9.2.2.10.1 Power Stage Gain, Zeroes, and Poles

The first step in compensating a fixed-frequency flyback is to verify if the converter operates in continuous conduction mode (CCM) or discontinuous conduction mode (DCM). If the primary inductance (L_P) is greater than the inductance for DCM-CCM boundary mode operation, called the critical inductance (L_Pcrit), then the converter operates in CCM. L_Pcrit is calculated with Equation 17.

Equation 17.

For loads greater than 10% of P_MAX over the entire input voltage range, the selected primary inductance has value larger than the critical inductance. Therefore, the converter operates in CCM and the compensation loop requires design based on CCM flyback equations.

The current-to-voltage conversion is done externally with the ground-referenced current-sense resistor (R_CS) and the internal resistor divider sets up the internal current-sense gain, A_CS = 1.65. The IC technology allows tight control of the resistor-divider ratio, regardless of the actual resistor value variations.

The DC open-loop gain (G_O) of the fixed-frequency voltage control loop of a peak-current-mode control CCM flyback converter shown in Figure 33 is approximated by first using the output load (R_OUT), the primary to secondary turns ratio (N_PS), and the maximum duty cycle (D) as shown in Equation 18.

Equation 18.

where

• R_OUT = V_OUT / I_OUT
• D is calculated with Equation 19
• τ_L is calculated with Equation 20
• M is calculated with Equation 21

Equation 19.

Equation 20.

Equation 21.

For this design, a converter with an output voltage (V_OUT) of 12 V, and 48 W relates to an output load (R_OUT) equal to 3 Ω at full load.

At minimum input bulk voltage of 75 V DC, the duty cycle reaches its maximum value of 0.615. The current sense resistance (R_CS) is 0.75 Ω and a primary to secondary turns-ratio (N_PS) is 10. The open-loop gain calculates to 14.95 dB.

A CCM flyback transfer function has two zeroes that are of interest. The ESR and the output capacitance contribute a left-half plane zero to the power stage, and the frequency of this zero (f_ESRz) is calculated with Equation 22.

Equation 22.

The f_ESRz zero for a capacitance bank of three 680-µF capacitors (for a total output capacitance of 2040 µF) and a total ESR of 13 mΩ is located at 6 kHz.

CCM flyback converters have a zero in the right-half plane (RHP) of their transfer function. RHP zero has the same 20 dB/decade rising gain magnitude with increasing frequency just like a left-half plane zero, but it adds phase lag instead of lead. This phase lag tends to limit the overall loop bandwidth. The frequency location (f_RHPz) in Equation 23 is a function of the output load, the duty cycle, the primary inductance (L_P), and the primary to secondary side turns ratio (N_PS).

Equation 23.

RHP zero frequency increases with higher input voltage and lighter load. Generally, the design requires consideration of the worst case of the lowest RHP zero frequency and the converter must be compensated at the minimum input and maximum load condition. With a primary inductance of 1.5 mH, at 75-V DC input, the RHP zero frequency (f_RHPz) is equal to 7.65 kHz at maximum duty cycle (full load).

The power stage has one dominant pole (ω_P1) which is in the region of interest, located at a lower frequency (f_P1) which is related to the duty cycle (D), the output load, and the output capacitance. There is also a double pole (f_P2) located at half the switching frequency of the converter. These poles are frequencies calculated with Equation 24 and Equation 25.

Equation 24.

Equation 25.

Subharmonic oscillation is the large signal instability that can occur in CCM flyback converters when duty cycles extend beyond 50%. The subharmonic oscillation increases the output voltage ripple and sometimes it even limits the power handling capability of the converter. Slope compensation to the CS signal is a technique used to eliminate the instability.

Ideally, the target of slope compensation is to achieve quality coefficient (Q_P = 1) at half of the switching frequency. The Q_P is calculated by Equation 26.

Equation 26.

where

• D is the primary side switch duty cycle
• M_C is the slope compensation factor, which is defined by Equation 27

Equation 27.

where

• S_e is the compensation ramp slope
• S_n represents the rising current slope of the transformer primary inductance

The optimal goal of the slope compensation is to achieve Q_P equal to 1, which means M_C must be 2.128 when D reaches it maximum value of 0.615.

The inductance current slope at the CS pin is calculated by Equation 28.

Equation 28.

The compensation slope is calculated by Equation 29.

Equation 29.

The compensation slope is added into the system through R_RAMP and R_CSF. A series capacitor (C_RAMP) is selected to approximate a high-frequency short circuit. Choose C_RAMP as 10 nF as the starting point, and make adjustments if required. R_RAMP and R_CSF form a voltage divider to scale the RC pin ramp voltage and inject the slope compensation into CS pin. Choose R_RAMP much larger than the R_T resistor so that it does not affect the frequency setting very much. In this design, R_RAMP is selected as 24.9 kΩ. The RC pin ramp slope is calculated with Equation 30.

Equation 30.

To achieve 46.3 mV/µs compensation slope, R_CSF resistor is calculated with Equation 31.

Equation 31.

The power stage open-loop gain and phase can be plotted as a function of frequency. The total open-loop transfer function, as a function of frequency, can be characterized by Equation 32.

Equation 32.

where

• ω_P1 and ω_P2 are based on the frequencies calculated by Equation 24 and Equation 25

The open-loop gain and phase Bode plots are graphed accordingly (see Figure 34 and Figure 35).

Figure 34. Converter Open-Loop Bode Plot: Gain

Figure 35. Converter Open-Loop Bode Plot: Phase