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Seemingly Simple Circuits, Part 3: Converter Waveforms

The sampling behavior within the feedback loop of a peak-current-loop converter can be discovered by way of its inductor current waveform, the central waveform of the converter. To find the inductor current, iL (t ), and duty-ratio, δ , which constitute the waveform equations , we depart from pure circuit analysis and apply behavioral analysis instead by assuming that the inductor current has the waveshape of a triangle-wave. With resistance in the circuit, it will actually be an exponential that is close to a ramp. By starting with a waveform common to all the PWM-switch configurations and more, the result is more general than circuit analysis. Its weakness is that because an ideal triangle-wave is assumed for iL , it is not as accurate as circuit simulation, though it accomplishes our objectives in converter modeling. The waveform is shown below.

Using the notational shorthand that x (kTs ) = x (k ) and following standard electrical engineering notation for total and incremental (small-signal) variables, the total current waveform is described by

where up-slope is mU during the on-time, down-slope during the off-time is –mD , and iI is the commanding input to the control loop in the form of a peak current, vI /RS . RS is omitted in this analysis because it is easily reinserted into the loop equations. The index of iI is that of the minimum or valley sample of iL ending cycle k , iL (k ). When input variable il (k) changes, the change affects where the peak of iL occurs, and is the value of the cycle after it changes. The slopes also can change with a change in either input voltage vG or output voltage, vO , so that total mX = Mx + mx . For now, they are held constant. The two rightmost expressions of the waveform equations can be solved for iL ;

Then solving for δ ,

and substituting into the first equation,

The incremental waveform equations for iL and δ are found by taking the differential of the above equations:

The incremental change in input, ii , is set by vVe of the Part 1 buck circuit. It is effectively coincident with the beginning of the change in inductor current, il . If the changes in il and ii each cycle are the same for each variable (il (k – 1) = il (k)), then il = ii . The iL (k ) values in the waveform equations are discrete inductor current sample values at the end of each cycle. These are the minimum or valley points of the waveform cycle, not the average. The current of interest to us is the average current because it is the quantity of current that is most useful in converter performance specification and is the desired output current. For now, the valley current is used for iL and refined later to cycle-average current. The other set of rather simple equations that describe the power stage of the converter are the slope equations . They relate converter inductor current slopes to converter parameters under steady-state operation. On the general assumption that the converter PWM-switch is in steady-state operation and inductor flux is balanced ( Δ λ =L ⋅ Δ iL = 0 over Ts ), the assumptions under which the PWM-switch incremental model was derived, then current slope

For steady-state response, Δ iL = 0 over a switching cycle of period Ts and

This balance is maintained by control, by having a stable current loop. The condition of flux balance over a cycle is

where subscript C denotes the switch common terminal in series with the inductor and vL is the voltage on the output terminal of the inductor. Then it follows from the above equation that for steady-state operation, vC = vL . Flux balance can be expressed as

Expressing the Δ λ s in PWM-switch terminal voltages,

where A and P are respectively the active and passive PWM-switch terminals. Then solving and for steady-state operation, substituting vC = vL ,

Applying the basic PWM-switch relationships,

The off-time inductor voltage corresponds with Δ iL for the cycle. Then

Returning to the slopes and relating them to PWM-switch voltages,

where total

These slopes are held constant as is between switching times of successive cycles (but not over the cycle). For constant slopes, then vOFF = voff = vap , a constant inductor voltage. The waveform equations can be expressed in <> instead of mU and mD by substituting the slope equations;

, The slope and waveform equations are the foundation for what follows, along with control and general circuit theory. To place in greater perspective where this development is going, it is time to introduce the general block diagram as a converter model, shown below.

Peak-Current Loop Converter Block Diagram

Peak-Current Loop Converter Block Diagram

Our focus is on the inner current loop with closed-loop transfer function, Tc . (The voltage control loop wraps around above the bottom loop, along with the effects of varying output and input voltages in Foff .) Included in the outer loop is the output load, ZO which affects voltage-loop behavior. RS is the current-loop sense resistor (times gain HC if a voltage amplifier follows it). In the forward path of the loop, the PWM block is the main part of the controller. The power stage is broken into two blocks, Gid = il /d and Goi = io /il . This is necessary to access iL in the power stage for current-loop feedback. In the next part of this article, the basic equations derived for the inductor current waveform are used to find the loop forward-path (G ) and feedback-path (H ) transfer functions. In doing this, the sampling effect appears.

2 comments on “Seemingly Simple Circuits, Part 3: Converter Waveforms

  1. Davidled
    February 2, 2015

    Based on the figure, the below equation is obtained by simply linear equation. iL = iL(k-1) + mu (t/Ts), iL = iL(k-1) – md(t/Ts) + P, where P could be point of intersection in Y coodinate (iL). I am curious how the first equations were derived in the blog. To obtain ii(k), we might need to know point of intersection in X coordinate (t/Ts) corresponding to ii (k) during cycle (k).   

  2. D Feucht
    February 2, 2015

    DaeJ,

    The problem might simply be the tiny type of the subscripts, where it is hard to distinguish a lower-case l from a lower-case i . The equation you are probably referring to begins with i subl (not i ) of k . If so, then this equation follows from the one for i sub L of k by taking the derivative of it. The last term is constant in i sub L (k ) and disappears from the differentiated result which is the incremental i sub l (k ).

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