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The state-plane, sometimes called the phase-plane, is a plot with two state-variables as coordinates. State-variables are variables whose initial conditions are required to determine the future behavior of the system. Typically they are anything that stores energy, like inductors and capacitors, or anything with a memory, such as a flip-flop. Common state-planes in switching-mode power supply design are inductor current versus capacitor voltage in the output LC filter (approximates load current and load voltage), the derivative of capacitor voltage (approximates the inductor current without the dc component) versus capacitor voltage, and the derivative of the output capacitor voltage versus the voltage (you do not have to be concerned with any currents and the derivative always crosses the voltage axis at right angles). Trajectories are unique for a set of initial conditions and never cross each other. Trajectories in the state-plane can easily be determined by most SPICE programs that are used to determine the time response of a circuit.
Figure 3-7 shows an approximate state-plane plot of the turn-on and turn-off events. The horizontal axis is the voltage across the output capacitor which approximates the output voltage. The vertical axis could one of several state-variables such as the inductor current, the derivative of the capacitor voltage, or the current through the capacitor. The inductor current, which approximates the load current is chosen. The steady-state turn-off conditions are 0 V and 0 A. The steady-state turn-on conditions are 12 V and 12 A (point one in the plot). The trajectories to reach these steady state conditions are shown in state-plane, Figure 3-7. 
Figure 3-7: Buck Converter Turn-On and Turn-Off State-Plane
Turn-on If there were no load, then the normalized turn-on trajectory would trace a clockwise circle reaching a positive peak current of Vin/Zo = (12 V)/(0.25 ohm) = 48 A, a peak voltage of twice the input voltage, 2*(12 V) = 24 V, and negative current equal to the positive current, return to zero volts, and then repeat itself until the circuit is disturbed, since there is nothing in the circuit that dissipates energy. Since there is a load to dissipate energy, the circle spirals in to the steady state value of 12 V and 12 A (point 1) as energy is lost to the load or other dissipative elements as shown in Figure 3-7. The state-plane turn-on trajectory tells us some things about this circuit that are worth remembering. On turn-on, buck-derived converters (and LC filters in general) tend to overshoot the final voltage, and can have an in-rush current much greater than the load current. Later in the design cycle it may be necessary to compensate for these inherent turn-on characteristics by adding soft-start circuits and in-rush current-limiting circuits to the design. Also, the turn-on characteristics are dependent on the type of switch. In our ideal example, the switch can pass the inductor current in either direction, which is true of a relay contact or a power MOSFET, where the body parasitic diode will conduct current in the reverse direction. Other switches may not behave this way. For example, the reverse current behavior of a bipolar transistor depends on the reverse beta of the transistor and the base drive current. For old alloy junction transistors, the reverse and forward beta can be the same. For modern planar bipolar transistors, the reverse beta may be less than one and the transistor can carry only a small fraction of the forward current in the reverse direction. If the switch can not carry current in the reverse direction, the voltage rings up to the first peak and stays there until the load discharges it. The bottom line is that you have to thoroughly know the behavior of the power switch to design reliable circuits. Later we will discuss switch topics such as secondary break-down, Safe Operating Areas (SOA), and parasitic elements that are important things to know about a switch. Here we see that it can be important to know how a switch behaves in quadrants other than its normal forward conduction quadrant. Turn-off The turn-of trajectory starts at 12 V and 12 A. The inductor keeps supplying 12 V to the load until it reaches zero amperes. Since the diode does not allow current to go below zero amperes, the voltage then drops along the zero ampere axis until it reaches its turn-off steady-state value of zero volts at zero amperes. There is no ringing in the waveform since the non-linear diode does not allow either voltage or current to go negative. Do not use this information for design without independent verification of the information. Editor: Jerrold Foutz |
