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One easy way to turn our prototype buck converter into a 5 V regulator is to sense the output and turn the switch on when the voltage is less than 5 V and turn the switch off when the voltage is greater than 5 V. This form of control is called by various names including bang-bang control, ripple regulators, and hysteretic control. It is instructive to exam this operation in the state-plane. For the following plots the values of L and C are the same (75 uH and 1200 uF) but load is one ohm instead of 0.25 ohm (less damping shows the effects better).
  Figure 3-8 shows bang-bang or hysteretic control with no hysteresis. The black line (1) is output voltage (voltage across the output capacitor) versus time, the blue line is inductor current (2) versus time, and the green line (3) is inductor current plotted versus capacitor voltage - the state plane for the two energy storage devices, L and C.
Figure 3-8: Bang-Bang Control - No Hysteresis
At time zero the switch representing the transistor is turned on and the voltage and current increase until the voltage reaches 5.0V. At this point the switch is turned off and the current starts to decrease. However, the voltage across the output capacitor continues to increase as long as current in the inductor is greater than the load current. The current drops to zero but does not reverse since the switch and diode will not allow this. When the voltage decays to 5.0V, the switch is turned again and voltage and current start to increase. This action is repeated with the amplitude of the voltage ripple and current ripple decreasing each cycle and the frequency increasing each cycle. In the limit the frequency becomes infinite and the ripple becomes zero and all that is left is a series dissipative regulator with an LC output filter. In this example, the initial cycle frequency is about 4.76 kHz, the voltage ripple is 30 mVp-p and the current ripple is 10Ap-p. This is seen on the state-plane in the outer limit of the green ellipse at (3). The ellipse is filled in as the ripple goes to zero and the frequency goes to infinity and we get a series dissipative regulator. This we do not want. How do we keep things switching? By adding either a time delay in the control loop (time hysteresis) or voltage hysteresis in the comparator that senses the output voltage. In practice there is always some time delay in real components and the regulator may settle into some stable switching frequency. However, this is not dependable and a controlled delay or voltage hysteresis must be added. Figure 3-9 shows the same circuit with a 1.2 microsecond delay added to the control loop.
Figure 3-9: Bang-Bang Control - Delay Hysteresis
The 1.2 microsecond delay in the feedback loop has stabilized the frequency at about 12.5 kHz with 25 mVp-p voltage ripple and 3 Ap-p current ripple. However, the overshoot is just as bad as the circuit without hysteresis. (This is shown in the state plane but not the voltage and current time plots which show the steady state, not the start-up transient.) How do we cure this? With a soft-start circuit. Figure 3-10 shows the startup with a soft-start circuit. The soft start is implemented in this case by slowly bringing up the reference voltage.
Figure 3-10: Bang-Bang Control with Soft Start Circuit
The red line (2) is the 5V reference voltage and ramps up from zero to 5V in 2 milliseconds. The black line (hidden 1) is the output voltage which follows the reference ramp except every time it makes a comparison to the reference and finds it is low, it turns on and overshoots the reference. At 2 ms it overshoots the desired 5V slightly and then after several oscillations settles into the steady state ripple. The green line (4) shows this action in the inductor-current versus output-voltage state plane. Figure 3-11 shows the switching details during steady state conditions. The black line (1) is the output voltage ripple, the blue line (2) is the ripple current in the inductor, and the green line (3) is the ripple current versus the ripple voltage in the state-plane. To this has been a vertical red line added at the 5V switching line. 
Figure 3-11: Switching Line Detail
When the upper green trace intercepts the red switching line, the comparator commands the switch to open. But due to the 1.2 microsecond delay added, the switching occurs some what later. The same with the lower green trace. This shows how the time hysteresis controls the output ripple and switching frequency (which is related in a somewhat complex way to the area traced by the trajectory). Note that although a time delay is used here, by putting two switching lines at the switch points (a comparator with voltage hysteresis) the ripple and frequency can be controlled by voltage hysteresis. Current hysteresis switching lines could also be used. Since all circuits have delays, the ripple and frequency are usually controlled by time delay and comparator hysteresis in real circuits. The switching line does not have to be fixed. For example, one converter uses as reference a fixed frequency triangle derived by chopping a dc reference and integrating the result into a triangle used as a reference for the comparator. This gives a fixed frequency of operation combined with the advantages of fast transient response and 90 degrees phase shift. The state-plane is a very powerful way to look at switching-mode power supplies. For example, here we see a steady state response for one set of line and load conditions. Other sets of line and load conditions have their steady states. The state-plane can be used to find the optimum switching strategy between the two steady state conditions, optimum in that there is only one switching point that will allow a transition from one steady-state to the other steady state with a single off-on switching set. The state-plane has been used in this manner to determine optimum controllers for switching-mode power supplies. All modern Spice programs, including the free demonstration programs, allow plotting the state-plane and provide a powerful tool for examining switching-mode power supplies. I highly recommend looking at them as part of your simulation and using them to learn more about switching-mode power supplies. Ask yourself how does the state-plane change if I change the input voltage? How does it change if I change the load? What does the transition look like in the state plane for these changes. One caution is that you usually have to control the maximum step size to get smooth plots. Using the usual defaults for Spice time plots usually produces jagged state-plane plots. For the plots in this tutorial, I usually set the maximum step size to the data step time. This increases the time somewhat needed to complete a simulation but makes much cleaner state-plane plots. Bang-bang control is very easy to implement. All circuits have a time delay, so a simple comparator comparing a reference to the output usually works -- and can always be made to work by adding an extra time delay. However, control is usually implemented using a comparator with hysteresis, hence the term hysteretic control. Whence the name bang-bang? This type of control is often used to control rocket and satellite control thrusters where a valve is either fully on or fully off. When tested in the atmosphere, these thrusters make a bang each time the are operated. When a control system is trying to lock on to a target, these thrusters can sound like the loudest machine-gun you have every heard. Definitely bang-bang! Besides simplicity, bang-bang or hysteretic controllers only have 90 degrees phase shift which makes them easy to stabilize, and a rapid transient response. With these advantages, why do most power supplies use pulse-width-modulation (PWM) control instead? A minor reason is that hysteretic converters are more prone to chaos than other converters. But the major reason is their tendency to synchronize or entrain with a switching load, a periodic input voltage, or random noise -- often with unwanted or disastrous results, such as ripple on a five volt logic converter increasing from a tens of millivolts to several volts. All converters can do this, but bang-bang controllers are far more susceptible to this problem than PWM controllers. Entrainment will be discussed in detail in an upcoming problem/solution discussion. Do not use this information for design without independent verification of the information. Editor: Jerrold Foutz |
