Electrical – Why does duty cycle seem relevant when calculating inductor current in boost converter circuit

You are not calculating what you think you are. You would normally use definite integrals to see the work done over a time period. An indefinite integral doesn't really make much physical sense.

The original equation of the energy in an inductor can be derived as the integral of the power needed to go from a 0 current to the final current I in some time T:

\$ Energy Stored = \int^T_0{P\cdot dt} = \int^I_0{L i'di'} = \frac{1}{2}LI^2\$

There is an important lesson here. This is solvable with general variables. This is possible because it doesn't matter how you get to a current of I, the same energy must be invested to get there.

If you go from 0A to 3A to 2A to 5A in some crazy complicated fashion, the expended energy will be the same as if you went linearly from 0A to 5A in 1 second.

If this weren't true, then it wouldn't make sense to talk about the energy stored in an inductor "at a particular current." You would have to know how you got there to know the answer.

With a boost regulator, although the same characteristic inductor equation is used, it is used in a different context and a different way than in the buck. In a boost, everything is kind of backward from the buck. For example in the buck, output ripple current is important because it's the dominant determinant of output ripple voltage. But, in the boost the inductor ripple current shows up on the input rather that the output. In fact, for part of the cycle, the inductor is not connected to the output and the capacitor alone must provide energy to the load, so the capacitor is the dominant determinant of output voltage ripple.

Another difference with the boost is the extreme change in circuit dynamics between the discontinuous conduction mode (DCM) and continuous conduction mode (CCM) of the inductor. With DCM you effectively end up with 1 pole to compensate, until the Nyquist frequency shows up. DCM mode is basically stable by itself. While with CCM you have 2 complex high Q poles, and a wandering right half plane zero (RFPZ)to contend with.

Because of the easier control dynamics, DCM is often preferred. The problem with DCM is that for a given power level, peak currents (switch, inductor, and capacitor) are higher. So, DCM is usually limited to lower powers, like under ~20W, with notable exceptions.

It is very important to choose DCM or CCM operation. You especially do not want to design for DCM operation, and then have it wander into CCM because that would be a stability nightmare.

The first step in design of a boost is finding the critical inductance or current, which defines the boundary between DCM and CCM. For that I'll use another equation that doesn't include duty cycle:

\$L_c\$ = \$\frac{V_{\text{out}}}{16 I_{\text{crit}} f_s}\$

In this case with \$V_{\text{out}}\$ = 12V, \$I_{\text{out}}\$ = 0.1A, and \$f_s\$ = 750kHz, \$L_c\$ would be \$10 \text{$\mu $H}\$. So, for DCM operation you wouldn't want an inductor value greater than \$10 \text{$\mu $H}\$. In fact you wouldn't want to get near that, so \$3.3 \text{$\mu $H}\$ or \$4.7 \text{$\mu $H}\$ something like that. Less is OK, but not more.

The MCP1650 is, from my 5 minute read of the datasheet, a hysteretic controller. These types of controllers don't use linear feedback techniques, but instead put out a fixed width pulse where either the frequency or pulse count is varied to regulate the output. When the output voltage gets low, the control will deliver a few pulses to raise back to regulation. So, you don't have to worry about compensating an error amplifier. But, because of the kick of fixed width pulses, hysteretic control is not the friend of high Q power modulators, like a boost in CCM can be. This type of control tends to be prone to ringing at the high Q frequency. For DCM, boost power modulator Q is ~0.5, and so doesn't tend to ring. That may be why the datasheet is tilted to DCM operation.