Fig. S.3S Block diagram having a right half-plane zero transfer fiinetion, as in Eq. (8.32), witli tuj = (0.

magnittide of the gain (х/ч>) is much greater than 1, u , = -(*/С1))Ы;,. The negative sign eanses a phase reversal at high frequency. The implication for the transient response is that the output initially tends in the opposite direction ofthe final value.

We have seen that the control-to-output transfer functions of the boost and buck-boost converters, Fig. 8.36, exhibit RFIP zeroes. Typical transient response wavefortns for a step change in duty cycle are illustrated in Fig. 8.37. For this exatnple, the converter initially operates in equilibriutn, a.t d - 0.4 and d -0.6. Equilibriutn inductor current ij(0, diode current io(/), and output voltage v(/) vt-avefonns are illustrated. The average diode current is

.o/r.

(8.134)

By capacitor charge balance, this average diwie current is equal to the dc load current when the converter operates in equilibriutn. Attime (= f the duty cycle is increased to 0.6. In consequence, tf decreases to 0.4. The average diode current, given by Eq. (8.134), therefore decreases, and the output capacitor begins to discharge. The output voltage magnitude initially decreases as illustrated.

2 о(0

-о I a-4--

с -ZZ. R < V

Fig. S.36 Two basic converters whose CCM control-to-output transfer functions exhibit RHP zeroes: (a) boost, (b) buck-boost.

Fig. 8,37 Wavcfoims of the conveners of Fig, 8,36, for il step re-spoiise In duty cycle. The average diode cuirent and output voltage initially decrease, as predicted by the RlIP zero. Eventually, the inductor current increases, causing the average diode current and the output voltage to iticrea.se.

d = OA

Tlie increased duty cycle causes the inductor current to slowly increase, and hence the average diode current eventually exceeds its original d 0.4 equilibrium value. The output voltage eventually increases in nnagnitude, to the new equilibriunn value corresponding to tf = 0.6.

The presence of a right half-plane zero tends to destabilize wide-bandwidth feedback loops, because during a transient the output initially changes in the wrong direction. The phase margin test for feedback loop stability is discussed in the next chapter; when a RHP zero is present, it is difficult to obtain an adequate phase nnargin in conventional single-loop feedback systems having wide bandwidth. Prediction of the right half-plane zero, and the consequent explanation of why the feedback loops controlling CCM boost and buck-boost converters tend to oscillate, was one of the early successes of averaged converter modeling.

(IRAPHICAL CONSTRUCTION OF IMPEDANCES AND TRANSFER FUNCTIONS

Often, we can draw approximate Bode diagrams by inspection, without large amounts of messy algebra and the inevitable associated algebra mistakes. A great deal of insight can be gained into the operation of the circuit using this method. It becomes clear which ct)mponents dominate the circuit response at various frequencies, and so .suitable approximations become obvious. Analytical expressions for the approximate corner frequencies and asymptotes can be obtained directly. Impedances and transfer functions of quite complicated networks can be constructed. Thus insight can be gained, so that the design engineer

S.3 Graphical Conslmction of hapsdances and Transfer Funaions

can modify the circuit to obtain a desired frequency response.

The graphical construction tnethod, also ktiowti as doitig algebra on the graph, involves use of a few simple rules for combiuitig the magnitude Bode plots of impedances atid transfer futictiotis.

8.3.1 Series Impedances: Addition of Asymptotes

A series connection represents the addition of impedances. If the Bode diagrams of the individual impedance magnitudes are kn[)wn, then the asymptotes of the .4eiies combination are found by simply taking the largest of the individual iuipedauce asymptotes. In many cases, the result is exact. In other ca.ses, such as when the itidividual asymptotes have the same slope, then the result is an approximation; nonetheless, the accuracy of the approximation can be quite good.

Consider the series-connected R-C network of Fig. 8.38. It is desired to construct the magnitude asymptotes of the total series impedance Z(s). where

R lOQ -ЛЛ/

Fig. 8.3S

example.

Series R-C network

(8.135}

Let us first sketch the magnitudes of the individual impedances. The 10 Ii resi.stor has an impedance magnitude of Ю ii => 20 (1ВЙ. This value is independent of frequency, and is given in Fig. 8.39. The capacitor has an impedance magnitude of l/oJC This quantity varies inversely with m, and hence its magnitude Bode plot is a line with slope -20 dB/decade. The line pas.ses through 1 Й 0 dBQ at the angular frequency (Л where

that is, at

(8,136)

(ia)c (i£i)(io*F}

= 10* md/see

(8,137)

-20dBQ

too Hi I kHz to kHz 100 kHz 1 MHz

Fig, H.39 Impedancu mnguitudes < f the individtJal eleraeiits in the netwrrrk of Fig. 8.3S.