ 0, and we will usually make this assumption. The number π is the phase lag (relative to the cosine). It is mea-sured in radians or degrees. The phase shift is −π. In many applica-tions, f (t) represents the response of a system to a signal of the form B cos(�t). The phase lag is then usually positive—the system response lags behind the signal—and this is one reason why we choose to favor the lag and not the shift by assigning a notation to it. Some engineers prefer to use π for the phase shift, i.e. the negative of our π. You will just have to check and see which convention is in use. The phase lag can be chosen to lie between 0 and 2ν. The ratio π/2ν is the fraction of a full period by which the function (1) is shifted to the right relative to cos(�t): f(t) is π/2ν radians behind cos(�t). Here are the instructions for building the graph of (1) from the graph of cos t. First amplify, or vertically expand, the graph by a factor of A; then shift the result to the right by π units; and finally compress it horizontally by a factor of �. t 0 P A x t Figure 1. Parameters of a sinusoidal function18 One can also write (1) as f(t) = A cos(�(t − t0)), where �t0 = π, or π (2) t0 = P 2ν t0 is the time lag. It is measured in the same units as t, and repre-sents the amount of time f (t) lags behind the compressed cosine signal cos(�t). Equation (2) expresses the fact that t0 makes up the same fraction of the period P as the phase lag π does of the period of the cosine function. There is a fundamental trigonometric identity, illustrated in the Mathlet Trigonometric Id, which rewrites the shifted and scaled co-sine function A cos(�t − π) as a linear combination of cos(�t) and sin(�t): (3) A cos(�t − π) = a cos(�t) + b sin(�t) The numbers a and b are determined by A and π: in fact, a = A cos(π) , b = A sin(π) This is the familiar formula for the cosine of a difference. Geometrically, (a, b) is the pair of coordinates of the point on the circle with radius A and center at the origin, making an angle of π counterclockwise from the positive x axis. ����������� (a, b) A (0, 0) � π � (a, 0) In the formula either or both of a and b can be negative; (a, b) can be any point in the plane. I want to stress the importance of this simple observation. Perhaps it’s more striking when read from right to left: any linear combination of cos(�t) and sin(�t) is not only periodic, of period 2ν/�—this much is obvious—but even sinusoidal—which seems much less obvious. And the geometric descriptions of the amplitude A and phase lag π is very useful. Remember them: A and π are the polar coordinates of (a, b)19 If we replace �t by −�t + π in (3), then �t − π gets replaced by −�t and the identity becomes A cos(−�t) = a cos(−�t+π)+b sin(−�t+π). Since the cosine is even and the sine is odd, this is equivalent to (4) A cos(�t) = a cos(�t − π) − b sin(�t − π) which is often useful as well. The relationship between a, b, A, and π is always the same. 4.2. Periodic solutions and transients. Let’s return to the model of the cooler, described in Section 2.2: x(t) is the temperature inside the cooler, y(t) the temperature outside, and we model the cooler by the first order linear equation with constant coefficient: x˙ + kx = ky. Let’s suppose the outside temperature varies sinusoidally (warmer in the day, cooler at night). (This involves choosing units for temperature so that the average temperature is zero.) By setting our clock so that the highest temperature occurs at t = 0, we can thus model y(t) by y(t) = y0 cos(�t) where y0 = y(0) is the daily high temperature. So our model is (5) x˙ + kx = ky0 cos(�t). The equation (5) can be solved by the standard method for solving first order linear ODEs (integrating factors, or variation of parameter). In fact, we will see in Section 10 that since the right hand side is sinusoidal there is an
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