With waveform equations derived for the valley points of the inductor current, we now construct the current-loop transfer function for the converter peak-current loop, and when we do, sampling effects appear. The valley-current transfer function can be derived by converting the discrete-time waveform equations to the z-domain by applying the time-shifting theorem:
where z = esTS, a time advance of one cycle, Ts. If you are unfamiliar with the z-domain, we pass through it quickly to the s-domain as a shortcut, using the above theorem and definition of z. The theorem lets us easily convert the discrete-time waveform equations to z-domain equations which are then transformed to the s-domain using the above definition of z. The waveform equation of il(k) from Part 3 is
The transformed quantities using the time-shifting theorem are
When in steady state, δ = D, and after being converted to the z-domain, can be put in the form of the transfer function
The z-domain transfer functions can be converted to the sampled (*) s-domain by substituting
This results in a sampled impulse train waveform -- a sampled il (s) -- denoted by il* (s). In general, from sampling theory,
x(k⋅Ts) are sample points each Ts but they are turned into an impulse train by the exponential.
The sampled, closed-loop incremental transfer function for inductor current for the sampled-loop model based on valley current is then
To recover the sampled transfer function, the sampled TCV is multiplied by the zero-order hold (ZOH) function,
which is the unitless (Ts-normalized) zero-order hold function that converts the sampled transfer function, TCV*(s), to a piecewise-continuous, quantized, or stepped function in time, TCV (s). The step changes in iL that can occur each cycle are the same as the output of a ZOH. Because the waveforms are stepped, the piecewise-continuous functions represent the actual waveforms.
In the time domain, h0(t) is a rectangular unit pulse of width Ts. One cycle of a function, f (t), is turned on by the pulse, then turned off Ts later, thereby forming a gated step having an average value during pulse k of
with a constant value equal to that of the integrated (1/s⋅Ts) impulse. A sampled function f *(t) convolved with rectangular pulse-train integrator h0(t) is converted to
which is not continuous but is the piecewise-continuous per-cycle-average stepped function of the continuous f (t).
The resulting valley-current transfer function for the current loop is a little tricky with the algebra. The extras steps have been included.
is Ridley’s feedback-path transfer function that accounts for loop sampling. In the sampled transfer function, il and C are not continuous and d(s) ≠ d*(s⋅ H0(s) but corresponds instead to the stepped cycle-averaged d (t). Thus the il /d transfer function is a piecewise-continuous (stepped) function. The steps have high-frequency content containing harmonics in the bands outside the Nyquist band.
In time, the stepped f(t) is shifted by –Ts/2 from that of f(t), following from the phase response of H0:
Consequently, the stepped f(t) lags behind f(t) by Ts/2 in time with a phase shift of ω
(–Ts/2) in radians. This is accounted for in frequency-domain loop modeling by use of H0(s) or its associated He(s).
The exponential in the transfer function can be eliminated by reducing it to a rational function using the two-point fit of Tymerski and Ridley,
The two-point fit is derived by solving for coefficients at s = 0 and ωs/2. The linear pole terms add to result in
From the linear pole coefficient it is evident that D < 1/2 for a LHP pole-pair and stability, and this is what is observed in actual converters. The damping is
Thus the damping varies with D.
In the next part, TCV will be decomposed so that the PWM and Gid power stage transfer functions can be recovered.