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, h 0 (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 fk (t ) 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 h 0 (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 ⋅ H 0 (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 H 0 :
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 H 0 (s ) or its associated H e (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 ω s /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.