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Abstract: As is well known, any closed-loop system at unity gain
1 Stability Metrics The metrics for measuring switching power supply stability are phase margin and gain margin. Phase margin refers to the phase corresponding to the point where the gain drops to0dB. Gain margin refers to the gain (actually attenuation) corresponding to the point where the phase is zero(. In actual switching power supply design, gain margin is only considered when designing flyback converters; when designing other converters, gain margin is generally not used. In switching power supply design, phase margin has two independent functions: one is to damp the dynamic process that occurs when the converter experiences a load step change; the other is to ensure system stability even when component parameters change. Phase margin can only ensure"small-signal stability."When a load step change occurs, the power supply inevitably enters the"large-signal stability"range. In engineering practice, we consider that at room temperature, standard input, and normal load conditions, the loop phase margin should be greater than45°. Under various parameter variations and error conditions, this phase margin is sufficient to ensure system stability. If the load variation or input voltage range is very large, consider that the loop phase margin should be greater than30°across all load and input voltage conditions. As shown in Figurel, which is a block diagram of a switching power supply control scheme, the switching power supply control loop consists of the following3parts.
(1)Power converter section, mainly including the square-wave-driven power switch, main power transformer, and output filter; (2)Pulse width modulation section, mainly including thePWMpulse width comparator and power amplification; (3)Sampling and control comparison amplifier section, mainly including output voltage sampling, comparison, amplification(such asTL431), error amplifier transmission(such as optocoupler), and the voltage comparator integrated inside thePWMIC(the compensation design of these amplifiers determines to the greatest extent the stability of the switching power supply system, and is the key and difficult point of the design).
2 Stability Analysis As shown in Figure1, suppose a disturbance wave is injected at nodeA. The energy contained in this square wave is distributed into an infinite series of odd harmonic components. If the response of the real system to ever-increasing harmonics is detected, it can be seen that the gain and phase shift also change with increasing frequency. If at a certain frequency the gain equalsland the total additional phase shift is180°(this phase shift plus the originally set180°phase shift gives a total phase shift of360°), then sufficient energy will return to the system input with the same phase as the original, and the disturbance will be sustained, causing the system to oscillate at this frequency. As shown in Figure2, under normal circumstances, the control amplifier will use feedback compensation componentsZ2to reduce the gain at higher frequencies, so that the switching power supply remains stable at all frequencies.
The Bode plot corresponds to the system response under small-signal(theoretically, a small signal is infinitely small)disturbances; however, if the disturbance is large, the system response may not be determined by the linear portion of the feedback, but may be determined by nonlinear portions, such as the slew rate and gain bandwidth of the op-amp, or the minimum and maximum duty cycles that can be reached in the circuit. When these factors affect the system response, the original system will behave nonlinearly, and the transfer function method can no longer be used. Therefore, although small-signal stability must be satisfied, it is not sufficient to ensure stable operation of the power supply. Therefore, when designing the power supply loop compensation, not only the response characteristics of the signal power supply system must be considered, but also the large-signal response characteristics of the power supply system must be handled properly. The quality of the large-signal response characteristics of the power supply system can be judged by the load step response characteristics and the input voltage step response characteristics. The load step response characteristics and the input voltage step response characteristics have a strong coupling relationship; if the load step response characteristics are good, then the input voltage step response characteristics will certainly be good. The theoretical analysis of the stability criteria for switching power supply loops is very complex because the transfer function changes with load conditions. The effective inductance values of various wire-wound power components usually change with load current. In addition, when considering large-signal transients, the control circuit operating mode changes to a nonlinear operating mode, and linear analysis alone will not provide a complete state description. The following provides a detailed introduction to judging switching power supply loop stability by analyzing the load step transient response waveform.
3 Stability Testing Test conditions: (1)non-inductive resistor; (2)load variation range of10%~100%; (3)adjustable load switching frequency(under the condition of obtaining the same ideal response waveform, the higher the switching frequency, the better); (4)limited load switch current slew rate of5A/μsor2A/μs; a transient response curve without a declared load current magnitude and slew rate is meaningless. Figure3(a)shows the transient load waveform. Figure3(b)shows a damped response, where the control loop oscillates after the transient edge. This indicates that a power supply with this response has very small gain margin and phase margin, and can only be stable under certain specific conditions. Therefore, this type of response should be avoided as much as possible, and the compensation network should be adjusted to roll off at a slightly lower frequency.
Figure3(c)shows an overdamped response, which, although relatively stable, does not have the best transient recovery performance. The roll-off frequency should be increased. Figure3(d)shows an ideal response waveform, close to the optimal case; in the vast majority of applications, the transient response is stable with excellent performance and sufficient gain margin and phase margin. For positive and negative spikes, a symmetrical waveform is also required, so it can be seen that the control section and the power supply section have a center line within the control, and their slew rates are the same when the load increases and decreases. The above introduced two stability criteria for switching power supply control loops, namely judging the phase margin of the switching power supply control loop under small signals through the Bode plot, and judging the stability of the switching power supply control loop under large signals through the load step transient response waveform. The following introduces four design methods for control loop stability.
4 Stability Design Methods 4.1 Analytical Method Analysis is performed based on closed-loop system theory, mathematics, and circuit models(computer simulation). In practice, when performing overall analysis, it is unlikely that all parameters can be precisely equal to the specified values, especially the inductance value; over the entire current variation range, the inductance value cannot remain constant. Similarly, the larger transient responses that can change the system's linear operation are also difficult to predict. 4.2 Trial-and-Error Method First, measure the transfer characteristics of the pulse width modulator and power converter sections well, then use"difference techniques"to determine the characteristics that the compensation control amplifier must have. It is unlikely that the actual amplifier can fully satisfy the optimal characteristics; the main goal is to achieve as close an approximation as possible. The specific steps are as follows: (1)Find the frequency corresponding to the zero crossing of the pole in the open-loop curve, and introduce a zero around the corresponding frequency in the compensation network; then, in the range up to the crossover frequency, the phase shift is less than315°(phase margin at least45°); (2)Find the frequency corresponding to theEsRzero in the open-loop curve, and introduce a pole around the corresponding frequency in the compensation network(otherwise these zeros will flatten the gain characteristic and it cannot decrease as desired); (3)If the low-frequency gain is too low to obtain the desired DC correction, then a pair of zero-pole can be introduced to increase the gain at low frequencies. In most cases,"fine-tuning"is required, and the best approach is to use the transient load measurement method. 4. 3 Empirical Method??The control loop uses an overcompensated control amplifier with a low-frequency dominant pole to form a closed loop to obtain initial stability. Then, the transient pulse load method is used to dynamically optimize the compensation network. This method is fast and effective. Its disadvantage is that the optimum performance cannot be determined. 4.4 Combined Calculation and Measurement Method Combining the above three points, it mainly depends on the skill and experience of the designer. For a power supply designed using the above methods, the Bode plot of the closed-loop switching power supply system can be measured using the following method. The measurement steps are as follows. As shown in Figure4, this is a commonly used method for measuring the gain and phase of the Bode plot of a closed-loop power supply system. The feature of this method is that no modification to the original circuit is required.
As shown in Figure4, the oscillator introduces a very small series voltageV3into the loop through transformerT1. The effective AC voltage flowing into the control amplifier is measured by voltmeterV1, and the AC voltage at the output is measured by voltmeterV2(capacitorsC1andC2serve to block DC current).V2/V1(in decibel form)is the voltage gain of the system. The phase difference is the phase shift of the entire loop(after accounting for the fixed180°negative feedback inversion)
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