1 Power Electronics by Dr. Carsten Nesgaard Small-signal converter modeling and frequency dependant behavior in controller synthesis

2 Agenda Small-signal approximation Voltage-mode controlled BUCK Converter transfer functions  dynamics of switching networks Controller design (voltage-mode control) Discrete time systems Measurements

3 Small-signal approximation An analytical evaluation of equipment performance Advantages of small-signal approximation of complex networks: An analysis of equipment dynamics Stability Bandwidth A design oriented equipment synthesis The linearization of basic AC equivalent circuit modeling corresponds to the mathematical concept of series expansion.

4 Small-signal approximation Limited to rather low frequencies (roughly f S /10) Drawbacks of small-signal approximation of complex networks: Inability to predict large-signal behavior Transients High frequency load steps Calculation complexity increases quite rapidly

5 Voltage-mode controlled BUCK Basic BUCK topology with closed feedback loop Not included in ‘Fundamentals of Power Electronics’ SE.

6 Voltage-mode controlled BUCK Converter waveforms:

7 AC modeling Converter states: 0 < t < d  T: d  T < t < (1 – d)  T KCL: KVL:

8 AC modeling Averaging and linearization (in terms of input and output variables): Inductor equation: Capacitor equation (same for both intervals): Input current equation: DC and 2 nd order terms are removed from the equations to the right.

9 AC modeling Resulting AC equivalent circuit: DC transformer relating input voltage and inductor current, thus behaving ‘almost’ like a real transformer.

10 Canonical AC model Rearranging the AC equivalent circuit found on the previous slide by the use of traditional circuit theory a universal model can be established: A similar model applies to a wide variety of other converter topologies. In Fundamentals of Power Electronics SE a table containing coefficients for the different sources can be found.

11 Converter transfer functions Basic control system State equation: Control equation: Output equation: A: State matrixB: Source matrix E: Control matrixF: Feedback matrix I: Identity matrixQ: Feed forward matrix M: Output-state-matrixN: Output-source-matrix d: Control variable u: Source variable x: State variable y: Output variable

12 Converter transfer functions Opening the loop and rewriting the system equations the following trans- fer functions can be obtained: Open loop transfer function: Closed loop transfer function:

13 State-space averaging State variables: Inductor currentx 1 Capacitor voltagex 2 Output variabley(dependent) In order to contain past information all variables are functions of time By definition the following apply: Source variableu(independent)

14 State-space averaging Averaging the equations previously found results in the following non-linearized matrices: The use of linearization requires: Insertion into the state equation results in:

15 State-space averaging The averaged and linearized matrices can now be identified: Comparing the above A matrix with the non-linearized A’ matrix found on the previous slide, it can be seen that no changes have occurred.

16 State-space averaging Averaging and linearizing the control variable d (PWM controller) in terms of state variables gives the following relation: is the sawtooth peak voltage is the EA gain, ‘a’ factor and comp. Collecting terms in accordance with the control equation, and realizing that multiplication by ‘s’ in the frequency domain is the equivalent to differentiation in the time domain, the matrices F and Q can be identified: and Since feedback is the only means of converter control applied, Q is (as expected) zero.

17 State-space averaging Summarizing the voltage-mode controlled BUCK matrices: 0 1 1/L 1/R LOAD  C -K/V P

18 Nyquist stability requirement for closed loop systems: P rh (G CL (s)) = P rh (G OL (s)) +  = 0 Where:P rh = number of right half-plane poles  = number of times the Nyquist contour of the open- loop transfer function circles the point (-1,0) G CL = Closed-loop transfer function G OL = Open-loop transfer function Stability Minimum open-loop transfer function gain margin:6 - 8 dB Minimum open-loop transfer function phase margin:30  - 60 

19 Voltage-mode controlled BUCK Circuit data: L = 300  H C = 69  F R ESR = 0,2  I Load,m = 1 A U 1 = 12 V y 2 = 5 V  I L =0,2 A f=50 kHz Additional data: V p =2,45 V a=0,5

20 Voltage-mode controlled BUCK A plot of the open-loop transfer function is shown below (K = 1):

21 Voltage-mode controlled BUCK A 3D plot of the converter filter transfer function is shown to the right. Note: The zero caused by R ESR increases the phase (green curve) as a function of frequency and R ESR. Unfortunately due to the same zero the filter attenuation drops (red curve).

22 PI-comp. Lag-comp. PD comp. Lead-comp. A PI-Lead-comp. (PID) will be used in this presentation Compensation

23 Compensation Widely accepted error amplifier configuration: Pole at f = 0 for increased DC gain Pole at f ESR for compensation Double zero at resonance peak for increased phase margin

24 Compensator and converter transfer functions: Compensation Amplitude: Phase: 0 dB/dec -40 dB/dec -20 dB/dec +20 dB/dec0 dB/dec Red : Converter transfer function Blue : Compensation transfer function f C  4.0 kHz   56,1 

25 Voltage-mode controlled BUCK Using the previously derived matrices an expression for the input impedance Z in can be established:

26 DCM reduces the converter transfer function to a first order system, since the time derivative of the small-signal inductor current is zero and thus disqualifies the inductor current as a state variable. Voltage-mode controlled BUCK A plot of the open-loop transfer function during Discontinuous Conduction Mode (red curve) and EA compensation (blue curve)

27 Discrete time systems Transient response and the relationship between the s-plane and the z-plane: Discrete time: Continuous time: At the sampling instants: Inserting into the expression to the left, it can be seen that the continuous time stability requirement maps onto the z-plane in form of the unit circle. Thus, the dynamics of the two systems are identical at the sampling instants:

28 Discrete time systems Arithmetic and operations: Integration and differentiation Plotting the frequency response Tustin’s rule Sampling rate

29 Discrete time systems Plot of the discrete compensation transfer function: ContContinuous time Disc_1Discrete time with sample frequency = 50 kHz (no prewarping) Disc_2Discrete time with sample frequency = 100 kHz (no prewarping)

30 GHContinuous time GD_2Discrete time with sample frequency = 50 kHz (no prewarping) MeasActual measurement Measurements Below is a comparison of the predicted continuous time loop gain, predicted discrete time loop gain and an actual measurement of the loop gain:

31 GHContinuous time MeasActual measurement Measurements The same transfer function as before, but during Discontinuous Conduction Mode: