Matrix power converters: spectra and stability

Transcription

1 Matrix power converters: spectra and stability Stephen Cox School of Mathematical Sciences, University of Nottingham supported by EPSRC grant number EP/E018580/1 Making It Real Seminar, Bristol 2009 Stephen Cox Matrix power converters: spectra and stability 1/29

2 Outline What is a (matrix) power converter? Stephen Cox Matrix power converters: spectra and stability 2/29

3 Outline What is a (matrix) power converter? How does a power converter work? Stephen Cox Matrix power converters: spectra and stability 2/29

4 Outline What is a (matrix) power converter? How does a power converter work? I. How to work out the spectrum of a power converter Stephen Cox Matrix power converters: spectra and stability 2/29

5 Outline What is a (matrix) power converter? How does a power converter work? I. How to work out the spectrum of a power converter II. Instability of a matrix converter Stephen Cox Matrix power converters: spectra and stability 2/29

6 Power converters Power converters convert electrical power from one voltage and frequency to another voltage and frequency (typically 50Hz) AC AC DC AC AC DC DC DC Stephen Cox Matrix power converters: spectra and stability 3/29

7 Power converters Power converters convert electrical power from one voltage and frequency to another voltage and frequency (typically 50Hz) AC AC DC AC AC DC DC DC Applications: aircraft, ships, laptops (5V 3.3V conversion) Stephen Cox Matrix power converters: spectra and stability 3/29

8 Power converters Power converters convert electrical power from one voltage and frequency to another voltage and frequency (typically 50Hz) AC AC DC AC AC DC DC DC Applications: aircraft, ships, laptops (5V 3.3V conversion) Switching power converters use high frequency switching (typically 500kHz), and their output is rather complicated Stephen Cox Matrix power converters: spectra and stability 3/29

9 Power converters Power converters convert electrical power from one voltage and frequency to another voltage and frequency (typically 50Hz) AC AC DC AC AC DC DC DC Applications: aircraft, ships, laptops (5V 3.3V conversion) Switching power converters use high frequency switching (typically 500kHz), and their output is rather complicated Filtering is necessary to remove the high frequency components (strict regulatory requirements on electromagnetic interference; power quality) Stephen Cox Matrix power converters: spectra and stability 3/29

10 Power converters Power converters convert electrical power from one voltage and frequency to another voltage and frequency (typically 50Hz) AC AC DC AC AC DC DC DC Applications: aircraft, ships, laptops (5V 3.3V conversion) Switching power converters use high frequency switching (typically 500kHz), and their output is rather complicated Filtering is necessary to remove the high frequency components (strict regulatory requirements on electromagnetic interference; power quality) So we need to understand the spectrum of the signals from the power converter (for filter design) and its stability Stephen Cox Matrix power converters: spectra and stability 3/29

11 Part I: the frequency spectrum of a power converter Stephen Cox Matrix power converters: spectra and stability 4/29

12 A DC DC power converter How to convert 5V DC to 3.3V DC (efficiently)? Stephen Cox Matrix power converters: spectra and stability 5/29

13 A DC DC power converter How to convert 5V DC to 3.3V DC (efficiently)? 5V V(t) 3.3V 5 0 V t Switch rapidly between 0V (34%) and 5V (66%) 66% is the duty cycle There is some filtering at the output, for smoothing purposes Stephen Cox Matrix power converters: spectra and stability 5/29

14 A DC AC power converter How to convert ±1V DC to x sin Ωt AC? Stephen Cox Matrix power converters: spectra and stability 6/29

15 A DC AC power converter How to convert ±1V DC to x sin Ωt AC? +1V V(t) x sin Ω t 1 V t 1V 1 Switch rapidly between ±1V, with slowly varying duty cycle (pulse width modulation, PWM) Stephen Cox Matrix power converters: spectra and stability 6/29

16 A DC AC power converter How to convert ±1V DC to x sin Ωt AC? +1V V(t) x sin Ω t 1 V t 1V 1 Switch rapidly between ±1V, with slowly varying duty cycle (pulse width modulation, PWM) How to vary the duty cycle? What is the spectrum of the output? Stephen Cox Matrix power converters: spectra and stability 6/29

17 A DC AC power converter How to convert ±1V DC to x sin Ωt AC? +1V V(t) x sin Ω t 1 V t 1V 1 Switch rapidly between ±1V, with slowly varying duty cycle (pulse width modulation, PWM) How to vary the duty cycle? What is the spectrum of the output? Standard engineering technique: Black s method (1953) Here is an alternative, more compact method Stephen Cox Matrix power converters: spectra and stability 6/29

18 A DC AC power converter: modulation strategy 1 1 V n 1 A n 1 B n 1 n n+1 A n B n A n+1 n+2 B n+1 t Choose switching times to satisfy A n = n (1+x sin ΩA n) B n = n (3 x sin ΩB n) Stephen Cox Matrix power converters: spectra and stability 7/29

19 A DC AC power converter: modulation strategy 1 1 V n 1 A n 1 B n 1 n n+1 A n B n A n+1 n+2 B n+1 t Choose switching times to satisfy A n = n (1+x sin ΩA n) B n = n (3 x sin ΩB n) We want to calculate the Fourier spectrum of this square wave 1 1 V desired output triangular carrier wave t n 1 n n+1 n+2 A n 1 B n 1 A n B n A n+1 B n+1 Switching times are given by intersections of a highfrequency carrier wave and the desired output Stephen Cox Matrix power converters: spectra and stability 7/29

20 A DC AC power converter: modulation strategy 1 1 V n 1 A n 1 B n 1 n n+1 A n B n A n+1 n+2 B n+1 t Choose switching times to satisfy A n = n (1+x sin ΩA n) B n = n (3 x sin ΩB n) 1 ψ (t;t,t ) ψ 1 2 Notation: ψ(t; t 1, t 2 ) = top-hat function 0 t 1 t 2 t Stephen Cox Matrix power converters: spectra and stability 7/29

21 A DC AC power converter: modulation strategy 1 1 V n 1 A n 1 B n 1 n n+1 A n B n A n+1 n+2 B n+1 t Choose switching times to satisfy A n = n (1+x sin ΩA n) B n = n (3 x sin ΩB n) 1 ψ 0 (t;t,t ) ψ 1 2 t 1 t 2 t Notation: ψ(t; t 1, t 2 ) = top-hat function Then converter output is [ ] v(t) = ψ(t; B n, A n+1 ) ψ(t; A n, B n ) n= Stephen Cox Matrix power converters: spectra and stability 7/29

22 A DC AC power converter: modulation strategy 1 1 V n 1 A n 1 B n 1 n n+1 A n B n A n+1 n+2 B n+1 t Choose switching times to satisfy A n = n (1+x sin ΩA n) B n = n (3 x sin ΩB n) 1 ψ 0 (t;t,t ) ψ 1 2 t 1 t 2 t Notation: ψ(t; t 1, t 2 ) = top-hat function Then converter output is [ ] v(t) = ψ(t; B n, A n+1 ) ψ(t; A n, B n ) n= Poisson resummation: f(n) = v(t) = n= n= n= e 2πniτ f (τ) dτ ] e [ψ(t; 2πniτ B(τ), A(τ + 1)) ψ(t; A(τ), B(τ) dτ Stephen Cox Matrix power converters: spectra and stability 7/29

23 A DC AC power converter: calculating the spectrum v(t) = n= ] e [ψ(t; 2πniτ B(τ), A(τ + 1)) ψ(t; A(τ), B(τ)) dτ where A(τ) = τ (1 + x sin ΩA(τ)), B(τ) = τ (3 x sin ΩB(τ)) Stephen Cox Matrix power converters: spectra and stability 8/29

24 A DC AC power converter: calculating the spectrum v(t) = n= ] e [ψ(t; 2πniτ B(τ), A(τ + 1)) ψ(t; A(τ), B(τ)) dτ where A(τ) = τ (1 + x sin ΩA(τ)), B(τ) = τ (3 x sin ΩB(τ)) Consider e 2πniτ ψ(t; A(τ), B(τ)) dτ Stephen Cox Matrix power converters: spectra and stability 8/29

25 A DC AC power converter: calculating the spectrum v(t) = n= ] e [ψ(t; 2πniτ B(τ), A(τ + 1)) ψ(t; A(τ), B(τ)) dτ where A(τ) = τ (1 + x sin ΩA(τ)), B(τ) = τ (3 x sin ΩB(τ)) Consider e 2πniτ ψ(t; A(τ), B(τ)) dτ We know that ψ(t; A(τ), B(τ)) = 1 when A(τ) < t < B(τ) Stephen Cox Matrix power converters: spectra and stability 8/29

26 A DC AC power converter: calculating the spectrum v(t) = n= ] e [ψ(t; 2πniτ B(τ), A(τ + 1)) ψ(t; A(τ), B(τ)) dτ where A(τ) = τ (1 + x sin ΩA(τ)), B(τ) = τ (3 x sin ΩB(τ)) Consider e 2πniτ ψ(t; A(τ), B(τ)) dτ We know that ψ(t; A(τ), B(τ)) = 1 when A(τ) < t < B(τ) So if we introduce A(t) and B(t) such that t = A(τ) τ = A(t), t = B(τ) τ = B(t) then ψ(t; A(τ), B(τ)) = 1 when B(t) < τ < A(t) Stephen Cox Matrix power converters: spectra and stability 8/29

27 A DC AC power converter: calculating the spectrum v(t) = n= ] e [ψ(t; 2πniτ B(τ), A(τ + 1)) ψ(t; A(τ), B(τ)) dτ where A(τ) = τ (1 + x sin ΩA(τ)), B(τ) = τ (3 x sin ΩB(τ)) Consider e 2πniτ ψ(t; A(τ), B(τ)) dτ We know that ψ(t; A(τ), B(τ)) = 1 when A(τ) < t < B(τ) So if we introduce A(t) and B(t) such that t = A(τ) τ = A(t), t = B(τ) τ = B(t) then ψ(t; A(τ), B(τ)) = 1 when B(t) < τ < A(t) [ ] Hence v(t) = (πni) 1 e 2πniB(t) e 2πniA(t) n= Stephen Cox Matrix power converters: spectra and stability 8/29

28 A DC AC power converter: calculating the spectrum But A(t) = t 1 4 (1 + x sin Ωt), B(t) = t 1 4 (3 x sin Ωt), so v(t) = n= e 2πnit πni [i n e (πnx/2)i sinωt ( i) n e (πnx/2)i sin Ωt ] and we have the Jacobi formula e iz sin θ = J m (z)e iθ m= Stephen Cox Matrix power converters: spectra and stability 9/29

29 A DC AC power converter: calculating the spectrum But A(t) = t 1 4 (1 + x sin Ωt), B(t) = t 1 4 (3 x sin Ωt), so e 2πnit ] v(t) = [i n e (πnx/2)i sinωt ( i) n e (πnx/2)i sin Ωt πni = mn n= v mn e iωmnt where v mn = (πni) 1 i n (( 1) m+n 1)J m (πnx/2) for n 0 and Ω mn = 2πn + mω Stephen Cox Matrix power converters: spectra and stability 9/29

30 A DC AC power converter: calculating the spectrum But A(t) = t 1 4 (1 + x sin Ωt), B(t) = t 1 4 (3 x sin Ωt), so e 2πnit ] v(t) = [i n e (πnx/2)i sinωt ( i) n e (πnx/2)i sin Ωt πni = mn n= v mn e iωmnt where v mn = (πni) 1 i n (( 1) m+n 1)J m (πnx/2) for n 0 and Ω mn = 2πn + mω Special case: v m0 = 1 2 xi for (m = ±1); v m0 = 0 otherwise Stephen Cox Matrix power converters: spectra and stability 9/29

31 A DC AC power converter: calculating the spectrum But A(t) = t 1 4 (1 + x sin Ωt), B(t) = t 1 4 (3 x sin Ωt), so e 2πnit ] v(t) = [i n e (πnx/2)i sinωt ( i) n e (πnx/2)i sin Ωt πni = mn n= v mn e iωmnt where v mn = (πni) 1 i n (( 1) m+n 1)J m (πnx/2) for n 0 and Ω mn = 2πn + mω Special case: v m0 = 1 2 xi for (m = ±1); v m0 = 0 otherwise The low-frequency part of the output is exactly x sin Ωt! Stephen Cox Matrix power converters: spectra and stability 9/29

32 Conclusions (on spectra) There is a compact way to determine the spectrum for switching power converters, much better than the engineering default (Black s method) Stephen Cox Matrix power converters: spectra and stability 10/29

33 Conclusions (on spectra) There is a compact way to determine the spectrum for switching power converters, much better than the engineering default (Black s method) We are currently developing the method for other power converters and modulation strategies Stephen Cox Matrix power converters: spectra and stability 10/29

34 Conclusions (on spectra) There is a compact way to determine the spectrum for switching power converters, much better than the engineering default (Black s method) We are currently developing the method for other power converters and modulation strategies Current spectra can also be found using the new method Stephen Cox Matrix power converters: spectra and stability 10/29

35 Part II: the stability of matrix power converters Stephen Cox Matrix power converters: spectra and stability 11/29

36 The matrix converter An AC power supply is generally delivered as a three-phase supply x sin Ωt x sin( Ωt+2 π/3) x sin( Ωt 2 π/3) 0V Stephen Cox Matrix power converters: spectra and stability 12/29

37 The matrix converter An AC power supply is generally delivered as a three-phase supply x sin Ωt x sin( Ωt+2 π/3) x sin( Ωt 2 π/3) 0V v A v B v C v a matrix converter v b v c A matrix converter converts between one three-phase power supply and another Inputs: A, B, C Outputs: a, b, c Stephen Cox Matrix power converters: spectra and stability 12/29

38 Matrix converter: modulation strategy Each output is connected in sequence to each of the input lines A a then B a then C a V a a simple modulation strategy Aa n An Bn n+1 Ba Ca t Stephen Cox Matrix power converters: spectra and stability 13/29

39 Matrix converter: modulation strategy Each output is connected in sequence to each of the input lines A a then B a then C a V a a simple modulation strategy n An Bn n+1 Choose switching times so that the low-frequency components of the outputs are the intended new three-phase supply Aa Ba Ca t Stephen Cox Matrix power converters: spectra and stability 13/29

40 Matrix converter: modulation strategy Each output is connected in sequence to each of the input lines A a then B a then C a V a a simple modulation strategy n An Bn n+1 Choose switching times so that the low-frequency components of the outputs are the intended new three-phase supply Full spectrum can be calculated: input/output voltage/current [SM Cox and SC Creagh, 2009, SIAM J Appl Math] Aa Ba Ca t Stephen Cox Matrix power converters: spectra and stability 13/29

41 Power converters: stability controller power supply filter power converter filter/ load But in reality there are filters at the input and output Stephen Cox Matrix power converters: spectra and stability 14/29

42 Power converters: stability controller power supply filter power converter filter/ load But in reality there are filters at the input and output Current drawn by the load leads to a discrepancy between the voltage seen by the controller and the power supply voltage Stephen Cox Matrix power converters: spectra and stability 14/29

43 Power converters: stability controller power supply filter power converter filter/ load But in reality there are filters at the input and output Current drawn by the load leads to a discrepancy between the voltage seen by the controller and the power supply voltage This in turn affects the output currents, which further influences the power converter controller, leading to instability Stephen Cox Matrix power converters: spectra and stability 14/29

44 Power converters: stability controller power supply filter power converter filter/ load But in reality there are filters at the input and output Current drawn by the load leads to a discrepancy between the voltage seen by the controller and the power supply voltage This in turn affects the output currents, which further influences the power converter controller, leading to instability Controller delays can also lead to (or modify) instability Stephen Cox Matrix power converters: spectra and stability 14/29

45 Power converters: stability controller power supply filter power converter filter/ load But in reality there are filters at the input and output Current drawn by the load leads to a discrepancy between the voltage seen by the controller and the power supply voltage This in turn affects the output currents, which further influences the power converter controller, leading to instability Controller delays can also lead to (or modify) instability This instability can be catastrophic in applications! Stephen Cox Matrix power converters: spectra and stability 14/29

46 Matrix converter: averaged model ω i R 2 i i digital controller i o ω o v eq R L 1 1 L 2 C v i matrix converter R v o L 3 3 voltage supply input filter load This diagram shows one input phase and one output phase: in practice there are three inputs and three outputs Stephen Cox Matrix power converters: spectra and stability 15/29

47 Matrix converter: averaged model ω i R 2 i i digital controller i o ω o v eq R L 1 1 L 2 C v i matrix converter R v o L 3 3 voltage supply input filter load Simplest model averages over the switching: duty cycles are specified, but individual switch commutations are not modelled V a Aa discretely switched voltages Ba Ca t averaged Stephen Cox Matrix power converters: spectra and stability 15/29

48 Matrix converter: averaged model ω i R 2 i i digital controller i o ω o v eq R L 1 1 L 2 C v i matrix converter R v o L 3 3 voltage supply input filter load Simplest model averages over the switching: duty cycles are specified, but individual switch commutations are not modelled Introduce space vectors, e.g. 2 3 ( v A (t) + e 2πi/3 v B (t) + e 2πi/3 v C (t) V a Aa discretely switched voltages Ba Ca t averaged ) = g(t) exp iω i t, so the complex function g(t) represents the input voltages, etc. Stephen Cox Matrix power converters: spectra and stability 15/29

49 Matrix converter: averaged model Then the governing ODE for g(t) is ( ) 2P X 3g + Yg(t) = YV eq (t) where X f = R 1 R 2 f +(R 1 L 2 +R 2 L 1 +R 2 L 2 ) ( iω i + d ) ( f +L 1 L 2 iω i + d ) 2 f dt dt and ( Yf = R 2 f + (R 1 R 2 C + L 2 ) iω i + d ) f dt ( + (R 1 L 2 + R 2 L 1 + R 2 L 2 )C iω i + d ) 2 ( f + L 1 L 2 C iω i + d ) 3 f dt dt and P = input power = output power Stephen Cox Matrix power converters: spectra and stability 16/29

50 Matrix converter: averaged model supply filter load R 1 = 0.55Ω R 2 = 300Ω R 3 = 8.2Ω L 1 = 0.90 mh L 2 = 1.16 mh L 3 = 1.3 mh V eq = V C = 4.5µF ω o = 2π 100 rad/s ω i = 2π 50 rad/s D. Casadei, G. Serra, A. Tani, A. Trentin, L. Zarri Theoretical and experimental investigation on the stability of matrix converters. IEEE Trans Industr Electr (2005). 50Hz input 100Hz output Stephen Cox Matrix power converters: spectra and stability 17/29

51 Matrix converter: averaged model Bifurcation parameter is power P or voltage transfer ratio q = reference output voltage input voltage Stephen Cox Matrix power converters: spectra and stability 18/29

52 Matrix converter: averaged model Bifurcation parameter is power P or voltage transfer ratio q = reference output voltage input voltage Results The steady-state operating point is g = constant Stephen Cox Matrix power converters: spectra and stability 18/29

53 Matrix converter: averaged model Bifurcation parameter is power P or voltage transfer ratio q = reference output voltage input voltage Results The steady-state operating point is g = constant The steady-state is stable provided q < q c Stephen Cox Matrix power converters: spectra and stability 18/29

54 Matrix converter: averaged model Bifurcation parameter is power P or voltage transfer ratio q = reference output voltage input voltage Results The steady-state operating point is g = constant The steady-state is stable provided q < q c At q = q c there is a Hopf bifurcation and the solution develops ripple at frequency approx 1651Hz (Typical switching frequency 12.5kHz) Stephen Cox Matrix power converters: spectra and stability 18/29

55 Matrix converter: bifurcation to ripple A weakly nonlinear expansion of the solution near the Hopf bifurcation shows that this bifurcation is strongly subcritical input voltage weakly nonlinear solution Hopf bifn solution with ripple steady state output voltage Results from AUTO (checked with home-made code) Stephen Cox Matrix power converters: spectra and stability 19/29

56 Matrix converter: bifurcation to ripple A weakly nonlinear expansion of the solution near the Hopf bifurcation shows that this bifurcation is strongly subcritical input voltage weakly nonlinear solution Hopf bifn solution with ripple steady state output voltage A start-up transient could lead to catastrophic failure, even at around 50% of the linear stability threshold Stephen Cox Matrix power converters: spectra and stability 19/29

57 Matrix converter: bifurcation to ripple Evolution of disturbances to the steady-state solution, just above threshold (28V>26.88V) 300 g t Catastrophic failure! Stephen Cox Matrix power converters: spectra and stability 20/29

58 Matrix converter: nonlinear ripple Eventually we cannot continue the solution branch, at input voltage Hopf bifn solution with ripple steady state output voltage Stephen Cox Matrix power converters: spectra and stability 21/29

59 Matrix converter: nonlinear ripple Eventually we cannot continue the solution branch, at input voltage Hopf bifn solution with ripple steady state output voltage 400 The solution develops an input current spike when the input voltage f(t) g(t) is close to zero 0 one period of the ripple Stephen Cox Matrix power converters: spectra and stability 21/29

60 Matrix converter: model foundations The sharp current spike occurs over a short fraction of the ripple period Stephen Cox Matrix power converters: spectra and stability 22/29

61 Matrix converter: model foundations The sharp current spike occurs over a short fraction of the ripple period This stretches the assumption that we can ignore the fast switching, and undermines our averaged model Stephen Cox Matrix power converters: spectra and stability 22/29

62 Matrix converter: model foundations The sharp current spike occurs over a short fraction of the ripple period This stretches the assumption that we can ignore the fast switching, and undermines our averaged model To understand the postinstability behaviour of the matrix converter we need to consider the details of the switching itself V a Aa discretely switched voltages Ba Ca t averaged Stephen Cox Matrix power converters: spectra and stability 22/29

63 Matrix converter: nonaveraged model ω i R 2 i i digital controller i o ω o v eq R L 1 1 L 2 C v i matrix converter R v o L 3 3 voltage supply input filter load Introduce vectors v A v i = v B v o = v C v a v b v c i i = i A i B i C i o = i a i b i c Stephen Cox Matrix power converters: spectra and stability 23/29

64 Matrix converter: nonaveraged model ω i R 2 i i digital controller i o ω o v eq R L 1 1 L 2 C v i matrix converter R v o L 3 3 voltage supply input filter load Input: X i i + Y(v i s) = 0 where X f = R 1 R 2 f + (R 1 L 2 + R 2 L 1 + R 2 L 2 ) df dt + L d 2 f 1L 2 dt 2 and Yf = R 2 f +(R 1 R 2 C +L 2 ) df dt +(R 1L 2 +R 2 L 1 +R 2 L 2 )C d2 f dt 2 +L 1L 2 C d3 f dt 3 and s represents the supply voltage Stephen Cox Matrix power converters: spectra and stability 23/29

65 Matrix converter: nonaveraged model ω i R 2 i i digital controller i o ω o v eq R L 1 1 L 2 C v i matrix converter R v o L 3 3 voltage supply input filter load Input: X i i + Y(v i s) = 0 where X f = R 1 R 2 f + (R 1 L 2 + R 2 L 1 + R 2 L 2 ) df dt + L d 2 f 1L 2 dt 2 and Yf = R 2 f +(R 1 R 2 C +L 2 ) df dt +(R 1L 2 +R 2 L 1 +R 2 L 2 )C d2 f dt 2 +L 1L 2 C d3 f dt 3 Output: v o = R 3 i o + L 3 di o dt Stephen Cox Matrix power converters: spectra and stability 23/29

66 Matrix converter: nonaveraged model Can characterise the state of the system by the vector x(t) = v i v i v i i o V a Aa n An Bn n+1 Ba Ca t Stephen Cox Matrix power converters: spectra and stability 24/29

67 Matrix converter: nonaveraged model Can characterise the state of the system by the vector x(t) = v i v i v i i o Then the switching model is the ODE system V a Aa Ca n An Bn n+1 Ba t dx dt = A(t)x(t) + f (t) where f represents the power supply and A(t) is a piecewise-constant matrix which encodes the linear voltage current relations and the discrete switching v A v B v C v a matrix converter v b v c Stephen Cox Matrix power converters: spectra and stability 24/29

68 Matrix converter: nonaveraged model The system dx = A(t)x + f has dt piecewise-constant coefficients, and so can be solved exactly between switch commutations V a Aa n An Bn n+1 Ba Ca t Stephen Cox Matrix power converters: spectra and stability 25/29

69 Matrix converter: nonaveraged model The system dx = A(t)x + f has dt piecewise-constant coefficients, and so can be solved exactly between switch commutations V a n An Bn n+1 But x is not continuous at switch commutations ( v i and v i ) Aa Ba Ca t Stephen Cox Matrix power converters: spectra and stability 25/29

70 Matrix converter: nonaveraged model The system dx = A(t)x + f has dt piecewise-constant coefficients, and so can be solved exactly between switch commutations V a n An Bn n+1 But x is not continuous at switch commutations ( v i and v i ) So at each switch commutation, we impose continuity of capacitor voltages and inductor currents, leading to jump conditions on x Aa Ba Ca t Stephen Cox Matrix power converters: spectra and stability 25/29

71 Matrix converter: nonaveraged model The system dx = A(t)x + f has dt piecewise-constant coefficients, and so can be solved exactly between switch commutations V a n An Bn n+1 But x is not continuous at switch commutations ( v i and v i ) So at each switch commutation, we impose continuity of capacitor voltages and inductor currents, leading to jump conditions on x Use regular sampling of the inputs to determine switching times Aa Ba Ca t Stephen Cox Matrix power converters: spectra and stability 25/29

72 Matrix converter: nonaveraged model The system dx = A(t)x + f has dt piecewise-constant coefficients, and so can be solved exactly between switch commutations V a n An Bn n+1 But x is not continuous at switch commutations ( v i and v i ) So at each switch commutation, we impose continuity of capacitor voltages and inductor currents, leading to jump conditions on x Use regular sampling of the inputs to determine switching times How does the behaviour of the nonaveraged model compare with that of the averaged model? Aa Ba Ca t Stephen Cox Matrix power converters: spectra and stability 25/29

73 Nonaveraged model results Start near steady-state operating point and simulate model for various voltage transfer ratios q and switching periods T q c averaged model steady state unstable T steady state stable Stephen Cox Matrix power converters: spectra and stability 26/29

74 Nonaveraged model results Start near steady-state operating point and simulate model for various voltage transfer ratios q and switching periods T q c averaged model steady state unstable T steady state stable As for the averaged model, q > q c gives instability to ripple But now q c = q c (T ) Stephen Cox Matrix power converters: spectra and stability 26/29

75 Nonaveraged model results v2 + v2 B + v2 A C t Above threshold there is catastrophic failure, as in the averaged model Stephen Cox Matrix power converters: spectra and stability 27/29

76 Nonaveraged model results q c averaged model steady state unstable T steady state stable Stephen Cox Matrix power converters: spectra and stability 28/29

77 Nonaveraged model results q c averaged model steady state unstable T steady state stable Instability threshold tends to that of averaged model as T 0 Stephen Cox Matrix power converters: spectra and stability 28/29

78 Nonaveraged model results q c averaged model steady state unstable T steady state stable Instability threshold tends to that of averaged model as T 0 Instability threshold q c (T ) is significantly below q c (0) for sufficiently rapid switching! Stephen Cox Matrix power converters: spectra and stability 28/29

79 Conclusions Power converters are an important technology, and mathematical modelling currently lags behind applications Stephen Cox Matrix power converters: spectra and stability 29/29

80 Conclusions Power converters are an important technology, and mathematical modelling currently lags behind applications Matrix converters are an emerging technology for converting AC AC Stephen Cox Matrix power converters: spectra and stability 29/29

81 Conclusions Power converters are an important technology, and mathematical modelling currently lags behind applications Matrix converters are an emerging technology for converting AC AC A model averaged over a switching period shows that matrix converters can become unstable in a subcritical bifurcation to a branch of solutions with ripple Stephen Cox Matrix power converters: spectra and stability 29/29

82 Conclusions Power converters are an important technology, and mathematical modelling currently lags behind applications Matrix converters are an emerging technology for converting AC AC A model averaged over a switching period shows that matrix converters can become unstable in a subcritical bifurcation to a branch of solutions with ripple Averaged model significantly overestimates threshold for instability; our full switching model gives the threshold Stephen Cox Matrix power converters: spectra and stability 29/29

83 Conclusions Power converters are an important technology, and mathematical modelling currently lags behind applications Matrix converters are an emerging technology for converting AC AC A model averaged over a switching period shows that matrix converters can become unstable in a subcritical bifurcation to a branch of solutions with ripple Averaged model significantly overestimates threshold for instability; our full switching model gives the threshold What about more realistic modulation strategies? Stephen Cox Matrix power converters: spectra and stability 29/29

84 Conclusions Power converters are an important technology, and mathematical modelling currently lags behind applications Matrix converters are an emerging technology for converting AC AC A model averaged over a switching period shows that matrix converters can become unstable in a subcritical bifurcation to a branch of solutions with ripple Averaged model significantly overestimates threshold for instability; our full switching model gives the threshold What about more realistic modulation strategies? Delays in the controller? Losses (etc) in the converter? Stephen Cox Matrix power converters: spectra and stability 29/29

85 Conclusions Power converters are an important technology, and mathematical modelling currently lags behind applications Matrix converters are an emerging technology for converting AC AC A model averaged over a switching period shows that matrix converters can become unstable in a subcritical bifurcation to a branch of solutions with ripple Averaged model significantly overestimates threshold for instability; our full switching model gives the threshold What about more realistic modulation strategies? Delays in the controller? Losses (etc) in the converter? Can the catastrophic failure be tamed? Stephen Cox Matrix power converters: spectra and stability 29/29