A Comparison of Ridley's Modulator Gain to that of F.D. Tan's Unified Modulator

Introduction

Definitions

S_e, the slope of the external compensation ramp in volts/sec at the comparator input.

S_n, the slope of the sensed current ramp in volts/sec at the comparator input.

T_s, the switching period.

V_ap, the dc voltage applied between the active and passive terminals of the switch element. Also, the voltage that appears across either switch when it is off.

R_i, the effective volts per amp from the sensed current to the comparator input.

L, the inductance of the converter as seen from the point where current is sensed.

Ridley's Modulator Gain Equation

Unified Modulator Gain Equation

The only difference between the two modulator gain equations is in the term:

-V_ap*R_i/(2*L)

which can be shown to represent the up-slope (and/or the down-slope) of the sensed current ramp at 50% duty cycle.

Therefore, the Unified modulator's gain goes to infinity, then changes sign as the duty cycle goes above 50% if there is no slope compensation ramp (S_e). This is shure to cause instablility, which is in agreement (intuitively) with the familiar known fact that the current loop will go unstable at 50% duty cycle without slope compensation. You can also see how slope compensation allows operation at 50% duty cycle or more without going unstable. However, even with slope compensation, the system will still go unstable at some duty cycle greater than 50%. So, you may be wondering, how does this "extra" term come about?

Derivation

By looking at these waveforms, one can see that the state-space averaged inductor current will be:

i_l = i_c - d*T_s*S_e/R_i - d*T_s*S_n/(2*R_i)

where,

i_l is the state-space averaged inductor current

i_c is the commanded inductor current

d is the duty cycle

That is, the inductor current will be less than the commanded value due to the slopes S_e and S_n and the time d*T_s.

Next, note that:

S_n = (1 - d)*V_ap*R_i /L

so that

i_l = i_c - d*T_s*S_e/R_i - d*T_s*(1 - d)*V_ap /(2*L)

Small signal information can be derived by perturbation, which yields

Di_l = Di_c - [S_e/R_i + (1-2*D)*V_ap/(2*L)]*T_s*Dd - [D*(1-D)*T_s/(2*L)]*Dv_ap

(Where capital letters denote DC (steady-state) quantities, and "D" precedes AC (perturbed) quantities.)

Solving for Dd,

Dd = 1/{(S_e + S_n - V_ap*R_i /(2*L))*T_s}*R_i*{(Di_c - Di_l) - [D*(1-D)*T_s/(2*L)]*Dv_ap}

From the above, we can see that the modulator gain,

F_m = Dd/{(Di_c - Di_l)*R_i}

is given by

F_m = 1/{(S_e + S_n - V_ap*R_i /(2*L))*T_s}

This completes the derivation.