Magnitude ratio

The amplitude of this normalized response, y/AKp, is called the magnitude ratio. 2... 

The magnitude ratio MR is defined as the ratio of the maximuni amplitude of the output over the maximum amplitude of the input ... 

For a given process, both the phase angle 0 and the magnitude ratio MR will change if frequency [Pg.417]

A magnitude G i f that is the same as the magnitude ratio MR that would be obtained by forcing the system with a sine wave input of frequency to... 

Therefore we have proved what we set out to prove (1) the magnitude ratio MR is the absolute value of G, with s set equal to ta> and (2) the phase angle is the argument of G( with s set equal to icD. 

There are three different kinds of plots that are commonly used to show how magnitude ratio (absolute magnitude) and phase angle (argument) vary with frequency CO. They are called Nyquist, Bode (pronounced "Bow-dee ), and Nichols plots. After defining what each of them is, we will show what some common transfer functions look like in the three different plots. 

The Nyquist plot discussed in the previous section presents all the frequency information in a compact, one-curve form. Bode plots require that two curves be plotted instead of one. This increase is well worth the trouble because complex transfer functions can be handled much more easily using Bode plots. The two curves show how magnitude ratio and phase angle (argument) vary with frequency. 

A commonly used maximum closedloop log modulus specification is 4 2 dB. The controller parameters are adjusted to give a maximum peak in the closedloop servo log modulus curve of -1-2 dB. This corresponds to a magnitude ratio of 1.3 and is approximately equivalent to an underdamped system with a damping coefficient of 0.4,... 

Order-of-magnitude (ratio estimate). Bnle-of-thnmb methods based on cost data from similar-type plants are used. The probable accuracy is —30 percent to -1-50 percent. 

Note further that the magnitude ratio is 1.0 at zero frequency and zero at infinite frequency and that the corresponding phase angles are zero and — 90 degrees respectively. 

As shown by Cohen and Johnson, Eq. (20) leads to frequency response characteristics which on a Bode diagram exhibit resonances both in the magnitude ratio and phase angle. The first resonance occurs at a period approximating the residence time of a slug of water in the inner pipe. 

In general, distributed process systems are characterized by magnitude ratios and phase angles which decrease without limit as frequency increases. If the sinusoidal forcing is applied in a distributed manner, the magnitude ratios and phase angles decrease in a periodic or resonating manner. 

Substituting = jw gives the frequency response characteristics, which are that the magnitude ratio is unity for all frequencies, and the phase lag (negative phase angle) increases with increasing frequency without limit. Since the characteristic of unlimited phase angle promotes system instability, time delays are undesirable and should be minimized whenever possible. 

Since the controller output must counteract the measured variable, these two quantities, 0 and 0O, are of opposite sign and hence are inherently 180° out of phase. In commercial proportional controllers this 180° phase shift and also any set gain Kp, are constant for all practical ranges of frequency. Thus the frequency response characteristics of a proportional controller are a magnitude ratio of Kp and a phase lag of 180°. 

From the Laplace transformation of Eq. (25), the transfer function for an integral controller is l/(s2 nt) and by substituting for s the corresponding frequency response characteristics are found to be a phase angle of —90° and a magnitude ratio of l/coTint. 

Consider the simple control loop of Fig. 2. The open-loop characteristics for a typical system exclusive of the controller, are a magnitude ratio and phase angle which fall off with increasing frequency. When the loop is closed through a controller, the magnitude ratio of the closed loop has a resonance peak if the controller gain is sufficiently great. 

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