Abstract
Purpose
The purpose of this paper is to evaluate the worstcase behavior of a given electronic circuit by varying the values of the components in a meaningful way in order not to exceed predefined currents or voltages limits during a transient operation.
Design/methodology/approach
An analytic formulation is used to identify the timedependent solution of voltages or currents using proper state equations in closed form. Circuits with linear elements can be described by a system of differential equations, while circuits composing nonlinear elements are described by piecewiselinear models. A sequential quadratic program (SQP) is used to find the worstcase scenario.
Findings
It is found that the worstcase scenario can be obtained with as few solutions to the forward problem as possible by applying an SQP method.
Originality/value
The SQP method in combination with the analytic forward solver approach shows that the worstcase limit converges in a few steps even if the worstcase limit is not on the boundary of the parameters.
Keywords
Citation
Schenk, M., Muetze, A., Krischan, K. and Magele, C. (2019), "Worstcase analysis of electronic circuits based on an analytic forward solver approach", COMPEL  The international journal for computation and mathematics in electrical and electronic engineering, Vol. 38 No. 5, pp. 16551666. https://doi.org/10.1108/COMPEL1220180531
Publisher
:Emerald Publishing Limited
Copyright © 2019, Mario Schenk, Annette Muetze, Klaus Krischan and Christian Magele.
License
Published by Emerald Publishing Limited. This article is published under the Creative Commons Attribution (CC BY 4.0) licence. Anyone may reproduce, distribute, translate and create derivative works of this article (for both commercial and noncommercial purposes), subject to full attribution to the original publication and authors. The full terms of this licence may be seen at http://creativecommons.org/licences/by/4.0/legalcode
1. Introduction
Electronic components always come along with certain tolerances; therefore, worstcase dimensioning of electronic circuits composed of such components has been gaining more and more importance. Some simulators [e.g. LTspice (Analog Devices, 2019) and PSpice (Orcad PSpice, 2019)] offer the possibility for a Monte Carlo (MC) analysis. This analysis provides statistical data on the impact of a device parameter’s variance. A major disadvantage of stochastic methods is that they require a high number of simulation runs to reach the worstcase limits. This can be improved advantageously by solving currents and voltages analytically. The proposed approach allows conducting a full search over the parameter space. This, in turn, provides the possibility of worstcase analyses of the different parameters of interest. (In our example cases the maximum current that may occur was chosen.) Additionally, it allows full flexibility in modeling the individual components and their parameters. Furthermore, simulation results are extremely compact and can theoretically be stored with arbitrary precision. Finally, the objective function f(x) needed for any optimizer is available in analytic form. In this paper, the analytic forward solver approach (AFSA) and the sequential quadratic program (SQP) are implemented in Maple (Maplesoft, 2019), a computer algebra system (CAS). Section 2 explores the analytic forward solver approach and the SQP approach. Section 3 describes the chosen example case application, a closedloop flyback converter and compares the performance of the SQP method with an MC analysis and evolution strategy. Examples for interesting references in the context of worstcase and sensitivity analyses are given by Chiariello et al. (2015), Khaligh et al. (2006); Lian (2012).
2. Method overview
2.1 Analytic forward solver approach
Today, many circuit simulators for electronic circuits with different approaches are available. The proposed approach uses analytic solution techniques and has been developed especially for optimization and worstcase dimensioning of smallscale electronic circuits. The advantages are:
the full flexibility in the modeling of the individual components and their parameters;
simulation results are extremely compact and can theoretically be stored with arbitrary precision; even with a high number of simulation runs, the generated data remain easy to handle; and
analytic methods show the potential of more efficient parameter studies.
While a comprehensive review of all existing circuit simulators is not within the scope of this paper, a short overview is nevertheless provided, for the sake of completion. The analytic forward solver approach supports analytic timedomain transient analysis for switched networks with piecewiselinear models and uses analytic methods for solving the systems of ordinary differential equations (ODEs). Symbolic simulators like ISAAC (Gielen et al., 1989), SAPWIN (Liberatore et al., 1995; Fontana et al., 2015) and Analog Insyde (Thomassian, 2007) exist. These simulators do not support fully analytic timedomain transient analysis for switched networks with piecewiselinear models. The simulator for integrated switchedmode power supplies circuits (SISMPSC) (Cliquennois and Trochut, 2007) is based on symbolic calculus tools and supports symbolic statespace equations (SSSE) but uses numerical methods for solving the systems of ordinary differential equations (ODEs). For the elemental circuit description the analytic forward solver approach uses a special CircuitModel instead of the widely used netlist as, for example, used in the different implementations of the core SPICE algorithm SPICE2 (Nagel, 1975), LTspice and PSpice. A CircuitModel describes the electronic circuit with symbolic ordinary differential equations, if state variables are present. In the case of no state variables, the electronic circuit is described by symbolic algebraic equations. The electronic circuit to be simulated may contain linear and nonlinear components. Linear parts are described directly with a SubCircuitModel, an extended symbolic statespace model (ESSSM) and nonlinear ones with a CircuitModel, comprising several SubCircuitModels itself, the associated boundary conditions and a state table. The fulfilled boundary conditions of the active SubCircuitModel are the reference (input for the state table) for the next SubCircuitModels. An example of a SubCircuitModel is shown in Figure 1.
The associated ESSSM is described by (1)(3). Equation (1) represents the system of differential (state) equations, (2) the signal of interest, and (3) the I/O interface. In this example, the inputcell y_{I}(t) of the I/O interface is not defined:
I/O interface:
A collection of predefined SubCircuitModels is provided for the individual circuit design. Connecting such simple predefined SubCircuitModels results in a new SubCircuitModel. This results in a large number of possible SubCircuitModels. The advantage of this approach is that the electronic circuit to be simulated can be built from such SubCircuitModels without transformation to a statespace model; only the I/O definitions must be substituted.
The ODE system solver module generally solves the ODE system from each SubCircuitModel. The proposed approach uses the builtin ODE solver from Maple. Out of the different solver methods and options available, the proposed approach uses the Laplace method. When the ODE system from each SubCircuitModel has been solved once, the solution is stored and, therefore, the ODE system does not need to be solved again. The analytic solution of the ODE System (1) for I_{L1}(t) from the SubCircuitModel in Figure 1 is shown (5). The initial conditions I_{L1}(0), V_{C1}(0) and I_{out}(t) were set to 0.
The timedomain transient analysis for a CircuitModel starts at the first SubCircuitModel, then, the boundary conditions for this SubCircuitModel are verified. The fulfilled boundary condition determines the next SubCircuitModel and the analytic solution for that time interval. This is repeated until the final circuit operating time to be simulated t_{Sim,} is reached.
2.2 Sequential quadratic program approach
The implemented SQP method (6) is based on an active set strategy with linear inequality constraints (7) (Fletcher, 2000):
x = column vector of the device parameters;
f(x) = objective function;
H = Hessian matrix of the objective function;
c = gradient of the objective function;
A = constant m × n matrix;
b = constant column vector b ∈ R^{m};
n = number of parameters; and
m = number of constraints.
The Hessian matrix H is updated in each iteration step until the optimal solution is found. To apply an SQP strategy, the objective function f(x) must be defined; for example, the inductor current (2) shown in Figure 1. The SQP strategy generally finds the minimum of the objective function f(x), in case of the maximum –f(x) must be used instead. The inequality constraints (7) are constructed from the device parameter bounds. In case of n device parameters, it is m = 2n. The general form of (7) in matrix form is described in (8).
2.3 Worstcase analysis of a resistor inductor capacitor (RLC) series resonant circuit
The schematic of the RLC series resonant circuit is shown in Figure 1. Table I summarizes the values of the components and the SQP parameters. The maximum peak inductor current Î_{L1} in steadystate should be determined as a function of two parameters, e.g. t and L_{1}. The objective function f(x) for the minimum peak inductor current is obtained by the evaluation of the component values from Table I in (5). The objective function for the maximum is:
To find the maximum peak inductor in steadystate, the lower bound for t was chosen to be 10 times larger than the period T = 1/f. The 3D plot of the objective function I_{L1 max}(t, L_{1}) (9) including the solution path of the SQP method is illustrated in Figure 2. The implemented SQP method converges to the maximum after 7 iterations (summarized in Table II) with Î_{L1} = 0.04 A and is also visualized in the contour plot in Figure 3. The solution of the SQP method is exactly the same as expected: at the resonant frequency, the capacitive and inductive reactances cancel each other and the current through the inductor L_{1} is only limited by the resistor R_{1}, hence Î_{L1} = V_{in}/R_{1} = 0.04 A.
3. Example case
The performance of the proposed approach is demonstrated by a worstcase analysis of a flyback converter in continuous conduction mode (CCM). The goal is to determine the maximum magnetizing current Î_{LM} from the transformer T_{1} in steadystate. This is especially important for the transformer design. The schematic of the closedloop flyback converter is shown in Figure 4 and is divided into three parts:
Power stage: includes a real transformer with the winding resistance R_{P,} the magnetizing inductance L_{M,} an ideal transformer T_{1}, a power switch Q_{1}, a current sense resistor R_{Sense,} the secondary rectifier D_{1} and the output filter C_{O.} The power switch Q_{1} is modeled as a voltage controlled ideal switch with two resistors Q_{1 RDS}(on) representing the resistance in the onregion and the resistance Q_{1 RDS(off)} for the cutoff region.
PWM controller: for the control method, peak current mode control with constant switching frequency F_{S} is chosen and is implemented at the PWM controller block. A detailed structure of the PWM controller is shown in Figure 5.
Compensator: a Type II compensator (Ridley, 2011; Basso, 2012) is used, containing the error amplifier EA_{1}, a diode D_{EA,} and a voltage reference V_{Ref} to model an adjustable shunt regulator (such as TL431). The error amplifier EA_{1} is modeled as an ideal amplifier with infinite gain. The diode D_{EA} is used to add an additional offset to the output level of EA_{1} and also ensures that the amplifier can only sink the current. The optocoupler IC_{1} is modeled in the forward linear region by multiplying I_{F} with a constant factor, the current transfer ratio (CTR) I_{C} = CTR I_{F} and in the saturation region by a constant voltage source V_{CE(sat).}
In general, many switching cycles are necessary until the system has reached the steadystate behavior in the case of switching mode power supply. In steadystate, a PWM signal with constant dutycycle d_{on} is generated in the way that the average output voltage
3.1 Steadystate analysis
The major advantage of the analytic forward solver approach is that the unknown dutycycle in steadystate can be calculated based on the analytic solutions of the state variables and/or signal of interest, e.g. (2). In addition, the closedloop flyback converter can be simplified to an openloop flyback converter or Power Stage. This means that no additional algorithms are needed for the steadystate analysis, e.g. as used in (Li and Tymerski, 2000; Wong et al., 2000; Setiawan et al., 2017; Moskovko and Vityaz, 2018). The flyback converter Power Stage operating in CCM has two SubCircuitModels (states): Figure 6 shows steadystate waveforms of the state variables for one switching cycle T_{S} = 1/F_{S}.
State 1: PWM high, MOSFET Q_{1} is switched on (saturation) and D_{1} switched off (reverse bias) with 0 < t ≤ t_{on}.
State 2: PWM low, MOSFET Q_{1} is switched off (cutoff) and D_{1} switched on (forward bias) with t_{on} < t < T_{S}.
The initial conditions of the state variables in state 1 can be expressed as:
Equation (14) expresses that the average voltage
3.2 Sequential quadratic program result
The solution for the initial condition I_{LM 20} of the system of equations (12)(14) corresponds to Î_{LM}. The objective function for the maximum magnetizing current is
Table III summarizes component values and parameters for the closedloop flyback converter and Table IV summarizes the values the SQP parameters.
The implemented SQP method converges to the maximum after 6 iterations with I^_{LM} = 2.583 A.
3.3 Transient simulation
To verify the results from the SQP method in steadystate, for each iteration k (summarized in Table V) a transient simulation of the closedloop flyback converter has been performed with the same parameters in Table IV and III as shown in Figure 7. The transient simulation results are obtained by applying the AFSA to the CircuitModel of the closedloop flyback converter. The simulation time t_{Sim} was set to 30 ms. The results from the SQP method are shown in Figure 7 as horizontal lines, these are marked with
3.4 Performance comparison
To compare the performance of the SQP method, an MC analysis and evolution strategy (1 + 1) ES (Beyer, 2001) has been implemented in Maple as well. The implemented MC analysis uses the continuous uniform distribution over the parameter ranges. All methods use the same objective function (15). Table VI summarizes the iterations for the methods and results. The (1 + 1) ES method and the SQP methods provide the same result; however, the number of iterations for convergence varies greatly. The MC method could not find the exact worstcase value even at higher iterations runs k. The total CPU time of the SQP method is significantly shorter than that of the other two (e.g. ≈1/9, 1/95, 1/7), illustrating the advantage of the AFSA.
3.5 A more complex example case
The example of the worstcase tolerance analysis of a flyback converter in CCM was shown. The example can be extended by e.g. the discontinuous conduction mode (DCM), which arises when the inductor current is zero. Here, the analytic forward solver approach provides the additional needed state for DCM. The initial conditions of the state variables can be expressed as it was done in the CCM example. Both models of the flyback converter can also be combined to one model which also supports switching between CCM and DCM. The CCM model becomes invalid when I_{LM 10} is negative.
4. Conclusion
The SQP method in combination with the AFSA shows that the worstcase limit converges in a few steps even if the worstcase limit is not on the boundary of the parameters. Based on the AFSA, it is possible to reduce a flyback converter in steadystate from a closedloop to an openloop system. These results are well in line with the transient simulation results obtained by applying AFSA to the open loop flyback converter. With respect to accuracy, the SQP method shows similar performance to (1 + 1) ES strategy. Superior performance in terms of total CPU time was shown.
Figures
Component values and SQP parameters RLC series resonant circuit of Figure 1
Components  Parameter  Lower bound  Upper bound 

values  x_{i}  
V_{in} = 2 V  t  t_{min} = 10 s  t_{max} = 10.5 s 
R_{1} = 50 Ω  L_{1}  L_{1 min} = 0.1 H  L_{1 max} = 30 H 
C_{1} = 15 mF  
ω = 2 πf  
f = 1 Hz  
I_{out}(t) = 0 A 
SQP Method on RLC series resonant circuit of Figure 1
k  0  1  2  3  4  5  6 

t^{(}^{k}^{)} in s  10.1  10.41  10.46  10.44  10.29  10.24  10.25 

15.0  15.0  9.735  5.999  0.4609  1.452  1.692 

8.062  –20.53  –24.58  –27.31  –36.25  –39.98  –40.0 
Component values and parameters for the closedloop flyback converter of Figure 4
Power stage  PWM controller  Compensator 

R_{P} = 0.2 Ω  V_{PWM low} = 0 V  V_{Ref} = 2.5 V 
T_{1 Ratio} = 0.25  V_{PWM high} = 10 V  R_{D1} = 110 kΩ 
Q_{1 Vth(on)} = 3.5 V 

R_{D2} = 10 kΩ 
Q_{1 RDS(on)} = 0.34 Ω  V_{DC Offset} = 0 V  R_{E} = 1 kΩ 
Q_{1 RDS(off)} = 0.1 GΩ  V_{FB0} = 6.6 V  R_{Bias} = 1 kΩ 
D_{1 RD} = 50 mΩ  R_{FB} = 1 kΩ  C_{Z} = 4.7 nF 
C_{O} = 2200 μF  D_{3 VF} = 1.4 V  C_{P} = 0 pF 
R_{L} = 10 Ω  R_{1} = 20 kΩ  D_{EA VF} = 2.5 V 
V_{O nom} = 30 V  R_{2} = 10 kΩ  IC_{1 VF} = 1.1 V 
ZD_{1 VZ} = 1 V  IC_{1 CTR} = 2  
IC_{1 VCE(sat)} = 0.5 V 
SQP Parameters for the closedloop flyback converter of Figure 4
Parameter x_{i}  Lower bound  Upper bound 

V_{DC}  V_{DC min} = 100 V  V_{DC max} = 300 V 
L_{M}  L_{M min} = 330 μH  L_{M max} = 470 μH 
F_{S}  F_{S min} = 95 kHz  F_{S max} = 105 kHz 
D_{1 VF}  D_{1 VF min} = 0.6 V  D_{1 VF max} = 1.2 V 
R_{Sense}  R_{Sense min} = 0.1425 Ω  R_{Sense max} = 0.1575 Ω 
SQP Method on closedloop flyback converter of Figure 4
k  0  1  2  3  4  5 


202.0  202.0  201.999  201.986  100.0  100.0 

380.0  330.0  330.0  330.0  330.0  330.0 

100.0  100.0  100.0  100.0  99.8984  95.0 

0.7  0.7  1.2  1.2  1.2  1.2 

0.14  0.14  0.1416  0.1575  0.1575  0.1575 

–2.215  –2.368  –2.387  –2.38702  –2.539  –2.583 
Performance comparison of different solutions methods
Method  MC  MC  (1 + 1) ES  SQP 

Î_{LM} in A  2.532  2.535  2.583  2.583 
#k iterations  1000  10000  712  6 
CPU time in s  20.67  211.6  14.88  2.21 
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