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Seemingly Simple Circuits, Part 4: Current-Mode Converter Transfer Function

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,

where

x (kTs ) 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.

where

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 *(sH 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.

6 comments on “Seemingly Simple Circuits, Part 4: Current-Mode Converter Transfer Function

  1. fasmicro
    January 6, 2015

    With all these complexities in the equation, no wonder people that make biomorphic and neuromorphics chips prefer designing at subthrehold physics where you can get some of these equations easily realized.  It solves the integration problems assciated with BJT while giving all the necessary exponentials

  2. Davidled
    January 7, 2015

    Hardware Chip might include Neuron and Synaptic weight based on learning phase in the specific environment.

  3. samicksha
    January 13, 2015

    Another important and interesting aspect in Biomorphic i saw is Central Pattern Generator chips, it claims to consume less than one microwatt of power and occupying less than 0.4 square millimeters of chip area can generate the basic competence.

  4. djoffe
    January 15, 2015

    I'm enjoying reading “Handbook of Analog Circuit Design”. Is there an errata list available?

     

    Thanks…

     

    Dan

  5. D Feucht
    January 15, 2015

    Dan,

    – not any more off-topic than biomorphic circuits!

    You found a first-edition copy of my analog circuits book. Please send me your email address via the address on my website (see below) and I'll send you one. There are many little errors, and a few embarrassing ones too! In the latest available print edition of the book, at

    https://sci.presswarehouse.com/Books/SearchResults.aspx?str=Feucht

    most of the errors have been corrected, yet I also have a list for that book-set. I estimate that the book is about four time-constants toward being error-free.

    I am also providing an open-source 430-page book (7.5 Mbytes email attachment), Transistor Amplifiers, from my website at

    http://www.innovatia.com

    that complements the ACD book-set. Any further comments, corrections, or questions arising from either book, please address to me via the above website. Thanks.

     

  6. chirshadblog
    January 15, 2015

    @djoffe: Well would be great if you can provide us with some sample work so that can be put into practical sessions and see the value behind it. 

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