Datasheet

AN3180 Sensitivity of zero-ripple current condition
Doc ID 17273 Rev 1 11/39
If the attenuation A is defined as the ratio of the residual ripple di
2
(t)/dt, given by Equation 7
or
8, to the ripple that would be there without the coupled inductor (di
1
(t)/dt =v
1
(t)/L
1
, equal
to the actual ripple on the cancellation winding, as L
1
is unchanged), it is possible to write for
the worst case scenario:
Equation 10
where Δv(t) = v
2
(t)-v
1
(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
In Figure 7 the attenuation A is plotted for different values of the relative voltage mismatch
Δv(t)/v
1
(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 L
2
(1-k
2
).
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)/v
1
(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
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
δ+
Δ
ρ===
)t(dt
)t(d
)t(
L
dt
)t(d
dt
)t(d
1
1
2
1
1
2
v
v(t)
i
v
i
i
A
()
2
2
2
k1
k
1
1
−
δ+
=ρ