4s) h

i5, , j(i; sC

(18,105)

(18.106)

In the ca.se of the NLC eonlroller, the parallel eombinalion /-jll/? beconies equal to Г2/2, and Eqs. (18.102) and (18.103) continue lo apply.

18.5 RMS VALUES OF RECTIFIER WAVEFORMS

To correclly specify the power stage elemenls of a near-ideal rectifier, il is necessary to compute Ihe root-mean-square values of their currents. A typical wavefomi such as the transistor current of the boost converter (Fig. 18.31) is pulse-width modulated, with botlt the duty cycle and the peak amplitude varying with the ac input voltage. When the switching frequency is much larger than the ac line frequency, then therms valtie cati be well-approximated as a double ititegral. The sqttare of the cttrrent is integrated first 10 find its average over a switching period, and the result is then integrated to fitid the average over the ac line period.

Computation of the rms and average values of the waveforms of a PWM rectifier can be quite tedious, and this can impede the effective design of the power stage components. In this section, several approximations are developed, which allow relatively simple analytical expressions to be written forthe rms and average values of the power stage currents, and which allow comparison of converter approaches [14,41]. The transistor current in the boost rectifier is found to be quite low.

The rms value of the transistor current is defined as

fyMt

(18.107)

where Г . is the period ofthe ac line waveform. The integral can be expressed as a sum of integrals over all ofthe switching periods contained in one ac hne period:

(18.108)

Fig. 18.31 Modulated transistor current waveform, boost rectifier.

where Г, is lhe switching period. The quatitily itiside lhe paretitheses is the value of averaged over the n* switching period. The summation can be approximated by an integral in the case when Г, is much less lhan ..This approximation coiresponds to taking the limit as tends to zero, as follows:

(18.10?)

So j(j(() is first averaged over one switching period. The result is then averaged over the ac line period, and the square root is taken of lhe result.

18.5.1 Boost Rectifier Exuimple

For the boost reciifier, the transistor current (y(f) is equal to the input current when the transistor conducts, and is zero when the transistor is off. Therefore, the average of jy(f) over one switching period is

(18.(10)

If the input voltage is given by

then the input current will be

= sin (o(

(18.111)

(18.112)

where is the emulated resistance. With a constant output voltage V, the iransistor duly cycle must obey lhe relationship

V I vj.t) 1 - d(t)

(18.113)

This assumes that the converter dynamics are fast compared to the ac line frequency. Stibstituiion of Eq. (18.111) inio(i8.ii3) and soluiion for f/(f) yields

rf(/)=l--sinu)( (ISllt)

Substitution of Eqs. ([8.112) and (18. [ (4) into Ее]. ([8.110) yields the following expression

One can now plug this expression into Eq. (1S.109):

KJn (Ш(>

(18.115)

(18.116)

:--Jsinw/t

sin(mf)rf

wllich tan be furlher simplilied to

(18.117)

sin(fflr)-sin:a r)

This involves integration of powers of sin(tO/) over a complete Table 18,2 .Solution ofthe integral of half-cycle. Ttie integral tan be evaluated with the help of the lot- Eq. (18.118). for seveial values ofrj lowini; formula:

2 24-6-( -l) ...

1-3-5...(h-1) , . 24-6---Г!

(18.118)

This type of integral commonly arises in rras calculations involving PWIVI rectifiers. The values ofthe integral for several choices of л are listed in Table 18.2. Evaluation oflhe integral in Eq. (18.117) using Eq. (18.118) leads to the following result:

/ -i JZlM.\i ./TT-S-b (1Й.119)

It can be seen thai Ihe rras transistor current is minimized by choosing the output voltage V to be as small as possible. The best that can be done is to choose K= which leads to

Larger values of V lead to a larger rms Iransistor current.

A similar analysis for the ims diode current leads to the following expression

f> .i - in.- rar.i V Зл V

The choice V =Vf maximizes the rras diode current, with the result

(18.120)

(18,121)