Switched mode power supply output filter configuration

• US 9,507,901 B2
• Filed: 12/01/2014
• Issued: 11/29/2016
• Est. Priority Date: 03/31/2014
• Status: Active Grant

• Current Assignee: Telefonaktiebolaget LM Ericsson
• Original Assignee: Telefonaktiebolaget LM Ericsson
• Inventors: Nilsson, Daniel, Mellteg, Magnus, Karlsson, Magnus
• Primary Examiner(s): Rossoshek, Helen
• Application Number:
  US14/418,334
• Publication Number:
  US 20150331984A1
• Time in Patent Office: 729 Days
• US Class Current: 1/1
• CPC Class Codes: G06F 2111/02 CAD in a network environment, e.g. collaborative CAD or distributed simulation
• CPC Class Codes: G06F 2111/06 Multi-objective optimisation, e.g. Pareto optimisation using simulated annealing [SA], ant colony algorithms or genetic algorithms [GA]
• CPC Class Codes: G06F 30/36 Circuit design at the analogue level
• CPC Class Codes: H01L 25/072 the devices being arranged next to each other
• CPC Class Codes: H02M 1/126 using passive filters
• CPC Class Codes: H02M 1/14 Arrangements for reducing ripples from dc input or output
• CPC Class Codes: H02M 1/44 Circuits or arrangements for compensating for electromagnetic interference in converters or inverters
• CPC Class Codes: H02M 3/07 using capacitors charged and discharged alternately by semiconductor devices with control electrode , e.g. charge pumps
• CPC Class Codes: H02M 3/156 with automatic control of output voltage or current, e.g. switching regulators

Abstract

Method determines a configuration of capacitors in an output filter of a switched mode power supply (SMPS). Candidate configurations are generated, each associated with a respective cost function value calculated based on number of capacitors of each kind in the candidate configuration and cost assigned to each kind of capacitor. The candidate configurations are ordered based on their cost function values. A binary search through the ordered candidate configurations determines a capacitor configuration associated with a lowest cost function value to allow a defined load transient response requirement of the SMPS. Whether a candidate configuration allows the defined load transient response requirement of the SMPS is determined using a model of a SMPS having the candidate configuration of the output filter simulating deviation of output voltage of the SMPS in response to change in load current of the SMPS and determining whether the defined load transient response requirement is allowed.

Claims

1. A method of determining a configuration of capacitors in an output filter of a switched mode power supply, SMPS, the method comprising:

generating candidate configurations of capacitors defining respective arrangements of capacitors of different kinds in the output filter, each candidate configuration being associated with a respective value of a cost function that is calculated based on a number of capacitors of each kind in the candidate configuration and a cost assigned to each kind of capacitor, the candidate configurations being ordered in order of increasing or decreasing values of the cost function;

performing a binary search on the ordered candidate configurations of capacitors to determine a capacitor configuration associated with a lowest value of the cost function from among candidate configurations of capacitors that are determined during the binary search to allow a load transient response requirement of the SMPS to be fulfilled,

wherein whether a candidate configuration allows the load transient response requirement of the SMPS to be fulfilled is determined using a model of a SMPS having the candidate configuration of the output filter to simulate a deviation of an output voltage of the SMPS in response to a change in a load current of the SMPS, and determining whether the load transient response requirement is fulfilled based on the simulated deviation of the output voltage, and

wherein performing the binary search comprises determining values of control law parameters for turning a feedback loop that is arranged to regulate the output voltage of the SMPS, in accordance with a control law defined by the control law parameters, the control law parameter values being determined by;

determining a respective estimate of a natural frequency of each of two zeros in a transfer function that corresponds to a model of the feedback loop by;

determining, for each of a plurality of points in a search space, wherein the coordinates of each of the points correspond to candidate values of the natural frequencies of the zeros, a deviation of the output voltage of the SMPS that occurs in response to a variation in the load current of the SMPS,

evaluating a goal function using the determined deviation of the output voltage of the SMPS at each of the plurality of points to obtain a measure of at least one of a size and a recovery time of the respective output voltage deviation,

estimating a location of a minimum of the goal function in the search space based on the evaluated values of the goal function, the location of the minimum of the goal function corresponding to the natural frequencies of the zeros, and

calculating, based on the determined estimates of the natural frequencies of the zeros, the control law parameter values for tuning the feedback loop; and

assembling the SMPS having the output filter with the capacitor configuration that is determined to be associated with the lowest value of the cost function from among the candidate configurations of capacitors.

2. The method according to claim 1, wherein the cost assigned to each kind of capacitor is indicative of at least one of a price and a physical size of the kind of capacitor.

3. The method according to claim 2, wherein the cost assigned to each kind of capacitor is indicative of a weighted sum of the price and the physical size of the kind of capacitor.

4. The method according to claim 3, wherein:

at least one of the candidate configurations comprises a plurality of different arrangements of capacitors of different kinds in the output filter, all of the different arrangements being associated with the same value of the cost function; and

generating candidate configurations of capacitors further comprises selecting, for each of the at least one candidate configuration having a plurality of different arrangements of capacitors of different kinds in the output filter, an arrangement of capacitors from the plurality of different arrangements that are to be retained for the binary search, the selection being performed by;

selecting a subset of the plurality of different arrangements based on a total of the costs assigned to the capacitors in each of the plurality of different arrangements;

comparing a distribution of capacitors among the different kinds of capacitors in each of the arrangements in the subset with a target distribution; and

selecting an arrangement of capacitors that has the most similar distribution of capacitors among the different kinds of capacitors to the target distribution.

5. The method according to claim 2, wherein:

at least one of the candidate configurations comprises a plurality of different arrangements of capacitors of different kinds in the output filter, all of the different arrangements being associated with the same value of the cost function; and

generating candidate configurations of capacitors further comprises selecting, for each of the at least one candidate configuration having a plurality of different arrangements of capacitors of different kinds in the output filter, an arrangement of capacitors from the plurality of different arrangements that are to be retained for the binary search, the selection being performed by;

selecting a subset of the plurality of different arrangements based on one of a total price and a total physical size of the capacitors in each of the plurality of different arrangements; and

selecting an arrangement from the subset based on the other of the total price and the total physical size of the capacitors in each of the arrangements in the subset.

6. The method according to claim 2, wherein:

at least one of the candidate configurations comprises a plurality of different arrangements of capacitors of different kinds in the output filter, all of the different arrangements being associated with the same value of the cost function; and

generating candidate configurations of capacitors further comprises selecting, for each of the at least one candidate configuration having a plurality of different arrangements of capacitors of different kinds in the output filter, an arrangement of capacitors from the plurality of different arrangements that are to be retained for the binary search, the selection being performed by;

comparing a distribution of capacitors among the different kinds of capacitors in each of the plurality of different arrangements with a target distribution; and

selecting an arrangement of capacitors that has the most similar distribution of capacitors among the different kinds of capacitors to the target distribution.

7. The method according to claim 2, wherein;

at least one of the candidate configurations comprises a plurality of different arrangements of capacitors of different kinds in the output filter, all of the different arrangements being associated with the same value of the cost function; and

generating candidate configurations of capacitors further comprises selecting, for each of the at least one candidate configuration having a plurality of different arrangements of capacitors of different kinds in the output filter, an arrangement of capacitors from the plurality of different arrangements that arc to be retained for the binary search, the selection being performed by;

selecting a subset of the plurality of different arrangements based on one of a total price and a total physical size of the capacitors in each of the plurality of different arrangements;

comparing a distribution of capacitors among the different kinds of capacitors in each of the arrangements in the subset with a target distribution; and

selecting an arrangement of capacitors that has the most similar distribution of capacitors among the different kinds of capacitors to the target distribution.

8. The method according to claim 1, wherein the goal function provides a measure of at least one of a maximum SMPS output voltage deviation during the variation in load current of the SMPS, and a recovery time during which the output voltage of the SMPS returns to within a tolerance band of output voltage values following the variation in the load current of the SMPS.

9. The method according to claim 8, wherein the goal function comprises

10. The method according to claim 8, wherein the goal function comprises

11. The method according to claim 8, wherein the goal function comprises

[∫

₀^Ttrans|V_out(t)−

V_DC|^Ndt]^1/N,

wherein V_out(t) denotes the deviation in the output voltage of the SMPS that occurs in response to the variation in the load current of the SMPS, T_trans denotes the duration of the variation in the load current of the SMPS, V_DC denotes a DC voltage from which the output voltage of the SMPS deviates in response to the variation in the load current of the SMPS, and N is an integer greater than or equal to 1.

12. The method according to claim 1, wherein the estimates of the respective natural frequencies of the zeros are determined by:

determining a respective natural frequency of each of two dominant poles in a transfer function corresponding to the model of the SMPS, the natural frequencies of the dominant poles being NF_pole1 and NF_pole2, where NF_pole1≦

NF_pole2;

defining a triangular search area for the zeros in the search space, the triangular search area having vertices at points (NF_zeromin, NF_zeromin), (NF_zeromax, NF_zeromax), and one of (NF_zeromin, NF_zeromax) and (NF_zeromax, NF_zeromin), where NF_zeromin=NF_pole1/r_min, NF_zeromax=NF_pole2·

r_max, and r_min and r_max are range factors each between 1.5 and 3;

simulating, for each of a plurality of points in the triangular search area, a deviation of the output voltage of the SMPS that occurs in response to the variation in the load current of the SMPS, using the model of the SMPS and the model of the feedback loop;

evaluating the goal function using the simulated deviation of the output voltage of the SMPS at each of the plurality of points within the triangular search area to obtain the measure of at least one of the physical size and the recovery time of the respective output voltage deviation; and

estimating the location of the minimum of the goal function within the triangular search area based on the evaluated values of the goal function.

13. The method according to claim 12, wherein the natural frequencies of the zeros are estimated by:

(i) calculating a value of the goal function for each of the vertices of the triangular search area;

(ii) replacing the vertex of the triangular search area yielding the greatest value of the goal function with a vertex within the search area that yields a smaller value of the goal function to generate a triangle within the search area that is defined by the vertex within the search area and the remaining vertices of the triangular search area;

(iii) evaluating the goal function at the new vertex;

(iv) replacing the vertex of the triangle yielding the greatest value of the goal function with a new vertex within the search area that yields a smaller value of the goal function to generate a new triangle within the search area;

(v) repeating steps (iii) and (iv) to generate a series of triangles within the search area until a stopping criterion is met; and

(vi) when the stopping criterion is met, estimating the location of the minimum of the goal function within the search area based on the evaluated values of the goal function to obtain an estimate of the natural frequencies of the zeros.

14. The method according to claim 13, wherein step (vi) comprises selecting a point in or on the triangle generated in the final performance of step (iv), the coordinates of the selected point providing the estimate of the natural frequencies of the zeros.

15. The method according to claim 12, further comprising:

evaluating a second goal function using a prescribed deviation of the output voltage of the SMPS and the determined deviation of the output voltage of the SMPS at each of the plurality of points in the triangular search area to obtain a measure of how much the determined deviation of the output voltage exceeds the prescribed deviation of the output voltage in response to the variation in the load current of the SMPS,

wherein the natural frequencies of the zeros are estimated by;

(i) calculating a value of the second goal function for each of the vertices of the triangular search area;

(ii) replacing the vertex of the triangular search area yielding the greatest value of the second goal function with a new vertex within the search area that yields a smaller value of the second goal function to generate a triangle within the search area that is defined by the new vertex within the search area and the remaining vertices of the triangular search area;

(iii) evaluating the second goal function at the new vertex;

(iv) replacing the vertex of the triangle yielding the greatest value of the second goal function with a new vertex within the search area that yields a smaller value of the second goal function to generate a new triangle within the search area;

(v) repeating steps (iii) and (iv) to generate a series of triangles within the search area until the second goal function is evaluated to be zero at the new vertex in step (iii) and has been evaluated to be zero at the remaining vertices of the triangle generated in the final performance of step (iv), the triangle generated in the final performance of step (iv) defining a restricted triangular search area;

(vi) calculating a value of the goal function for each of the vertices of the restricted search area;

(vii) replacing the vertex of the restricted search area yielding the greatest value of the goal function with a new vertex within the restricted search area that yields a smaller value of the goal function to generate a triangle within the restricted search area that is defined by the new vertex and the remaining vertices of the restricted search area;

(viii) evaluating the goal function at the new vertex from step (vii);

(ix) replacing the vertex of the triangle yielding the greatest value of the goal function with a new vertex within the restricted search area that yields a smaller value of the goal function to generate a new triangle within the restricted search area;

(x) repeating steps (viii) and (ix) to generate a series of triangles within the restricted search area until a stopping criterion is met; and

(xi) when the stopping criterion is met, estimating the location of the minimum of the goal function within the restricted search area based on the evaluated values of the goal function to obtain an estimate of the natural frequencies of the zeros.

16. The method according to claim 15, wherein step (xi) comprises selecting a point in or on the triangle generated in the final performance of step (ix), the coordinates of the selected point providing the estimate of the natural frequencies of the zeros.

17. The method according to claim 15, wherein step (viii) further comprises evaluating the second goal function at the new vertex from step (vii) and, when the second goal function is evaluated to be greater than zero at the new vertex from step (vii), the method comprises repeating steps (iv) to (viii), otherwise continuing to step (ix).

18. The method according to claim 1, wherein determining a respective estimate of the natural frequency of each of two zeros in the transfer function further comprises:

evaluating a second goal function using a prescribed deviation of the output voltage of the SMPS and the determined deviation of the output voltage of the SMPS at each of the plurality of points in the search space to obtain a measure of how much the determined deviation of the output voltage exceeds the prescribed deviation of the output voltage in response to the variation in the load current of the SMPS,

wherein the location of the minimum of the goal function is estimated based on the evaluated values of the goal function and the evaluated values of the second goal function.

19. The method according to claim 18, wherein the prescribed deviation of the output voltage of the SMPS follows a prescribed variation over time.

20. The method according to claim 1, wherein the feedback loop comprises a PID regulator, and the model of the feedback loop models the feedback loop as a PID regulator, the control law parameters comprising the P, I and D control parameters of the PID regulator.

21. A non-transitory computer-readable storage medium storing computer program instructions which, when executed by a processor, cause the processor to perform a method as set out in at least claim 1.

22. An apparatus for determining a configuration of capacitors in an output filter of a switched mode power supply, SMPS, the apparatus comprising:

a processor; and

a memory coupled to the processor and storing program code executed by the processor, the program code comprising;

a candidate configuration generator module arranged to generate candidate configurations of capacitors defining respective arrangements of capacitors of different kinds in the output filter, the candidate configuration generator module comprising;

a cost calculation module arranged to evaluate, for each candidate configuration, a cost function based on a number of capacitors of each kind in the candidate configuration and a cost assigned to each kind of capacitor; and

an association module arranged to associate each candidate configuration with a respective value of the cost function calculated by the cost calculation module,

wherein the candidate configuration generator module is arranged to order the candidate configurations in order of increasing or decreasing values of the cost function; and

a search module arranged to perform a binary search on the ordered candidate configurations of capacitors to determine a capacitor configuration associated with a lowest value of the cost function from among candidate configurations of capacitors that are determined during the binary search to allow a load transient response requirement of the SMPS to be fulfilled, the search module comprising;

an output voltage deviation determining module arranged to simulate a deviation of an output voltage of the SMPS in response to a change in a load current of the SMPS, using a model of a SMPS having a candidate configuration of the output filter; and

an evaluation module arranged to determine whether the candidate configuration of capacitors allows the load transient response requirement to be fulfilled based on the simulated deviation of the output voltage,

wherein the search module is further arranged to determine values of control law parameters for tuning feedback loop that is arranged to regulate the output voltage of the SMPS, in accordance with a control law defined by the control law parameters, the search module further comprising;

a natural frequency estimator arranged to determine a respective estimate of a natural frequency of each of two zeros in a transfer function that corresponds to a model of the feedback loop, the natural frequency estimator comprising the output voltage deviation determining module and the evaluation module, wherein;

the output voltage deviation determining module is arranged to simulate, for each of a plurality of points in a search space, wherein the coordinates of each of the points correspond to candidate values of the natural frequencies of the zeros, a deviation of the output voltage of the SMPS that occurs in response to a variation in the load current of the SMPS, using the model of the SMPS and the model of the feedback loop;

the evaluation module is arranged to evaluate a goal function using the simulated deviation of the output voltage of the SMPS at each of the plurality of points to obtain a measure of at least one of a size and a recovery time of the respective output voltage deviation; and

the natural frequency estimator is arranged to estimate a location of a minimum of the goal function in the search space based on the evaluated values of the goal function, the location of the minimum of the goal function corresponding to the natural frequencies of the zeros; and

a control law parameter calculator arranged to calculate, based on the determined estimates of the natural frequencies of the zeros, the control law parameter values for tuning the feedback loop; and

a module that stores in the memory a configuration for the SMPS having the output filter with the capacitor configuration that is determined to be associated with the lowest value of the cost function from among the candidate configurations of capacitors.

23. The apparatus according to claim 22, wherein the cost assigned to each kind of capacitor is indicative of at least one of a price and a physical size of the kind of capacitor.

24. The apparatus according to claim 23, wherein the cost assigned to each kind of capacitor is indicative of a weighted sum of the price and the physical size of the kind of capacitor.

25. The apparatus according to claim 24, wherein:

at least one of the candidate configurations comprises a plurality of different arrangements of capacitors of different kinds in the output filter, all of the different arrangements being associated with the same value of the cost function; and

the candidate configuration generator module is arranged to select, for each of the at least one candidate configuration having a plurality of different arrangements of capacitors of different kinds in the output filter, an arrangement of capacitors from the plurality of different arrangements that are to be retained for the binary search, the selection being performed by;

selecting a subset of the plurality of different arrangements based on one of the total price and the total physical size of the capacitors in each of the plurality of different arrangements;

comparing a distribution of capacitors among the different kinds of capacitors in each of the arrangements in the subset with a target distribution; and

selecting an arrangement of capacitors that has the most similar distribution of capacitors among the different kinds of capacitors to the target distribution.

26. The apparatus according to claim 24, wherein:

at least one of the candidate configurations comprises a plurality of different arrangements of capacitors of different kinds in the output filter, all of the different arrangements being associated with the same value of the cost function; and

the candidate configuration generator module is arranged to select, for each of the at least one candidate configuration having a plurality of different arrangements of capacitors of different kinds in the output filter, an arrangement of capacitors from the plurality of different arrangements that are to be retained for the binary search, the selection being performed by;

selecting a subset of the plurality of different arrangements based on a total of the costs assigned to the capacitors in each of the plurality of different arrangements;

comparing a distribution of capacitors among the different kinds of capacitors in each of the arrangements in the subset with a target distribution; and

selecting an arrangement of capacitors that has the most similar distribution of capacitors among the different kinds of capacitors to the target distribution.

27. The apparatus according to claim 23, wherein:

at least one of the candidate configurations comprises a plurality of different arrangements of capacitors of different kinds in the output filter, all of the different arrangements being associated with the same value of the cost function; and

the candidate configuration generator module is arranged to select, for each of the at least one candidate configuration having a plurality of different arrangements of capacitors of different kinds in the output filter, an arrangement of capacitors from the plurality of different arrangements that are to be retained for the binary search, the selection being performed by;

selecting a subset of the plurality of different arrangements based on one of a total price and a total physical size of the capacitors in each of the plurality of different arrangements; and

selecting an arrangement from the subset based on the other of the total price and the total physical size of the capacitors in each of the arrangements in the subset.

28. The apparatus according to claim 23, wherein:

at least one of the candidate configurations comprises a plurality of different arrangements of capacitors of different kinds in the output filter, all of the different arrangements being associated with the same value of the cost function; and

the candidate configuration generator module is arranged to select, for each of the at least one candidate configuration having a plurality of different arrangements of capacitors of different kinds in the output filter an arrangement of capacitors from the plurality of different arrangements that are to be retained for the binary search, the selection being performed by;

comparing a distribution of capacitors among the different kinds of capacitors in each of the plurality of different arrangements with a target distribution; and

selecting an arrangement of capacitors that has the most similar distribution of capacitors among the different kinds of capacitors to the target distribution.

29. The apparatus according to claim 22, wherein the goal function provides a measure of at least one of a maximum SMPS output voltage deviation during the variation in load current of the SMPS, and a recovery time during which the output voltage of the SMPS returns to within a tolerance band of output voltage values following the variation in the load current of the SMPS.

30. The apparatus according to claim 29, wherein the goal function comprises

31. The apparatus according to claim 29, wherein the goal function comprises

32. The apparatus according to claim 29, wherein the goal function comprises

[∫

₀^Ttrans|V_out(t)−

V_DC|^Ndt]^1/N,

wherein V_out(t) denotes the variation in the output voltage of the SMPS that occurs in response to the variation in the load current of the SMPS, T_trans denotes the duration of the variation in the load current of the SMPS, V_DC denotes a DC voltage from which the output voltage of the SMPS deviates in response to the variation in the load current of the SMPS, and N is an integer greater than or equal to 1.

33. The apparatus according to claim 22, wherein:

the natural frequency estimator is further arranged to;

determine a respective natural frequency of each of two dominant poles in a transfer function corresponding to the model of the SMPS, the natural frequencies of the dominant poles being NF_pole1 and NF_pole2, where NF_pole1≦

NF_pole2; and

define a triangular search area for the zeros in the search space, the triangular search area having vertices at points (NF_zeromin, NF_zeromin), (NF_zeromax, NF_zeromax), and one of (NF_zeromin, NF_zeromax) and (NF_zeromax, NF_zeromin), where NF_zeromin=NF_pole1/r_min, NF_zeromax=NF_pole2·

r_max, and r_min and r_max are range factors each between 1.5 and 3;

the output voltage deviation determining module is arranged to simulate, for each of a plurality of points in the triangular search area, a deviation of the output voltage of the SMPS that occurs in response to the variation in the load current of the SMPS, using the model of the SMPS and the model of the feedback loop;

the evaluation module is arranged to evaluate the goal function using the determined deviation of the output voltage of the SMPS at each of the plurality of points within the triangular search area to obtain the measure of at least one of the physical size and the recovery time of the respective output voltage deviation; and

the natural frequency estimator is arranged to estimate a location of the minimum of the goal function in the triangular search area based on the evaluated values of the goal function.

34. The apparatus according to claim 33, wherein the natural frequency estimator is arranged to calculate an estimate of the respective natural frequencies of the zeros by:

(i) calculating a value of the goal function for each of the vertices of the triangular search area;

(ii) replacing the vertex of the triangular search area yielding the greatest value of the goal function with a vertex within the search area that yields a smaller value of the goal function to generate a triangle within the search area that is defined by the vertex within the search area and the remaining vertices of the triangular search area;

(iii) evaluating the goal function at the new vertex;

(iv) replacing the vertex of the triangle yielding the greatest value of the goal function with a new vertex within the search area that yields a smaller value of the goal function to generate a new triangle within the search area;

(v) repeating steps (iii) and (iv) to generate a series of triangles within the search area until a stopping criterion is met; and

(vi) when the stopping criterion is met, estimating the location of the minimum of the goal function in the search area based on the evaluated values of the goal function to obtain an estimate of the natural frequencies of the zeros.

35. The apparatus according to claim 34, wherein the natural frequency estimator is arranged to select in step (vi) a point in or on the triangle generated in the final performance of step (iv), the coordinates of the selected point providing the estimate of the natural frequencies of the zeros.

36. The apparatus according to claim 22, wherein:

the evaluation module is further arranged to evaluate a second goal function using a prescribed deviation of the output voltage of the SMPS and the determined deviation of the output voltage of the SMPS at each of the plurality of points in the search space at which the deviation of the output voltage of the SMPS has been determined, to obtain a measure of how much the determined deviation of the output voltage exceeds the prescribed deviation of the output voltage in response to the variation in the load current of the SMPS; and

the natural frequency estimator is arranged to estimate the location of the minimum of the goal function in the search space based on the evaluated values of the goal function and the evaluated values of the second goal function.

37. The apparatus according to claim 33, wherein:

the evaluation module is further arranged to evaluate a second goal function using a prescribed deviation of the output voltage of the SMPS and the determined deviation of the output voltage of the SMPS at each of the plurality of points in the triangular search area to obtain a measure of how much the determined deviation of the output voltage exceeds the prescribed deviation of the output voltage in response to the variation in the load current of the SMPS,

the natural frequency estimator is arranged to calculate an estimate of the respective natural frequencies of the zeros by;

(i) calculating a value of the second goal function for each of the vertices of the triangular search area;

(ii) replacing the vertex of the triangular search area yielding the greatest value of the second goal function with a new vertex within the search area that yields a smaller value of the second goal function to generate a triangle within the search area that is defined by the new vertex within the search area and the remaining vertices of the triangular search area;

(iii) evaluating the second goal function at the new vertex;

(iv) replacing the vertex of the triangle yielding the greatest value of the second goal function with a new vertex within the search area that yields a smaller value of the second goal function to generate a new triangle within the search area;

(v) repeating steps (iii) and (iv) to generate a series of triangles within the search area until the second goal function is evaluated to be zero at the new vertex in step (iii) and has been evaluated to be zero at the remaining vertices of the triangle generated in the final performance of step (iv), the triangle generated in the final performance of step (iv) defining a restricted triangular search area;

(vi) calculating a value of the goal function for each of the vertices of the restricted search area;

(vii) replacing the vertex of the restricted search area yielding the greatest value of the goal function with a new vertex within the restricted search area that yields a smaller value of the goal function to generate a triangle within the restricted search area that is defined by the new vertex and the remaining vertices of the restricted search area;

(viii) evaluating the goal function at the new vertex from step (vii);

(ix) replacing the vertex of the triangle yielding the greatest value of the goal function with a new vertex within the restricted search area that yields a smaller value of the goal function to generate a new triangle within the restricted search area;

(x) repeating steps (viii) and (ix) to generate a series of triangles within the restricted search area until a stopping criterion is met; and

(xi) when the stopping criterion is met, estimating the location of the minimum of the goal function within the restricted search area based on the evaluated values of the goal function to obtain an estimate of the natural frequencies of the zeros.

38. The apparatus according to claim 37, wherein the natural frequency estimator is arranged to select in step (xi) a point in or on the triangle generated in the final performance of step (ix), the coordinates of the selected point providing the estimate of the natural frequencies of the zeros.

39. The apparatus according to claim 37, wherein the natural frequency estimator is further arranged to evaluate in step (viii) the second goal function at the new vertex from step (vii) and, when the second goal function is evaluated to be greater than zero at the new vertex from step (vii), to repeat steps (iv) to (viii), and otherwise to perform steps (ix) to (xi).

40. The apparatus according to claim 36, wherein the prescribed deviation of the output voltage of the SMPS follows a prescribed variation over time.

41. The apparatus according to claim 22, wherein the feedback loop comprises a PID regulator and the model of the feedback loop models the feedback loop as a PID regulator, the control law parameters comprising the P, I and D control parameters of the PID regulator.