Transfer function for optocoupler

Question

I read application report from Texas Instruments:
Practical Feedback Loop Design Considerations for Flyback Converter Using UCC28740 (SLUAA66)

There is a schematic (Figure 4. Optocoupler Frequency Response Analysis Test Circuit) on page 3.
And transfer function is on page 4.

I'm trying to figure out how to get that transfer function.
I know that \$ I_c(s) = CTR * I_d(s) \$ and calculated \$ V_{out}(s) = \frac{R_{pullup}}{(R_{pullup} s C_{opto}+1)} I_c(s) \$
so \$ V_{out}(s) = CTR \frac{R_{pullup}}{(R_{pullup} s C_{opto}+1)} I_d(s) \$
where
\$ I_d(s) \$ - current through diode of optocoupler
\$ I_c(s) \$ - current through transistor/collector of optocoupler

but now I do not know how to continue, how to calculate \$ I_d(s) \$ and so how to get to complete transfer function.

My idea was:
\$ I_1(s) \$ - current through R1 + C1
\$ I_2(s) \$ - current through R2
\$ V_d(s) \$ - voltage of node at the diode anode
V2 is shorted to ground

\$ I_d(s) = I_1(s) - I_2(s) \$
\$ I_1(s) = \frac{V_{in}(s) - V_d(s)}{ R1 + \frac{1}{s C1}} \$
\$ I_2(s) = \frac{V_{d}(s)}{ R2} \$

There is something I am missing. How to continue?

Answer 1

As explained in the comment section, the entire transfer function of the circuit shown in Figure 4 - which is, by the way, a figure I drawn in my seminar from 2009, slide 20 - is irrelevant for the loop stability exercise. What you want to extract is the high-frequency pole from which you infer the collector-emitter parasitic capacitance of the optocoupler. But I understand you want to look at the transfer function yourself and, for the sake of the exercise, I will apply the fast analytical circuits techniques or FACTs as described in my last book on the subject.

In this circuit, the stimulus is the input voltage and the first response is the LED current. Then, this current is brought to the collector via the current transfer ratio (CTR) and the pull-up resistance: the response is the output voltage \$V_{out}\$, observed at the collector. Since capacitor \$C_1\$ blocks the dc component, the 0-Hz gain is zero and we can study the circuit for \$s\$ approaching infinity instead. This gives us a high-frequency gain. Then, we can determine the time constant of this circuit by zeroing the input source and determining the resistance "seen" from the capacitor's connecting terminals:

With these elements in hand, the output voltage is simply the LED current scaled by CTR and \$R_{pullup}\$ with a pole implying \$R_{pullup}\$ and the parasitic capacitance (or the combination including the 1.3-nF capacitor):

If the LED dynamic resistance \$r_d\$ is small enough, the mid-band gain simplifies to \$CTR\frac{R_{pullup}}{R_1}\$.