[Shishuo Knowledge] BUCK power stage circuit frequency domain calculation and simulation

The block diagram of the BUCK circuit power stage circuit, as shown in Figure 1 , includes the parasitic series resistances RL and Rc of the inductor and capacitor.

Figure 1 Power stage block diagram of BUCK circuit

Figure 2 Power stage frequency domain small signal transfer function of BUCK circuit

Assuming that the BUCK circuit operates in the voltage-controlled CCM continuous mode, Figure 2 shows the small signal transfer function model derived based on the switching average method, which expresses the transfer function from the duty cycle to the output (including the modulation part of the PWM controller). Note that the sawtooth wave amplitude Vm here is 1 .

Figure 3 Bode plot to solve equations

Figure 4 Definition of power stage circuit component parameters

Here are the circuit component parameters of the BUCK circuit analyzed in the previous article as the basis for frequency domain analysis.

According to the basic concept of Bode diagram shown in Figure 3 , the amplitude and phase curves of the power stage transfer function are obtained.

Figure 5 Power stage transfer function gain curve

Figure 6 Power stage transfer function phase curve

FIG5 and FIG6 are logarithmic coordinate curves of the Bode diagram obtained based on the small signal transfer function shown in FIG2 . Next , the relevant parameters of this curve are analyzed.

In the low-frequency section, the gain curve is a fixed gain until the double-pole frequency composed of LC . We can find that the low-frequency gain is about 19db .

Figure 7 Power stage transfer function Low frequency gain calculation

This gain is also called DC Gain or PWM gain. The modulation part of the PWM controller has a great influence on the power stage transfer function. They can change the order and characteristics of the power stage transfer function. Therefore, the modulation part of the general PWM controller Will belong to the power stage circuit transfer function part, where it affects the low-frequency gain of the power stage transfer function.

In fact, DC Gain can be directly used in the power stage transfer function, ignoring the inductor parasitic resistance value, making s=0 , that is, the value of DC Gain can be obtained , as shown in Figure 8,

Figure 8 Concept and calculation of DC Gain

According to the equation in Figure 8 above, it can be found that when Vin is 9V and Vramp is 1 , the value is 19.085db .

In the BUCK power stage circuit, from the Bode plot curve, the phase of the curve is from 0C at low frequency to the double pole formed by the output LC filter circuit, and the phase is reduced to -180C. We calculate and obtain its turning frequency as shown in Figure 9 .

Figure 9. Turn frequency of LC filter circuit

In addition, the output ESR and output capacitance produce a zero point, and the phase will further increase from -180C to -90C caused by the previous double pole . This turning frequency can also be calculated by the formula, as shown in Figure 10 .

Figure 10 Output capacitor ESR corner frequency

From the gain curve, it can also be seen that the gain is a fixed gain in the low frequency band, and with the LC turning frequency, the gain drops, and then rises further due to the ESR turning frequency.

Figure 11 Solving for gain crossover frequency and phase margin

By solving the 0 point of the gain curve, we can know that its crossover frequency is 16.35k , and the phase of the power stage transfer function at this time is - 151.62C, which means that the original phase margin is about 28.38C , so subsequent steps are needed Loop compensation gives it sufficient phase margin.

Figure 12 BUCK power stage small signal Bode diagram simulation schematic diagram

The simulation is carried out based on the circuit parameters defined in the first part. Here, a 1V sawtooth wave is used to compare with the given error signal. The BUCK is operated in the CCM voltage control mode. After adding a 10mV small signal disturbance, the Bode diagram simulation is performed. The measurement is from Bode plot of op amp output to output voltage section .

Figure 13 BUCK circuit voltage mode CCM time domain waveform verification

Before the small signal frequency domain simulation, the time domain waveform verification is performed first. As shown in Figure 13 , the output voltage and inductor current waveforms are consistent with expectations.

Figure 14 BUCK circuit power stage transfer function Bode diagram

Figure 14 shows the power stage Bode diagram simulation results. From the measurement results, the crossover frequency of the gain curve is 16.32k and the phase is -150.9C. Therefore, the phase margin is also obtained as 29C , which is consistent with the aforementioned calculation results.

Figure 15 Measured low frequency gain DC Gain

Judging from the measured gain curve, the low-frequency gain is 19.35db , which is basically consistent with the calculated results, as shown in Figure 15.

Figure 16 BUCK circuit output LC filter corner frequency measurement

From Figure 16 , it can be measured that the gain 3db drop point frequency is near 5.6k , which is close to the LC corner frequency calculated previously .

Figure 17 Gain Curve Crossing Slope Measurement 1

From Figure 17 , it can be measured that the gain curve crosses the 0db line at -40db/10 times between 4k and 40k . This is exactly the role of the double pole formed by the LC .

Figure 18 Gain curve slope measurement 2

From Figure 18 , it can be measured that the slope of the gain curve between 40k and 400k is about -20db/10 times the frequency. It can be seen from here that this is the slope of the gain curve after the ESR zero point takes effect and the LC double pole is superimposed. , pay attention to the ESR zero point, which is near 39k according to the previous calculation.

Figure 19 LC filter circuit structure

Figure 19 shows a typical LC filter circuit, which is similar to the output filter of the voltage mode CCM BUCK circuit. The transfer function from input to output is shown in Figure 20 .

Figure 20 LC filter circuit transfer function

Figure 21 Gain curve of LC filter input to output transfer function

Figure 22 Phase curve from input to output of LC filter

From the transfer function in Figure 20 , the gain and phase curves of the LC filter can be obtained, as shown in Figure 21 and Figure 22 respectively. It can be seen that the gain curve is 0db at low frequency , and the output signal will not be attenuated or amplified due to passing through the filter. , which is different from the power stage transfer function of the BUCK circuit, and the turning frequency of the LC and the ESR turning frequency are consistent with the turning frequency of the power stage of the voltage mode CCM of the BUCK circuit .

From the phase curve in Figure 22 , the phase curve is consistent with the phase curve of the BUCK circuit voltage mode CCM .

From a frequency domain perspective, the output filter network of the BUCK circuit can also be regarded as a low-pass filter average network. Since the turning frequency of this filter is about 5k, which is far lower than the switching frequency of the BUCK circuit of 500k, it is applied to the filter The input voltage on the device, that is, the BUCK input voltage that has been chopped by the switch at high frequency, has a high-frequency component that will be filtered out by the LC filter, and the low-frequency component, that is, the DC component, will be output through the LC filter, which is the average DC voltage. .

In summary, through Mathcad calculation of the power stage transfer function of the BUCK circuit, the corresponding Bode plot curve is obtained, and based on this, the ESR corner frequency, LC corner frequency, low-frequency gain, crossover frequency, phase margin, etc. are obtained, and based on the power stage circuit SIMPLIS small-signal loop simulation was performed , and corresponding curve measurements were made based on the simulation results, such as low-frequency gain, crossover frequency, phase margin, and slope of the gain curve, etc., which were basically consistent with the calculation results.

Finally, the main differences between the transfer function of the LC filter and the voltage mode CCM transfer function of the BUCK circuit are briefly analyzed . At the same time, the LC output filter network characteristics of the BUCK circuit are analyzed from the frequency domain perspective, providing a corresponding reference for the subsequent BUCK circuit loop compensation design.