over frequency. The closed-loop response then also contains three or more poles near the crossover frequency, and these poles cannot be completely characterized by a single g-factor. Additional work is required to find the behavior of the exact 7/(1 + Л and 1/(1 +7) near the crossover frequency, but nonedieless it can be said that a small phase margin leads to a peaked closed-loop response.

9.4.3 Tran-sient Response VS. Damping Factor

One can solve for the unit-step response of the 77(1 + T) transfer function, by muhiplying Eq. (9.23) by Us and then taking the inverse Laplace transform. The result for Q > 0.5 is

r.., + tan[,/4Q

(9.26)

For Q < 0.5, the result is

>- (1), (Uj-CUj

(9.27)

with

1 + /l-4Q-

These equations are plotted in Fig. 9.14 for various values of Q.

According to Eq. (9.23), when > 4/(j. the -factor is less than 0.5, and the closed-kjop

Cd.ff radians

Fie. 9Л4 Unit-step response of die secondHjrdcr system, Eqs. (У.26) and (9.27), fur various values of Q.

response coniains a low-frequency and a high-frequeiicy real pole. The transient response in this case, Eq. (9.27), contains decaying-ex.poiiential functions of time, ofthe form

/!(,((.<*) (9.2Й)

This is called the overdamped ca.se. With very low Q. the low-frequency pole leads to a slow step response.

F[)i/2 = the й-factor is equal to 0.5. The closed-loop respt)nse contains two real poles at frequency 2/(,. This is called the critically damped case. The transient response is faster than in the over-damped case, because the lowest-frequency pole is at a higher fiequency. This is the fastest response that dtKS not exhibit overshoot. At = Tt radians (/ - XI2f), the voltage has reached 82% ai its final value. At iiij = 2n radians (/ - 1 .), the voltage has reached 98.6% of its final value.

For /2 < 4. the 0-factor is greater than 0.5. The closed-loop response contains ctjmplex poles, and the transient response exhibits sinusoidal-type waveforms with decaying amplitude, Eq. (9.26). The rise time ofthe step response is faster than in the critically-damped ca.se, but the waveforms exhibit overshoot. The peak value of l(() is

poakv(()=l-He- >/Iy fJ

This is called the iinderdamped case. A -factor of 1 leads to an oversho[)t of 16.3%, while a -factor of 2 leads to a 44.4% overshoot. Large -factors lead to overshoots approaching 100%.

The exact transient response of the feedback loop may differ from the plots of Fig. 9.14, because of additional poles and zeroes in T, and because of differences in initial conditions. Nonetheless, Fig. 9.14 illustrates how high-Q poles lead to overshoot and ringing. In most power applications, overshoot is unacceptable. For example, in a 3.3 V computer power supply, the voltage must not be allowed to overshoot to 5 or 6 volts when the supply is turned on-this would likely destroy all of the integrated circuits in the ctmputer! So the Q-fiictor must be sufficiently low, often 0.5 or less, corresponding to a phase margin of at least 76 .

9.5 REGULATOR DESK;N

Lets now consider how to design a regulator system, to meet specifications or design goals regarding rejection of disturbances, transient response, and stability. Typical dc regulator designs are defined using specifications such as the following:

1. £ffcf ofload currem variations on she output voltage regulation. The outpul voltage must remain within a

specified range whea ihe load current varies la ii prescribed way. This amnunts to ii limii on the maximum

magnitude ofthe closed-loop oulpul impedance ofEq. (9.f>), repeated bielow

,.=(1

If, over some frequency range, the open-loop output impedance 2 , has magnitude that exceeds the limit, iheii the loop gain Fmusl be sulficienlly large in magnitude over ihe same frequeiicv range, such lhat the magnitude ofthe closed-lnnp output impedance given in Eq. (.ЗЦ) is less than the given limit.

Effect of input voltage variations (for exatnple, at the second harmonic ofthe ac line frequency) on tiie output voltage regulation. Specific maximum limits are usually placed on the amplitude of variations in the

output voltage at the second harmonic of the ac line frequency (120 Hi or 100 Hi). If we know ihe magnitude of the rectification voltage ripple which appears at the converter input (as v), then we can calculate lhe resulting output voltage ripple (in v) using the closed loop line-to-output transfer funcdon of Eq. (9.5), repeated below

Сф) J j=n 1 + T{s)

(9.31)

Tlte output voltage ripple can be reduced by increasing the magnitude of the loop gain at the ripple frequency. Ill a lypical good design, V is 30 dB or more at 120 Hz, so that the transfer function of Eq. (9.31) is at least an order of magnitude smaller than the open-loop line-to-output transfer function G. Ц

3. Transient response time. When a specified large disturhance occurs, such as a large step change in load current or input voltage, the output voltage may undergo a transient. During this transient, lhe output voltage typically deviates from its specified allowable range. Eventually, the feedback loop operates to return the output voltage within tolerance. The time required to do so is the transient response time; typically, the response time can he shortened by increasing the feedback loop crossover frequency.

4. Overshoot and ringing. As discussed in Section 9.4.3, the amount of overshoot and ringing allowed in the transient response may be limited. Such a specification implies thai lhe phase margin must be sufficiently

large.

Each of these requireraetit.s irapo.se.s constraints on the loop gain T[s). Therefore, the design of the control system involves modifying the loop gain. As illustrated in Fig. 9.2, a compensator network is added for this purpose. Several well-known strategies for design of the compensator transferfunction 0,.(л) are discussed below.

9.5.1 Lead {PD) compensator

This type of compensator transfer function is used to improve the phase margin. A zero is added to the loop gain, at a frequency/, sufficiently far below the crossover frequency /(.isuch that the phase margin of T{s) is increased by the desired amount. The lead compensator is also called a proportional-plus-derivative, or PD, controller-at high frequencies, the zero causes the compensator to differentiate the eiTor signal. It often finds application in systems originally containing a two-pole response. By use of this type of compensator, the bandwidth of the feedback loop (i.e., the crossover frequency f) can be

Fig. 9,15 Magnitude and phase asymptotes of the pd oompetisator transfer function (7, of Eq. (9.32).