Check my calculations and Bode plot

I am trying to recreate the Bode plot and phase margin calculations of ST Application Note 2823 "Current source for LED driving based on the L5973AD".

• The analysis there (p17) gives: F_C as 36 kHz and ζ_M as 84°
• My calculations produce: F_C as 36110.825275 and ζ_M as 82.552834°

My question: 1) have I done this correctly? 2) How do I evaluate if these values are "good"?

This is the circuit (p7), but understood as L5973AD (datasheet), with f_sw of 500 kHz.

(image from application note given above)

Following the main formula (eq. 9) I'm building it up in SymPy and substituting values right at the end.

$$ G_{loop} = A_{ota} \cdot G_{pwm} \cdot G_{pwr} \cdot \alpha_{led} \tag{9} $$

We do each part: $$ A_{ota} = \frac{G_{m}}{C_{5} \cdot s + C_{EA} \cdot s + \frac{1}{R_{5} + \frac{1}{C_{4} \cdot s}} + \frac{1}{R_{o}}} \tag{15} $$

$$ G_{pwr} = \frac{1}{\left(\frac{1}{R_{led} + R_{sense}} + \frac{1}{ESR + \frac{1}{C_{out} \cdot s}}\right) \cdot \left(DCR + L \cdot s + \frac{1}{\frac{1}{R_{led} + R_{sense}} + \frac{1}{ESR + \frac{1}{C_{out} \cdot s}}}\right)} \tag{14} $$ $$ \alpha_{led} = \frac{R_{1} \cdot R_{sense}}{\left(R_{1} + R_{6}\right) \cdot \left(R_{led} + R_{sense}\right)} \tag{13} $$

Substituting in (9) we get $$ G_{loop} = \frac{G_{m} \cdot G_{pwm} \cdot R_{1} \cdot R_{sense}}{\left(R_{1} + R_{6}\right) \cdot \left(R_{led} + R_{sense}\right) \cdot \left(\frac{1}{R_{led} + R_{sense}} + \frac{1}{ESR + \frac{1}{C_{out} \cdot s}}\right) \cdot \left(DCR + L \cdot s + \frac{1}{\frac{1}{R_{led} + R_{sense}} + \frac{1}{ESR + \frac{1}{C_{out} \cdot s}}}\right) \cdot \left(C_{5} \cdot s + C_{EA} \cdot s + \frac{1}{R_{5} + \frac{1}{C_{4} \cdot s}} + \frac{1}{R_{o}}\right)} \tag{99} $$

Sympy simplify gives

$$ G_{loop} = \frac{G_{m} \cdot G_{pwm} \cdot R_{1} \cdot R_{o} \cdot R_{sense} \cdot \left(C_{4} \cdot R_{5} \cdot s + 1\right) \cdot \left(C_{out} \cdot ESR \cdot s + 1\right)}{\left(R_{1} + R_{6}\right) \cdot \left(\left(DCR + L \cdot s\right) \cdot \left(C_{out} \cdot ESR \cdot s + C_{out} \cdot s \cdot \left(R_{led} + R_{sense}\right) + 1\right) + \left(R_{led} + R_{sense}\right) \cdot \left(C_{out} \cdot ESR \cdot s + 1\right)\right) \cdot \left(C_{4} \cdot R_{5} \cdot s + C_{4} \cdot R_{o} \cdot s + R_{o} \cdot s \cdot \left(C_{5} + C_{EA}\right) \cdot \left(C_{4} \cdot R_{5} \cdot s + 1\right) + 1\right)} \tag{9} $$

Constant values

# only values are given in fig 7 Substitute C4 = 6.8e-08 Substitute C5 = 3.3e-11 Substitute R5 = 330.0 Substitute C_EA = 3e-12 (p16, in eq 17) Substitute C_out = 1e-07 (p 15 in eq 14) Substitute DCR = 0.155 (not given, typ value guessed) Substitute ESR = 0.1 (not given, typ value guessed) Substitute G_m = 0.0023 (p16, in eq 17) Substitute G_pwm = 26.3 (p13, eq 10) Substitute L = 0.0001 (p15 in eq 14) Substitute R1 = 20000.0 (p8 eq 5) Substitute R6 = 6800.0 (p8 eq 5) Substitute R_led = 1.3 (p13, eq 11) Substitute R_o = 800000.0 (p16, in eq 17) Substitute R_sense = 1.5 (p8 eq 4) 

Now we have the single remaining s: $$ G_{loop} = \frac{54170.1492537313 \cdot \left(1.0 \cdot 10^{-8} \cdot s + 1\right) \cdot \left(2.244 \cdot 10^{-5} \cdot s + 1\right)}{\left(2.8 \cdot 10^{-8} \cdot s + \left(2.9 \cdot 10^{-7} \cdot s + 1\right) \cdot \left(0.0001 \cdot s + 0.155\right) + 2.8\right) \cdot \left(2.88 \cdot 10^{-5} \cdot s \cdot \left(2.244 \cdot 10^{-5} \cdot s + 1\right) + 0.05442244 \cdot s + 1\right)} $$

We extract the polynomial coefficients of s:

num, den = G_loop.as_numer_denom() numf = list(map(float, num.as_poly(s).coeffs())) denf = list(map(float, den.as_poly(s).coeffs())) 

And using Python's Control library doc

G = ct.tf(numf, denf) gm, pm, _, _ = ct.margin(G) 

G is this, polynomials of s:

 1.216e-08 s^2 + 1.216 s + 5.417e+04 --------------------------------------------------------------- 1.874e-20 s^4 + 1.644e-12 s^3 + 5.451e-06 s^2 + 0.161 s + 2.955 

Gain and phase margin:

gain margin 3143.446534 phase margin 82.552834 

Further functions give a frequency margin of 36110.825275 and a (visually) matching Bode plot.

Bode plot actually made with sympy.physics.control.lti.TransferFunction and .bode_plot