T S R Q R Q = (R(ST) | ) | = (R(SQ) | ) | T S R Q CEC 220 Revisited

Power Converter Control Power Pole PWM Output Filter kDkD Compensator VDVD V IN Load Properties + - Desired Output Voltage “Error” Signal Loop Gain Adjust Frequency Response Tweak (if required) When characterizing the overall behavior of the feedback system, it is desirable to manipulate the describing equations such that the node we are trying to control (the “output”, or a measurement of the output) appears alone, subtracted from a control input term which represents what we want the output to be. When the output is equal to the control input, the error term is zero.

Linearization Before we can apply linear feedback theory, the models of our devices must exhibit linear relationships between input and output. Unfortunately, we often see relationships like: Which is non-linear. To get around the non-linearity, we “linearize” by modeling the behavior as linear in a small region near a fixed operating duty cycle, say D 0, where Variation of actual duty cycle from designated operating point. We can express Let where (for a buck-boost converter)

Example: v out D D0D0 V0V0

T1T1 X1X1 T2T2 T3T3 TUTU T5T5 TVTV T4T4 11 22 UU 33 44 55 Y1Y1 X4X4 X3X3 X5X5 X2X2 XVXV XUXU VV Y2Y2 Y3Y3 Y5Y5 Y4Y4 YVYV YUYU Consider all the X inputs to be zero except X i. Then Behavior of Signals Propagating Around Loops

Define H i,q as the transfer function from input i to output q. If all of the block Transfer functions, T k, are linear, we can apply superposition, and any output can be expressed as the sum of the individual responses from all inputs. T i,q is the product of all block transfer functions in the forward (clockwise) direction from X i to Y q T L is the product of all block transfer functions in the loop, also referred to as the Loop Gain.

For simplicity, the summing junctions in the foregoing general analysis all indicate addition. This is commonly referred to as a positive feedback loop. Loops are often (in fact, usually) implemented with the loop signal subtracted at one or more summing junctions. If the number of such subtractions is odd, then the loop is considered to have negative feedback. Thus there are two forms used for transfer functions, depending on whether the loop exhibits positive or negative feedback: Positive Feedback vs Negative Feedback Positive Feedback Negative Feedback

A Simple, but Very Common Example: F(s) R(s) X1X1 + - Y1Y1 Gain, with low-pass delay  Unity feedback F(s) H(s) log  |H, F| dB 20 dB/dec Much Faster Response!

Observations on Negative Feedback: There is a unique transfer function, H i,q, relating each input to each output. Each and every H i,q, has the same denominator term: 1 + T L. The block Transfer functions are generally functions of our complex variable s. Therefore, T L will have a magnitude and a phase, and it is quite possible that for some value of s, T L = -1 = e j . When this occurs, the magnitude of every transfer function becomes infinite (pole of the transfer function). If a pole occurs for a value of s in the right half-plane, the loop is unstable. If a pole occurs for some s = j , the loop exhibits spontaneous oscillation at frequency , which is a precursor to instability. A Bode plot of the loop gain will reveal this tendency.

Control Objectives 1.Zero steady state error 2.Fast Response to disturbances Change in Load Conditions Change in Input Voltage 3.Low Overshoot 4.Low Noise Susceptibility Power Pole Output Filter kDkD Controller/PWM VDVD v out (t) V IN Load + - v pp (t)

Power Pole, Dynamic Average Model kDkD Controller/PWM V D (s) V OUT (s) V IN (s) + - Filter/Load F(s) The Transfer function is: Which will exhibit overshoot, damping, potential instability, as determined by gain and phase margin... T PP (s) T C (s)

The loop Gain, G L (s) is a complex function. If its magnitude is greater than one when the phase is -  radians (phase lag =  ) for some value of s in the Right Half-Plane, the denominator will go to zero, resulting in instability. If its magnitude is equal to one when the phase is -  radians (phase lag =  ), for some value of s = j , the loop will oscillate at frequency . Examining the gain and phase plots vs frequency, we look for the frequency at which the magnitude falls to unity (0 dB). The difference between the actual phase lag and  radians is called the phase margin. The amount by which the magnitude deviates below 0 dB at the frequency where the phase lag reaches  radians is called the gain margin. The smaller these margins are, the greater is the overshoot and tendency for instability due to uncontrollable variations. --  0 dB Gain Margin Phase Margin Gain Phase