phase_shifts

# phase_shifts - and X = A 2 B 2 tan φ = sin φ cos φ = B A...

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P HASE -S HIFTED H ARMONICS For a number of different types of problems in this course, we will find it useful to convert a linear combination of two harmonic functions, sin ϖ t and cos ϖ t, x(t) = Asin ϖ t + Bcos ϖ t (1) into a single harmonic with a phase shift: x(t) = Xsin ϖ t + φ ( 29 (2) The textbook offers an approach for making this conversion. Here we will discuss an alternate way that is likely to be easier to remember. Equation (1) can be rewritten as (multiply and divide the right hand side by A 2 + B 2 : x(t) = A 2 + B 2 A A 2 + B 2 sin ϖ t + B A 2 + B 2 cos ϖ t (3) Say we define the following: cos φ = A A 2 + B 2 sin φ = B A 2 + B 2 (4) Substituting (4) into (3): x(t) = A 2 + B 2 cos φ sin ϖ t + sin φ cos ϖ t [ ] = A 2 + B 2 sin ϖ t ( 29 = X sin ϖ t ( 29 (5) where in the second equation above we used sin(a+b) = cos(a)sin(b)+cos(b)sin(a)

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Unformatted text preview: and: X = A 2 + B 2 tan φ = sin φ cos φ = B/ A 2 + B 2 A/ A 2 + B 2 = B A (6) That is, the equations in (6) provide the equations to convert from A and B to the amplitude X and phase angle φ . Remarks There are alternate ways to write a sum of harmonics in terms of an amplitude and phase: x(t) = Asin ϖ t + Bcos ϖ t = Xsin ϖ t-ψ ( 29 = Xcos ϖ t +η ( 29 = Xcos ϖ t-γ ( 29 In each case, the amplitude X is the same as given in (6) above. However, the phase angles ψ,η and γ are different from the phase angle φ in (6). As an exercise with this approach, derive the phase angles ψ,η and γ in terms of A and B....
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phase_shifts - and X = A 2 B 2 tan φ = sin φ cos φ = B A...

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