Magnitude and phase

Tlte value of / can thus be varied in magnitude and phase displacement to suit a particular location of installation or pi otective scheme by introducing suitable R and /Y into the neutral circuit. When the impedance is inductive, the fault current will also be inductive and will offset the ground capacitive current /". In such a grounding, the main purpose is to offset the fault current as much is possible to immunize the system from the ha/ai ds of an arcing ground. This is achieved by providing an inductor coil, also known as an arc suppression coil, of a suitable value in the neutral circuit. 

Script file fig627.m produces the Nichols chart for Example 6.4 when K = 4, as illustrated in Figure 6.27. The command ngrid produces the closed-loop magnitude and phase contours and axis provides user-defined axes. Some versions of MATLAB appear to have problems with the nichols command. 

The current transformers must have identical ratios. The assumption is that all the feeders have voltages which are equal in magnitude and phase. If the feeders are physically close together, it may be possible to use one current... 

Nichols chart Nichols chart is a frequency parametric plot of open-loop function magnitude vs. phase angle. The closed-loop magnitude and phase angle are overlaid as contours. 

We do not need to expand the entire function into partial fractions. The functions Gb G2, etc., are better viewed as simply first and at the most second order functions. In frequency response analysis, we make the s = jto substitution and further write the function in terms of magnitude and phase angle as ... 

Example 8.1. Derive the magnitude and phase lag of the following transfer function ... 

The magnitude and phase angle of these terms with the use of Eq. (8-9) are... 

We will use the time constant form of transfer functions. The magnitude and phase angle of... 

These are sample MATLAB statements to plot the magnitude and phase angle as in Fig. E8.3 ... 

The magnitude and phase angle plots are sort of "upside down" versions of first order lag, with the phase angle increasing from 0° to 90° in the high frequency asymptote. The polar plot, on the other hand, is entirely different. The real part of G(jco) is always 1 and not dependent on frequency. 

From Example 8.5, we know that the magnitude of the dead time function is 1. Combining also with the results in Example 8.2, the magnitude and phase angle of G(jco) are... 

However, the result is immediately obvious if we consider the function as the product of a first order lag and an integrator. Combining the results from Examples 8.2 and 8.7, the magnitude and phase angle are... 

X Example 8.13. Derive the magnitude and phase lag of the transfer functions of phase-lead and phase-lag compensators. In many electromechanical control systems, the controller Gc is built with relatively simple R-C circuits and takes the form of a lead-lag element ... 

Here, z0 and p0 are just two positive numbers. There are obviously two possibilities case (a) z0 > po, and case (b) z0 < p0. Sketch the magnitude and phase lag plots of Gc for both cases. Identify which case is the phase-lead and which case is the phase-lag compensation. What types of classical controllers may phase-lead and phase-lag compensations resemble ... 

Again for illustration purpose, we supposedly have chosen Kc such that KcKvKpKm = 5, and xp is the mixing process time constant. Find, without trial-and-error and without further approximation, the maximum distance L that the photodetector can be placed downstream such that the system remains stable. (There are two ways to get the answer. The idea of using magnitude and phase... 

Frequency methods can give us the relative stability (the gain and phase margins). In addition, we could construct the Bode plot with experimental data using a sinusoidal or pulse input, i.e., the subsequent design does not need a (theoretical) model. If we do have a model, the data can be used to verify the model. However, there are systems which have more than one crossover frequency on the Bode plot (the magnitude and phase lag do not decrease monotonically with frequency), and it would be hard to judge which is the appropriate one with the Bode plot alone. 

In Example 8.12, we used the interacting form of a PID controller. Derive the magnitude and phase angle equations for the ideal non-interacting PID controller. (It is called non-interacting because the three controller modes are simply added together.) See that this function will have the same frequency asymptotes. 

As an option, we can omit the subplot command and put the magnitude and phase plots in individual figures. 

DFT methods are valuable for determining the magnitude and phase of a complex mixture of frequency components simultaneously such as might be encountered in the multiplexed systems for collection of several frequencies. Once the discrete Fourier coefficients have been computed the uncorrected values of m and [Pg.91]

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. 

Now let us look at the Bode plots of some common transfer functions. We have already calculated the magnitudes and phase angles for most of them in the previous section. The job now is to plot them in this new coordinate system. 

One major problem crystallographers have to deal with is the so-called phase problem, which states that of the two components of an irrational Figure (magnitude and phase) only the magnitude can be measured. A technique called molecular replacement is an approach to deal with this problem [131]. 

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