AN3180 Datasheet, PDF (11/39 Pages) STMicroelectronics – A 200 W ripple-free input current

AN3180
Sensitivity of zero-ripple current condition
If the attenuation A is defined as the ratio of the residual ripple di₂(t)/dt, given by Equation 7
or 8, to the ripple that would be there without the coupled inductor (di₁(t)/dt =v₁(t)/L₁, equal
to the actual ripple on the cancellation winding, as L₁ is unchanged), it is possible to write for
the worst case scenario:
Equation 10
di₂ (t)
A=
dt
di₁ (t)
=
L₁
v₁(t)
di₂ (t)
dt
=
ρ
⎜⎜⎝⎛
Δv(t)
v₁ (t)
+
δ _{⎟⎟⎠⎞}
dt
where Δv(t) = v₂(t)-v₁(t) is the absolute voltage mismatch, δ = k n_e - 1 is the zero-ripple
condition mismatch (absolute and relative values coincide) and the factor ρ is given by:
Equation 11
ρ
=
1
(1+ δ)²
k²
1− k²
In Figure 7 the attenuation A is plotted for different values of the relative voltage mismatch
Δv(t)/v₁(t) and of the coupling coefficient k, as a function of the zero-ripple condition. From
the inspection of these plots, it is apparent that a low coupling coefficient is essential for a
good attenuation even if the zero-ripple condition is not exactly met. To achieve attenuations
always greater than 10-12 dB even with a tolerance of ±10 % on the value of δ and 10 %
voltage mismatch, the coupling coefficient k must be around 0.7. Lower k values would lead
to a higher insensitivity of the zero-ripple condition due to mismatches but lead to more
turns for the DC winding, which could become an issue in terms of inductor construction.
Note that in case of under-compensation (δ<0) the residual ripple can be higher than the
original value: this is due to a too low value of the “residual” inductance L₂ (1-k²).
Extending these concepts to the smoothing transformer of Figure 3, where the voltage
externally applied to the AC winding is affected by the voltage ripple on the capacitor C_S
(due to its finite capacitance value as well as its ESR), the voltage mismatch Δv(t)/v₁(t) and,
as a result, also the attenuation A, become frequency-dependent. Magnetic flux distribution
modifications with frequency, which also affect δ, are a second-order effect and are
neglected.
References 1 and 2 provide a complete analysis of the smoothing transformer frequency
behavior. Here it is convenient only to summarize the results:
1. The smoothing transformer is capable of a third-order attenuation of current ripple, i.e.
it is equivalent to an inductor combined with an additional LC filter
2. The transfer function generally includes three poles and two zeros; if the zero-ripple
condition is fulfilled (δ=0), the two zeros go to infinity and the smoothing transformer
becomes a third-order all-pole filter
3. Modeling winding resistance and capacitor ESR produces a little damping of the pole-
zero resonances but do not significantly affect the imaginary component of their
locations
4. The effect of the zero-ripple condition mismatch (δ≠0) is to move the zero pair towards
the poles, therefore creating some “notch” frequencies where greater attenuation is
achieved, but degrading the overall attenuation produced at higher frequencies
Doc ID 17273 Rev 1
11/39