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Introduction to Frequency Response Roundup before the confusion

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Frequency Sine wave

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Frequency Response Analysis Any signal can be broken into sine and cosine waves of different frequencies and amplitudes (Fourier) A dynamic system’s response can be described in terms of the frequency content of the input and output signals The transfer function describes the effect of the system on changing an input into an output

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Frequency response The process is described in terms of changes in amplitude (size) and phase (shift in time) of the signal from input to output. Input amplitude Output amplitude Process Output amplitude / Input amplitude = amplitude ratio (AR)

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Period Shift Phase shift = Shift/Period * 2π or Phase shift = Shift/Period * 360° A delay is a negative phase shift (as we will see later)

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For a linear system, AR and phase vary with frequency, but not amplitude, of the input signal. Frequency is usually measured in radians per second (or per minute) and is denoted by

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Response of First Order System to Sine Wave K=1, =1 sec, =0, =1 rad/s

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Response of First Order System to Sine Wave K=1, =1 sec, =0, =3 rad/s

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Response of First Order System to Sine Wave K=1, =1 sec, =0, =10 rad/s

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Response of First Order System to Sine Waves K=1, =1 sec, =0

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What’s going on (Physically)? Faster sine waves have the same amplitude, but smaller integral before returning to zero. First order systems have a capacitance, integrating the difference between the input and output flows. Smaller integral means smaller physical changes. There is an initial transient before the output sine wave asserts itself. We will ignore this transient. We are only interested in the sustained behaviour.

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Amplitude Ratio and Phase Shift using Transfer Functions 1.Replace S with j in the transfer function: G(s) G(j ) 2.Rationalize G: make it equal to a + jb, where a and b may be functions of (G is now a complex number that is a function of ) 3.AR = |G| = sqrt(a 2 + b 2 ) 4. = tan -1 (b/a)

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