And we see that at the frequency that is the desired cross-over frequency,

the magnitude response is well below zero dB,

which really means that our current-loop compensator

will have to provide an additional gain in the loop, so

that we can bring cross-over frequency to ten kilohertz.

The amount of gain needed to do so

can be found very simply by noting that around the cross-over frequency,

which is well above the zero frequency and well above the frequency of the poles,

we have a simple asymptotic behavior of the uncompensated

current-loop gain that can be found as follows.

You see in the expression for uncompensated current control loop,

we have a low frequency gain, zero and a pair of poles.

In the range of frequencies around the crossover frequency,

we are well pass the zero frequency, so

that we can approximate the numerator with just s over w zi.

We are well pass the frequency of the pair of poles.

So in the denominator, we can approximate the transfer

function with just s square over omega_0 square.

So that in this range of frequencies,

Ti(s) is going to behave as Tiu0

times s over omega_zi over s squared

over omega_0 squared and

you see that the result is equal

to Tiu0 omega_0 squared over omega_zi

over s or a constant over s.

And indeed, we see that in the part of the magnitude response,

we have a rolloff at minus 20 dB per decade in that range of frequencies.

And we have indeed,

a simple expression, a constant over s, that represents the behavior of

the uncompensated current-loop gain around the cross-over frequency.

This is summarized right here.

So, asymptotic behavior of the uncompensated loop gain around

the cross-over frequency can be represented as a constant over s.

When we plugin analytical expressions for these three values that

we have in the numerator, we obtain a very simple expression for

that constant, which depends only on the gain and the pulse-width modulator,

the equivalent current sensing resistance Rf,

the dc value of the output voltage, and the value of the inductance L.