Open Loop Gain

These rules will become even clearer, a little later, when we actually try to work out the open-loop gain of a converter. 

H(s) is the feedback transfer function, and we can see this goes to a summing block (or node) — represented by the circle with an enclosed summation sign. 

Note The summing block is sometimes shown in literature as just a simple circle (nothing enclosed), but sometimes rather confusingly as a circle with a multiplication sign (or x) inside it. Nevertheless, it still is a 

One of the inputs to this summation block is the reference level (the input from the viewpoint of the control system), and the other is the output of the feedback block (i.e. the part of the output being fed back). The output of the summation node is therefore the error signal. 

In general, the plant can receive various disturbances that can affect its output. In a power supply these are essentially the line and load variations. The basic purpose of feedback is to reduce the effect of these disturbances on the output voltage. 

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The schematic and Bode plot for the single-pole method of compensation are given in Figure B-16. At dc it exhibits the full open-loop gain of the op amp, and its gain drops at -20dB/decade from dc. It also has a constant -270 degree phase shift. Any phase shift contributed by the control-to-output characteristic... 

This ean only happen if the open-loop gain eonstant K K is infinite. In praetiee this is not possible and therefore the proportional eontrol system proposed in Figure 4.23 will always produee steady-state errors. These ean be minimized by keeping the open-loop gain eonstant K K as high as possible. 

Maximum value of the open-loop gain constant for the stability of a closed-loop system... 

If a point. vi lies on a loeus, then the value of the open-loop gain eonstant K at that point may be evaluated by using the magnitude eriterion. 

Note that method (b) provides both the erossover value (i.e. the frequeney of oseillation at marginal stability) and the open-loop gain eonstant. 

Value of open-loop gain constant K Applying the magnitude criterion to the above point... 

Gain Margin (GM) The gain margin is the inerease in open-loop gain required when the open-loop phase is —180° to make the elosed-loop system just unstable. 

In general, type zero are unsatisfaetory unless the open-loop gain K ean be raised, without instability, to a suffieiently high value to make 1/(1 + A p) aeeeptably small. Most eontrol systems are type one, the integrator either oeeurring naturally, or deliberately ineluded in the form of integral eontrol aetion, i.e. PI or PID. 

The Nichols chart shown in Figure 6.26 is a rectangular plot of open-loop phase on the x-axis against open-loop modulus (dB) on the jr-axis. M and N contours are superimposed so that open-loop and closed-loop frequency response characteristics can be evaluated simultaneously. Like the Bode diagram, the effect of increasing the open-loop gain constant K is to move the open-loop frequency response locus in the y-direction. The Nichols chart is one of the most useful tools in frequency domain analysis. 

The open-loop transfer function is third-order type 2, and is unstable for all values of open-loop gain K, as can be seen from the Nichols chart in Figure 6.33. From Figure 6.33 it can be seen that the zero modulus crossover occurs at a frequency of 1.9 rad/s, with a phase margin of —21°. A lead compensator should therefore have its maximum phase advance 0m at this frequency. Flowever, inserting the lead compensator in the loop will change (increase) the modulus crossover frequency. 

Figure 6.35 shows the Bode gain and phase for both compensated and uncompensated systems. From Figure 6.35, it can be seen that by reducing the open-loop gain by 5.4dB, the original modulus crossover frequency, where the phase advance is a maximum, can be attained. 

Using MATLAB to design a system, it is possible to superimpose lines of constant ( and ajn on the root locus diagram. It is also possible, using a cursor in the graphics window, to select a point on the locus, and return values for open-loop gain K and closed-loop poles using the command... 

Requires K12K21 = 0. Open-loop gain is the same as the closed-loop gain. The controlled variable (or loop) / is not subject to interaction from other manipulated variables (or other loops). Of course, we know nothing about whether other manipulated variables may interact and affect other controlled variables. Nevertheless, pairing the i-th controlled variable to they-th manipulated variable is desirable. 

The open-loop gain is zero. The manipulated variablej has no effect on the controlled variable i. Of course nij may still influence other... 

No doubt there are interactions from other loops, and from (10-37), some of the process gains must have opposite signs (or act in different directions). When Ay = 0.5, we can interpret that the effect of the interactions is identical to the open-loop gain—recall statement after (10-36). When Ay > 0.5, the interaction is less than the main effect of nij on Cj. However, when Ay < 0.5, the interactive effects predominate and we want to avoid pairing nij with c . 

The overall ability of a power supply to attenuate disturbances at its input is expressed as its PSRR (power supply rejection ratio). In graphs, PSRR is usually plotted as a function of frequency. We will invariably find that the rejection ratio is very low at higher frequencies. One reason for this is that the Bode plot cannot really help because the open-loop gain is very small at these frequencies. The other reason is, even a tiny stray parasitic capacitance (e.g., across the power switch and inductor) presents such a low impedance to noise frequencies (whatever their origin) that almost all the noise present at the input migrates to the output unimpeded. In other words, the power stage attenuation (which we had earlier declared to be Vo/Rin) is also nonexistent for noise (and maybe even ripple) frequencies. The only noise attenuation comes from the LC filter (hopefully). 

The dominant pole of this temperature control system is also determined by the thermal time constant of the microhotplate, which is approximately 20 ms. The open-loop gain of the differential analog architecture (Aql daa) is given by Eq. (5.8) ... 

Using the typical values given, we estimated that the open-loop gain G can be adjusted from 10 to 10. From Eq. (11.23) we find... 

Because usually 1/G < 10 the tip closely follows the contour of the constant tunneling current surface of the sample. The higher the open-loop gain G, or the stronger the feedback, the more accurately the tip follows the con-stant-Zr contour. 

For low-frequency signals, the open-loop gain A is very large, typically 10 10<-. 

The open-loop gain decreases with increasing frequency. 

Figure 7.22 Bode plot of open-loop gain of OA (lines A and B), cell attenuation (lines C and D), and their product (lines E and F). See the text for details.

Consider the control loop shown in Fig. 7.44. Suppose the loop to be broken after the measuring element, and that a sinusoidal forcing function M sin cot is applied to the set point R. Suppose also that the open-loop gain (or amplitude ratio) of the system is unity and that the phase shift xj/ is -180°. Then the output JB from the measuring element (i.e. the system open-loop response) will have the form ... 

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