The purpose of this lecture is to go through a complete numerical example that illustrates how to use transfer functions of the converter, how to evaluate them and how to understand the origins of dynamics of CMP controlled converters using analytical and simulation tools. The example that we're going to look at is a boost converter that takes 24 V as the input and produces a 60 V as the output across a load of 20 ohms. The inductor employed is 100 microhenries, the filter capacitor is 33 microfarads, the switching frequency is 100 kHz. We assume that the equivalent current-sensing resistance is 0.1 ohms. And when you look at the input and output voltage specs, neglecting losses, the steady-state duty cycle is going to be 0.6. So, the converter is CPM controlled with the senses inductor current compared to the control input vc. As a first step, let's look at the steady-state solution. First of all, we need to decide what artificial ramp to use. We will follow the usual practice and choose ma=m2, in the boost converter the slope m2=(Vout-Vg)/L. So remember, the slope m2 is the slope of the inductor current with a negative sign in front of m2, so it is decaying inductor current during the d prime Ts interval. The amplitude of the artificial ramp corresponding to the selected value of the artificial ramp slope in the CPM controller is simply equal to the slope times the length of the switching period times the scale factor to obtain the voltage amplitude of the artificial ramp. So Rf Ma Ts, multiplying the values for the numerical example gives 0.36 volts for the amplitude of the artificial ramp. The average value of the input current, which in the boost converter is equal to the average value of the inductor current is, assuming negligible losses, equal to the output power divided by the input voltage, that's equal to 7.5 amps. So now given the average value of the inductor current, we can also find out how large should the control input be to produce the desired output. The slope of the inductor current during the DTs subinterval is equal to input voltage divided by L. The inductor current is ramping up at the slope M1. So starting with the average value of the inductor current, which is now known, plus the slope times half of the DTs interval, plus the slope of the artificial ramp, times the length of the DTs interval that gives us the value predicted for the steady state Vc control input V sub c, to produce the desired operating point. Now that we have a complete setup and the steady state solution is known, we can proceed to find the control to output transfer function analytically. And we will do that using first the simple model and then the more accurate model and compare the two. The control to output transfer function is output voltage hat, the small signal perturbation in the output voltage with respect to the small signal perturbation in the control voltage. That ratio can also be written as i sub c hat over vc hat times v hat over ic hat. And the reason I write it in this form is that the front-end part of that can be recognized as 1 over Rf, where Rf is the equivalent current sensing resistance. In the remaining part v hat over ic hat will then be recognized as the control to output transfer function in the reference tables that we have available for the basic converters. So here is the simple model. Shown here on top is the equivalent circuit diagram of the simple model, together with the reference table with parameter values shown for the three basic converters. Of course, we take the boost set of parameters, plug that into the simple model, and compute control to output transfer function, which then for a simple model is actually very simple. The control-to-output transfer function is simply going to be equal to f2 times the parallel combination of r2, C and R with a scale factor of 1 over Rf in front. f2 and r2 come from the table which when plugged into the Gvc expression gives us the final result for the control-to-output transfer function predicted by the simple model in the form of low-frequency gain, the right-half-plane zero, and a single pole. The low-frequency gain expression is shown right here, it's equal to 40 or 32 dB. The lower frequency pole frequency is 1 over 2 pi, one half times RC and that's 482 hertz and right-half-plane zero frequency is at 5.1 kilohertz. So the control to output transfer function predicted by the simple model is a single pole transfer function with a right half plane zero. You can sketch the magnitude response that would look like this, a low frequency pole followed by a right-half-plane zero. This would be fp1, this would be frequency of fz, and the low frequency gain equal to Gvc0 which is 32dB, with a slope of -20dB per decade here. So we will take that as a starting point and compare to a more complete transfer function that we obtained using a more detailed, more accurate, small single model for the CPM control boost converter. Here's the more accurate small-signal model and the solution to that model is what follows. The approach that we discussed in the last lecture, the result of which is the complete set of transfer functions that are obtained right here, that we can use as a reference. Let's evaluate these transfer functions. So for the duty-cycle-control transfer functions from duty-cycle d hat to output voltage v hat or from duty cycle d hat to perturbation in the inductor current we have these two expressions. They share the same denominator with a Q factor that is determined by the operating point in terms of the duty cycle, the steady state value of the duty cycle, and the parameter values of the power stage. That Q factor, not surprisingly, comes out to be relatively large, close to 5. So large value of the Q factor is very typical for the pair of poles in the duty cycle control converters. The center frequency of the pair of poles is around 1 kHz. Similarly, we obtained the low frequency gain and the zero frequency in the Gid transfer function. Now those two transfer functions are input in block diagram form to the complete small signal model for the CPM controlled boost converter. And we are now in position to evaluate the transfer functions for the CPM controlled converter. We will make use of the results from the table, from the reference tables that we have derived earlier. The gain in the CPM modulator and the gain F sub v are shown right here and computed numerically. And then, the transfer functions for control-to-output are obtained in the form that shows a pair of poles and the right half plane zero following the more accurate model derivation. The low frequency gain comes out to be 35.5 or 31dB. The pair of poles has a frequency f sub c equal to 4.2 kilohertz but the Q factor of that pair of poles is just 0.13. And as we discussed earlier, the Q factor of the pair of poles in the CPM controlled converter tends to be very low which means that we can employ the low Q approximation to find the dominant low frequency pole. And also the high-frequency pole in the transfer function, that's what we're going to do next. So here is the Low-Q approximation applied to this transfer function. So in the final form, factored poles are shown in the denominator with the right-half-plane zero in the numerator and a low-frequency gain in front. Numerically, the low frequency gain is 31 dB, the low frequency pole approximated by the product of the center frequency of the pair of poles and the Q factor comes out to be around 542 hertz. The high-frequency pole is very high, at 32 kilohertz. Remember this is a converter operating at 100 kilohertz, so this second high frequency pole comes at a value that is comparable to the switching frequency. And finally, the right-half plane zero is at 5.1 kilohertz, the same location as in the duty cycle controlled converter and the same location as what we have found using the simple model. It is interesting to actually compare the more accurate model to the simple model for the CPM controlled boost converter. The low frequency gains are slightly different because the more accurate model properly takes into account the inductor current ripple and the presence of the artificial ramp. The low frequency pole frequency is also very close. The more accurate model includes the very high frequency dynamic associated with the pole at fp2. The simple model has no predictions at all for the second pole. And finally, the right-half-plane zero is exactly the same in both models. We should also note that the more accurate model is not able to predict the instabilities when the artificial ramp slope is equal to zero and is not very good at predicting frequency responses in cases when the the artificial ramp is very small. But it is very good in practical cases where adequate amount of artificial ramp is added to ensure stability and well behaved operation of the peak current mode controller. Let's employ again average circuit simulation for this boost converter, so here is again, the average circuit model using the CPM sub-circuit in combination with a CCM -DCM1 average switch model. This combination allows us to examine DCM operation of the converter, if so desired. At a particular operating point that we have set up to examine in this lecture, we operate in continuous conduction mode. The DC value of the input voltage, of the control voltage, is set to 1.05, just slightly higher value than what was computed by theoretical steady state analysis. To account for small amount of losses associated with R-sub-L in the circuit, and obtain the desired value of output voltage equal to close to 60 volts. The inductor current is sensed with the resistance 0.1 ohm, and the M1 and M2 slope voltages are set up as we have discussed earlier. So here is the CPM Boost control-to-output transfer function Gvc that's predicted by simulation and you can see the magnitude response in thick line and the phase response in thin line. The phase response showing ultimate phase lag of minus 270 degrees associated with the presence of two poles and the right half-plane zero in the transfer function. You can see that the actual response obtained by simulation matches what we expect based on the more accurate model including low frequency gain, the location of the two poles and the location of the right half plane zero. So our analysis matches the simulation results which means that we can use either the two tools as a design aid in the process of designing CPM control converters. So to conclude, we have design-oriented analytical and simulation tools for peak current mode controlled converters and we have examined those tools in a number of examples. At this point, you can with the boost example that was examined here in detail with all the numerical results shown, you could proceed to design a voltage control loop and then verify DC, transient and frequency responses over ranges of operating points or parameter variations. So all tools that we have seen earlier in simulations of average models can be applied now to CPM controlled converters.