Switching Power Supply Interest Group 15th Task
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Question 14
: 1. Figure (06) is a typical application circuit in the UC1525A manual. Transformer T1 has two secondary windings, which are used to drive a half-bridge circuit. If a full-bridge circuit is to be driven, the transformer should obviously have four mutually insulated secondary windings. What should be the phase relationship of these four secondary windings? In a
normal full-bridge circuit, the two power switches on the diagonal should be turned on and off at the same time, while the two power switches on the other diagonal should be turned on and off in opposite phases. Therefore, when using transformer coupling to drive a full-bridge, the four windings of the transformer secondary that drive the two tubes on one diagonal should have the same phase, while the windings that drive the two tubes on the other diagonal should have opposite phases.
2. Figure (08) is a typical application circuit in the IR2104 chip manual. This chip is used to drive a half-bridge circuit. Can two IR2104 chips, with their signal input terminals IN connected together, drive a full-bridge circuit consisting of four power MOS tubes?
No. This connection will only make the two upper tubes on both sides of the full bridge turn on and off at the same time, and the two lower tubes turn on and off at the same time, but the phase is opposite to that of the upper tubes. The four power switch tubes will not be damaged, but the full bridge circuit has no output. For the
15th activity, please read the first three sections of Chapter 12 "Stabilization of the Feedback Loop" in "Switching Power Supply Design Third Edition", namely Section 12.1 "Introduction", Section 12.2 "System Oscillation Principle" and Section 12.3 "Design of Error Amplifier Amplitude-Frequency Characteristic Curve".
The stable design of the feedback loop is a more difficult part, and it is worth spending more time to learn.
The various components in the switching power supply, including power switch tubes and diodes, etc., all work in a very extreme nonlinear state, either on or off. The small signal analysis method we learned in the analog circuit course is completely useless for the power switch tubes and diodes in the switching power supply.
However, the switching power supply usually requires a stable voltage output, that is, the output DC voltage does not change with the changes in the AC mains and the load. Some switching power supplies require a stable output current, such as the power supply for LEDs. The switching power supply achieves output stability by means of negative feedback, which must contain a linear circuit. If the negative feedback of the switching power supply is not designed well, its output voltage or current will produce strong oscillations, that is, the output voltage rises to a very high level all of a sudden, and then drops to a very low level all of a sudden, which may burn the load or even the switching power supply itself.
The "Switching Power Supply Design 3rd Edition" discusses the typical closed-loop feedback loop of the switching power supply, and Figure (01) in this article is copied from Figure 12.1 of the book. However, I am afraid that most netizens who want to learn about switching power supplies do not understand this loop, so they cannot understand the compensation measures mentioned later in the book.
Figure (01)
Let's look at Figure (02). Generally speaking, a feedback loop is composed of multiple links such as A1, A2...An, as shown in Figure (02), and forms a closed loop. The output signal of each link is the input signal of the next level. At any point in the closed loop. For example, at point B in Figure (02), the loop is disconnected, and the corresponding impedance is connected as a load and a signal source with internal resistance at the disconnection point, which constitutes the open loop corresponding to this closed loop. For Figure (02), it is necessary to connect a load equal to the input impedance of link A2 on the left side of point B, and at the same time connect a signal source with an output impedance equivalent to A1 on the right side of point B.
So, what is transmitted between the links in Figure (02)? Or what do the arrows in Figure (02) represent? Signals are transmitted between the links. The ratio of the output signal of each link to the input signal includes both the change in amplitude and the change in phase. Generally speaking, this ratio is a complex number, and this ratio is a function of frequency and changes with the change in frequency.
So, what is a link? Let's take an example with Figure (01). In Figure (01), the voltage divider circuit composed of resistors R1 and R2 can be regarded as a link. This link only divides the output voltage Vo of the switching power supply to obtain another signal Vs, which changes the amplitude of the input signal but does not change the phase of the signal. The error amplifier EA in Figure (01) can also be regarded as a link, although the circuit of an op amp is much more complicated than the voltage divider of two resistors R1 and R2. The error amplifier EA may change the amplitude of the signal or the phase of the signal, which depends on the specific amplifier circuit. As for the PWM modulation circuit and base drive circuit in Figure (01), as well as the forward conversion circuit together with the transformer and rectifier, it is best to regard them as a link. In this link, we are not very clear about the amplitude change and phase change of its output signal relative to the input signal, and this issue will be mentioned later. The filter inductor Lo and filter capacitor Co in Figure (01) together with the equivalent series resistance Resr of the capacitor are also a link. This link will obviously produce a certain phase shift, but the amplitude change (relative to the average value of Vy) is not large.
Figure (02)
Figure 12.2 of the third edition of Switching Power Supply Design (Figure (03) in this article gives the amplitude-frequency curves of several typical links (called "networks" in the book), but does not give the phase-frequency curves for the first two. The third circuit consisting of an inductor and a capacitor gives a phase shift curve after this figure.
Figure (03)
Now let's take a look at the amplitude-frequency curve and phase-frequency curve of the first two links in Figure (03).
Figure (04) is the amplitude-frequency curve and phase-frequency curve of a simple first-order RC low-pass circuit, which is actually the amplitude-frequency and phase-frequency curve of the first link in Figure (03). Figure (04) shows a circuit, where IN is the signal input side and OUT is the signal output side. At the same time, it is assumed that the signal source is an ideal voltage source (internal resistance is zero) and the input impedance of the subsequent stage is infinite. The horizontal axis of the amplitude-frequency curve and the phase-frequency curve in the figure is frequency, which is a logarithmic scale. The vertical axis of the amplitude-frequency curve is the ratio of the output signal voltage amplitude to the input signal voltage amplitude, and then the logarithm is taken, in decibels (dB). The vertical axis of the phase-frequency curve is the phase shift of the output signal relative to the input signal. A positive value indicates a phase advance, and a negative value indicates a phase lag. The vertical axis of the phase-frequency curve is a linear scale. This kind of curve is usually called a Bode diagram, and some books also translate it as a Bode diagram. Bode plots are widely used in the compensation design of switching power supply feedback circuits.
One benefit of using a logarithmic scale is that it simplifies the drawing of amplitude-frequency curves and phase-frequency curves. We all know that the capacitive reactance of a capacitor is inversely proportional to the frequency. In the coordinates of a linear scale, the inverse function is a hyperbola, and a hyperbola is difficult to draw accurately. In a logarithmic scale, the inverse function is a straight line. Obviously, a straight line is easy to draw accurately.
The curve below the circuit in Figure (04) is the amplitude-frequency curve, and below that is the phase-frequency curve of the circuit. The actual curve is roughly as shown in the red curve.
Figure (04)
This curve is calculated based on the change in the voltage divider ratio of resistor R and capacitor C with frequency. After the curve is scaled logarithmically, it can be represented by two straight lines. The horizontal part is marked as 0dB, that is, the voltage ratio is 1. When the frequency is very low, the capacitor is equivalent to an open circuit, and the entire input signal voltage is applied to the output end through the resistor, so the gain is 1, which is obvious. As the frequency gradually increases, the capacitive reactance decreases. At a certain frequency, the capacitive reactance is equal to the resistance, and the voltage across the capacitor is 0.707 times the input voltage. The actual curve is 3dB lower than 0dB, as shown in the red curve. This frequency is recorded as f0, and is recorded as fp in Figure (03). This frequency is obviously uniquely determined by the values of resistor R and capacitor C, and its calculation formula has been marked in Figure (03). This frequency is often called the turning frequency. As the frequency continues to increase, the capacitive reactance continues to decrease, and eventually the capacitive reactance is much smaller than the resistance. The output voltage decreases approximately inversely with the frequency. This is the inclined straight line on the right, with a slope of 20dB/dec (dec means ten times), which is the slope of -1 mentioned in the "Switching Power Supply Design Third Edition". It can be seen that these two straight lines are asymptotic lines of the actual curve. When these two straight lines are used to replace the actual curve, the maximum error occurs at the frequency of f0, which is 3dB. So the two straight lines are a good approximation of the actual curve.
As for the phase-frequency curve, it is also calculated based on the voltage division of the resistor R and the capacitor C. A simple qualitative analysis is as follows: When the frequency is very low, the capacitive reactance of the capacitor is very large, the resistor and the capacitor are connected in series, and the current of the series circuit is mainly determined by the capacitive reactance. The current leads the voltage by 90°, and the voltage across the capacitor is basically in phase with the input signal, so the phase shift of the phase-frequency curve is close to 0. When the frequency is very high, the resistance is much larger than the capacitive reactance. The current in the series circuit is mainly determined by the resistance. The current has a small phase shift to the input signal voltage, and the voltage across the capacitor lags behind the current by 90°, so the output voltage signal lags behind the input signal voltage by nearly 90°. The actual phase-frequency curve is roughly as shown in the red curve. This curve can also be approximated by three straight lines. From very low frequency to 0.1f0, it is basically a horizontal straight line, that is, there is no phase shift. At frequency f0, the phase shift is that the output lags behind the input by 45°. As the frequency increases, the phase shift continues to increase. After the frequency increases to 10 times f0, the phase shift basically no longer increases. When the frequency is very high, the maximum phase shift is 90° behind, that is, the lowest horizontal straight line of the phase-frequency curve in Figure (04). The actual curve is represented by a straight line, and the approximation is quite good. This can only be achieved when the horizontal axis (frequency) is a logarithmic scale.
Figure (05)
The circuit in Figure (05) is a first-order RC high-pass circuit, which is formed by swapping the positions of the resistors and capacitors in the circuit in Figure (04). Its amplitude-frequency curve is the curve at the bottom of the circuit, and the phase-frequency curve is the curve at the bottom. It can be seen that the amplitude-frequency curve in Figure (04) is swapped left and right to form the curve in Figure (05). The only difference is that the phase in Figure (05) is ahead rather than behind.
When the frequency is very low, the capacitive reactance of the capacitor tends to infinity. As the frequency increases, the capacitive reactance of the capacitor decreases, and the frequency at which the capacitive reactance equals the resistance is also called the turning frequency. The calculation of the turning frequency of this circuit is exactly the same as that of Figure (04). As the frequency increases, the capacitive reactance eventually decreases to zero, and the output signal also increases to be close to the input signal, that is, 0dB.
When the frequency is very low, the capacitive reactance of the capacitor C in Figure (05) is very large, much larger than the resistance of the resistor R. The capacitor and the resistor are connected in series in the circuit, and the current is mainly determined by the capacitive reactance, so the current leads the signal voltage by nearly 90°. When the frequency is very high, the capacitive reactance is very small. The current in the circuit of capacitor and resistor in series is mainly determined by the resistor, so the current leads the signal voltage very little, close to 0. This forms the phase-frequency curve in Figure (05).
After the two links in Figure (02) are cascaded, what are the total amplitude-frequency curve and phase-frequency curve?
When two links are cascaded, in general, the amplitude change should be multiplied, but the phase shift is added.
The second advantage of using a logarithmic scale on the vertical axis of the Bode plot amplitude-frequency curve is that it is very convenient to calculate when two links are cascaded. Because the vertical axis uses a logarithmic scale, multiplication becomes addition.
Figure (06)
For example, in the circuit in Figure (06), R1C1 forms the first first-order low-pass circuit, and R2C2 forms the second first-order low-pass circuit. In order to prevent the second RC circuit from affecting the first RC circuit due to insufficient input impedance, a follower A is added between the two RC sections. The follower does not produce a phase shift, and the voltage gain is 1. Assume that the first RC circuit has a turning frequency of f1, and its amplitude-frequency curve is shown in the green curve. Assume that the second RC circuit has a turning frequency of f2, and its amplitude-frequency curve is shown in the blue curve. Then the total amplitude-frequency curve of Figure (06) is the black curve in the figure.
When the frequency is lower than f1, both the green and blue curves are horizontal. 0dB plus 0dB is still 0dB, so the black curve is also horizontal. When the frequency is higher than f1 but lower than f2, the green curve decreases at a slope of 20dB/dec, but the blue curve remains horizontal. When the two are added together, the black curve should obviously decrease at a slope of 20dB/dec between the frequencies f1 and f2. When the frequency is higher than f2, both the green and blue curves decrease at a slope of 20dB/dec. Then the black curve is the sum of the vertical coordinates of the green and blue curves, and should decrease at a slope of 40dB/dec. The method of drawing with straight lines is very intuitive and simple.
The phase-frequency curve is actually the sum of two phase shifts. Therefore, the vertical axis is a phase-frequency curve with a linear scale. Simply add the phase shifts of the two links. The phase-frequency curve is not drawn in Figure (06).
This approximate calculation method of converting multiplication into addition straight line drawing has been fully applied in Figures 12.13, 12.16 and 12.19 of "Switching Power Supply Design 3rd Edition".
Let's take a look at the circuit in Figure (07). This circuit is also very simple, consisting of resistors R1, R2 and capacitor C. In this circuit, R1 is much larger than R2, so we might as well set R1＝9R2. The curve in Figure (07) is drawn based on R1＝9R2.
When the frequency in Figure (07)
is very low, the capacitive reactance of the capacitor is very large, close to an open circuit. Therefore, the output voltage amplitude is close to the input signal voltage amplitude, and the phase shift is close to 0. In this frequency range, the amplitude-frequency curve is generally a horizontal straight line. When the frequency increases to f1, the capacitive reactance of the capacitor decreases and begins to work, causing the output signal amplitude to begin to decrease, and the amplitude decreases at a slope of 20dB/dec. However, as the frequency continues to increase, the capacitive reactance continues to decrease, and the resistor R2 begins to increase relative to the capacitive reactance. When the frequency increases to f2, the effect of resistor R2 becomes apparent. The output signal amplitude is generally determined by the voltage divider ratio of R1 and R2. As the frequency increases further, the capacitive reactance continues to decrease, and the output signal amplitude stabilizes at the voltage divider ratio of R1 and R2, and is no longer affected by the capacitive reactance of the capacitor. Therefore, according to R1=9R2 in Figure (07), the amplitude-frequency curve decreases by 20dB and then turns horizontal. The rightmost side of the amplitude-frequency curve in Figure (07) becomes a horizontal straight line.
The qualitative analysis of the phase-frequency curve is roughly the same. When the frequency is very low, the capacitance of capacitor C is very large, which is equivalent to an open circuit, and there is almost no current in resistor R2, so the phase shift of the output signal to the input signal is close to 0. But when the frequency is very high, the capacitance of capacitor C is very small, close to a short circuit, and the output signal is close to the voltage divider of resistors R1 and R2, so the phase shift of the output signal to the input signal is also close to 0.
Compared with Figure (07) and Figure (04), an important feature is that when the frequency is very high, the circuit of Figure (07) will reduce the amplitude, but will not produce a -90° phase shift like Figure (04). This is very important and has been widely used in switching power supply feedback circuits.
I have previously posted a post "Miscellaneous Talks on Sinusoidal Oscillator Circuits (I)", which quoted a passage starting from page 434 of the fifth edition of Kang Huaguang's "Electronic Technology Basics Analog Part", as shown in Figure (08).
First of all, we should note that the block diagram in Figure (08) is the same as the block diagram in Figure (02) of this article, except that Figure (08) combines multiple links into only two links: "basic amplifier circuit" and "feedback network".
Figure (08)
Figure (08) talks about how to generate self-oscillation, but the switching power supply in Figure (01) of this article is absolutely not allowed to generate self-oscillation. If the switching power supply generates self-oscillation, the output voltage Vo will rise to a very high level all of a sudden, and then drop to a very low level all of a sudden, which may burn the load or even burn the switching power supply itself.
Equations (9.5.2) and (9.5.3) in Figure (08) talk about how to generate self-oscillation. So if you don't want to generate self-oscillation, you just need to not satisfy Equations (9.5.2) and (9.5.3) in Figure (08). In other words, at a phase shift of 360°, the loop gain is less than 1. In other words, when the loop gain is equal to 1, the phase shift is less than 360°.
But for a switching power supply, "no self-oscillation" is not enough. If the phase shift of a switching power supply feedback loop reaches 355° (5° away from 360°) when the gain is 1 (this frequency is called the crossover frequency in the "Switching Power Supply Design Third Edition"), then this switching power supply is likely to produce a strong damped oscillation in the output voltage Vo when it is impacted (for example, a sudden change in load resistance), and the output voltage rises to 50% above the rated value, and then drops to less than 50% of the rated value. Such damped oscillations will gradually decay, and after several cycles, the output voltage will return to normal values. Such a phenomenon is not self-oscillation, but no one can stand such attenuated oscillations.
However, if the gain of the error amplifier EA in the typical switching power supply circuit in Figure (01) is quite high and does not change with the change of signal frequency, then the typical switching power supply circuit in Figure (01) will almost certainly be as mentioned above, with a phase shift close to 360° when the loop gain is 1 (at the crossover frequency), thereby generating damped oscillations when it is impacted. This is because the filter inductor Lo and filter capacitor Co in Figure (01) are a second-order circuit, as shown in Figure 12.3 of Chapter 12 of "Switching Power Supply Design, Third Edition". When the frequency is higher than f0, a lagging phase shift of nearly 180° is generated. Adding the 180° phase shift caused by negative feedback, the total phase shift is close to 360°. The gain of the error amplifier EA is quite high, so it is almost certain that if the circuit is not compensated for frequency, the output voltage Vo will produce damped oscillation when it is impacted. This is exactly why Chapter 12 of "Switching Power Supply Design, Third Edition" requires that the total open-loop phase shift at the crossover frequency be less than 360°, that is, the phase margin, which is usually at least greater than 45°. Of course, it is better if the phase margin can reach 60°.
Then can the amplification factor of the error amplifier EA be reduced to make the open-loop gain close to 1 but less than 1 to prevent self-oscillation or damped oscillation under impact? The answer is no. If the open-loop gain is always less than 1, then negative feedback will not exist, and the stability of the output voltage will be out of the question. In the 12th activity, we talked about the design example of a forward transformer in Section 3.2.5 of "Mastering Switching Power Supply Design (2nd Edition)", which requires the input DC voltage Vdc to vary in the range of 90 to 270V, with a maximum to minimum ratio of 3 times. However, the output voltage is generally only allowed to vary by 1% when the input voltage changes. This requires the feedback loop to have an open-loop gain of at least 50dB at very low frequencies. In fact, the input voltage of this type of switching power supply is often obtained from the AC mains rectifier and filter, and the AC mains rectifier and filter always have a certain ripple, and the ripple frequency is at least twice the industrial frequency, which is 100Hz in China. This industrial frequency ripple is not allowed to appear in the output of the switching power supply, so at a frequency of 100Hz, the open-loop gain of the feedback
loop is also required to be 50dB or even higher. If a first-order low-pass RC circuit as shown in Figure (04) is inserted into the feedback loop, and the RC value is relatively large, it can indeed make the open-loop gain high enough at very low frequencies, and low enough at high frequencies, so that the feedback loop does not produce self-excitation. But this method has a great disadvantage: such a first-order low-pass circuit will bring an additional maximum phase shift of 90° to the feedback loop, thereby increasing the risk of self-excitation. Therefore, in order to avoid self-excitation and leave enough phase margin, the corner frequency f0 of the first-order low-pass circuit must be designed to be very low, which reduces the ability of the switching power supply to quickly respond to shocks (such as sudden load changes). However, in many actual switching power supply feedback loops, there is still a simple capacitor design using the circuit in Figure (04). It must be said that this is a design that does not have a deep understanding of the feedback loop.
A slightly better design is to insert a circuit like Figure (07) into the feedback loop. This circuit can make the gain drop to a certain level when the frequency is high to a certain value (approximately equal to the ratio of R2 to R1 in the figure), but at higher frequencies, it only produces a relatively small additional phase shift, unlike Figure (04) which ultimately produces an additional phase shift close to 90°. It is indeed a bit surprising that the result is so different from Figure (04) just by adding a resistor R2. The insertion of this small resistor changes the phase-frequency curve not only in the circuit of Figure (07), but also in the LC filter composed of the output filter inductor and filter capacitor of the switching power supply, as shown in the curve of Figure 12.5 of the "Switching Power Supply Design Third Edition". There, the equivalent series resistance Resr of the filter capacitor plays the role of correcting the phase-frequency curve so that its slope becomes -20dB/dec in the high frequency band.
However, in order to minimize the loss of the output filter capacitor Co, capacitor manufacturers are always improving the process and producing capacitors with the smallest equivalent series resistance Resr. This makes the curve of Figure 12.5 (b) close to the curve of Figure 12.5 (a) where the output capacitor does not contain the equivalent series resistance Resr. The slope of the high frequency band is still -40dB/dec, and the phase shift is still close to 180° behind.
However, we must not connect a resistor in series with the filter capacitor in order to change the phase-frequency curve of the LC filter at the output end of the switching power supply. When a resistor is connected in series with the filter capacitor, a considerable amount of power must be dissipated in this resistor, resulting in heat generation and reducing the efficiency of the switching power supply, and the ripple of the output Vo will increase. A circuit like Figure (07) can only be inserted into a low-power circuit before and after the error amplifier EA. The signal amplitude there is not large, and the impedance of the front and rear stages is relatively high. Inserting a circuit like Figure (07) will not cause much energy loss, and will not increase the ripple of Vo.
Of course, a circuit like Figure (07) in which the gain decreases at high frequencies and the phase shift gradually approaches 0 at high frequencies can also be placed in the local feedback circuit of the op amp, as shown in C1R2 in Figure 12.7 of "Switching Power Supply Design Third Edition".
Chapter 12 of "Switching Power Supply Design Third Edition" lists three criteria as follows:
The first criterion is that the total open-loop phase shift at the crossover frequency (open-loop gain is 1, i.e. 0dB, and the gain curve passes through zero) is less than 360°, i.e. the phase margin, and is usually at least greater than 45°.
The second criterion is that in order to prevent the rapid change of the phase of the -2 gain slope circuit, the slope of the total open-loop gain of the system at the crossover frequency should be -1. The total gain is the logarithm sum of the gains of all links in the loop.
The third criterion is to provide the required phase margin, which is specified as 45° here.
For designs that meet these three criteria, see Chapter 12, Section 12.3 "Design of the Error Amplitude-Frequency Characteristic Curve" of "Switching Power Supply Design, Third Edition".
Question 15
1. From Figure (05) in this article, we know that a circuit like Figure (05) will produce a leading phase shift. So in order to make the open-loop characteristics of the switching power supply feedback have a 45° phase margin at the crossover frequency, can a circuit like Figure (05) be used to achieve sufficient phase margin at the crossover frequency?
2. Figure (08) in this article is copied from a paragraph starting on page 434 of Kang Huaguang's "Electronic Technology Basics Analog Part" Fifth Edition. That paragraph clearly states: This block diagram is a feedback amplifier. We also said that the block diagram in Figure (08) is the same as the block diagram in Figure (02) in this article. Then can the switching power supply shown in Figure (01) of this article and other switching power supplies of other topologies also be considered as an amplifier? If Figure (01) of this article is a feedback amplifier, where is its input signal?
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