Electrical Engineering (2024) 106:7831–7838 https://doi.org/10.1007/s00202-024-02463-z ORIG INAL PAPER Transient forward harmonic adjoint sensitivity analysis Julian Sarpe1 · Andreas Klaedtke2 · Herbert De Gersem1 Received: 12 July 2023 / Accepted: 4 May 2024 / Published online: 28 May 2024 © The Author(s) 2024 Abstract This paper presents a transient forward harmonic adjoint sensitivity analysis (TFHA), which is a combination of a transient forward circuit analysis with a harmonic balance-based adjoint sensitivity analysis. TFHA provides sensitivities of quantities of interest from time-periodic problems with many design parameters, as used in the design process of power-electronics devices. The TFHA shows advantages in applications where the harmonic balance-based adjoint sensitivity analysis or finite difference approaches for sensitivity analysis perform poorly. In contrast to existing methods, the TFHA can be used in combination with arbitrary forward solvers, i.e., general transient solvers. Keywords Direct sensitivity analysis · Adjoint sensitivity analysis · Harmonic balance method · Nonlinear circuit analysis 1 Introduction The development cost of devices containing electronic cir- cuits is directly linked to the number of prototyping cycles. A means of reducing the number of prototype cycles is the use of electric circuit analysis tools, such as Spice, that help predict the behavior of the circuits without the need to create any hardware. For time-harmonic systems, the analysis can be performed either in time or frequency domain. Transient circuit analysis treats nonlinear devices by linearizing their behavior at every time step. In frequency domain, nonlin- ear devices cause a mutual dependence of different spectral components. Thus, a naive approach for frequency domain analyses of nonlinear circuits is not generally possible [1, 2]. Nonetheless, a nonlinear circuit problem can be solved in fre- quency domain using the harmonic balance (HB)method [1]. The HBmethod approximates the solution of the steady state B Julian Sarpe julian_johannes.buschbaum@tu-darmstadt.de Andreas Klaedtke andreas.klaedtke@de.bosch.com Herbert De Gersem degersem@temf.tu-darmstadt.de 1 Institute for Accelerator Science and Electromagnetic Fields (TEMF), Technische Universität Darmstadt, Schloßgartenstraße 8, 64289 Darmstadt, Germany 2 Corporate Sector Research and Advance Engineering, Robert Bosch GmbH, Robert-Bosch-Campus 1, 71272 Renningen, Germany for a finite number of harmonics with Newton iteration [3]. The HB method avoids the issue of long transients in time domain simulations. However one also has to consider the drawbacks of HB. The more harmonics that are necessary to approximate the nonlinear device behavior, the more degrees of freedom (DoF) the equation system has. Moreover, the convergence of the Newton procedure is suffering in strongly nonlinear systems [3, 4]. A systematic approach for circuit analysis is the employ- ment of sensitivity analysis, which aims to reduce opti- mization cycles in the early stages of development by systematically optimizing certain design parameters [5]. A common global sensitivity analysis is performed by comput- ing Sobol indices from a polynomial chaos expansion (PCE) surrogate model [6]. However, PCE and similar methods are limited by the number of designparameters due to the curse of dimensionality [7] that makes this type of analysis unfeasible for large parameter spaces. Another category of sensitivity analysis techniques involves computing the gradient of the Quantity of Interest (QoI) w.r.t. one or more design param- eters. Gradient-based methods are also referred to as local sensitivity analysis [7]. A more global sensitivity analysis can be carried out by a higher order adjoint sensitivity method [8, 9]. This is particularly advantageous when large parameter varia- tions are considered, but a global sensitivity analysis is too costly [9]. A higher order adjoint sensitivity analysis requires the solutionofmultiple adjoint problems, resulting in ahigher numerical cost when compared to first order adjoint sensi- 123 http://crossmark.crossref.org/dialog/?doi=10.1007/s00202-024-02463-z&domain=pdf 7832 Electrical Engineering (2024) 106:7831–7838 tivity analysis. For the class of problems considered in this paper, which only exhibit small parameter variations, a first order sensitivity analysis is sufficient. Adjoint sensitivity analysis is the method of choice for a setting with many design parameters [5, 10]. Transient adjoint sensitivity analysis [10, 11] aswell as sensitivity anal- ysis based on HB solvers [12, 13] have been published and used before. In more recent publications, transient adjoint sensitivity analyses have been applied for a larger variety of nonlinear problems such as electromagnetics, electroqua- sistatics or nonlinear damped mechanical systems [14–16]. Transient adjoint sensitivity analysis is an efficient approach for the analysis of strongly nonlinear systems. However, when dealing with time-dependent sensitivities, the inherent part of solving multiple adjoint problems creates a computa- tional bottleneck [5]. This issue does not exist for HB-based sensitivity analysis. Combining the advantages of transient analysis and HB-based adjoint sensitivity analysis, it is pos- sible to utilize an HB-based adjoint sensitivity analysis while at the same time avoiding convergence problems and many necessary Newton iterations. This paper proposes the transient forward adjoint har- monic sensitivity analysis (TFHA) as a combination of a nonlinear transient forward analysis with a harmonic adjoint sensitivity analysis. The TFHA is an efficient and robust method for sensitivity analysis in power-electronics appli- cations as demonstrated in practical examples in Sect. 5. This paper is structured as follows: Sect. 2 gives a brief introduction to the used circuit analysis methods and nec- essary considerations for the analysis of nonlinear circuits. Section3 gives an overview over the mentioned sensitivity analysis methods. Section4 introduces the TFHA as a novel sensitivity analysis method. After that, TFHA is applied to several model problems in Sect. 5. Conclusions on the appli- cability of the TFHA are drawn in Sect. 6. 2 Nonlinear circuit analysis To perform the sensitivity analysis, a simulation framework for the considered nonlinear circuits must be introduced first. All presented methods are based on the modified nodal anal- ysis (MNA), which will be introduced first. 2.1 Modified nodal analysis MNA has been the method of choice for most circuit prob- lems since its introduction in 1975 [17]. MNA extends nodal analysis to accommodate impedance devices such as inductances or voltage sources in addition to the admittance devices. MNA can be utilized for a transient circuit analysis. The transient MNA is given as a differential algebraic equation (DAE) system F(x, ẋ, t)= AC ẋ(t)+ AGx(t)− i s(t) = 0, x(t = 0) = 0, (1) where AC contains capacitor and inductor contributions, AG contains conductance contributions, and i s contains the inde- pendent sources. If the circuit contains nonlinear devices, a linearization of the system is required. This is done by using the Newton method [18] and the introduction of the Jacobian matrix J JG(x, t) = AG − ∂ inl(x(t), t) ∂x(t) , (2) where inl denotes the vector of voltage dependent currents for nonlinear admittance devices. For impedance and energy storage devices, the linearization is performed analogously. For transient analyses, the linearization is performed for each time step t . 2.2 Harmonic balancemethod A linear system can directly be transferred to frequency domain [5]: F(x, ω) = jωACx(ω) + AGx(ω) − i s(ω) = 0, (3) where the underlining indicates the phasors of the corre- sponding quantities and ω is the circular frequency. The resulting system matrices are combined into a single matrix A = jωAC + AG which gives the constitutive equation for the default MNA in frequency domain: F(x) = Ax − i s = 0. (4) The HB method is a formalism to approximate the solution of a nonlinear system iteratively with a Newton iteration in frequency domain [1]. The Jacobian (5) is defined at multiple frequencies in order to represent the nonlinear behavior J(x,ω) = A(ω) − ∂ inl(x,ω) ∂x , (5) where ω is the vector of all considered frequencies in the system. Approximation with multiple harmonic frequen- cies increases the number of degrees of freedom (DoF) the stronger the nonlinearities [3]. Resultingly, the HB method is very performant in weakly nonlinear systems that exhibit long transient times. This advantage is particularly pro- nounced in systems where the transient effects feature time constants that are by orders of magnitude larger than the periods of steady state operation, resulting in the HB New- ton iteration to converge faster than the transient simulation 123 Electrical Engineering (2024) 106:7831–7838 7833 takes. This performance advantage is diminished for strongly nonlinear systems due to the exploding number of DoFs. The holomorphy of the function vector (4) needs to be ensured in the computational domain to define the Jaco- bian matrix. The Paley–Wiener theorem [19] states that the Fourier transform of a bounded function F ∈ L2 is holo- morphic if the function variable is in the upper half plane, i.e., when t is restricted to �+, which can be assumed in real-world applications without the loss of generality. 3 Sensitivity analysis This section summarizes known methods to obtain the sensi- tivity for a QoIU w.r.t. one or multiple design parameters p. TheQoI can be any quantitywithin the circuit such as an edge voltage or current. Design parameters in our application are devices within the circuit, such as inductances, capacitances or resistances. 3.1 Transient direct sensitivity analysis U (x) depends on the circuit solution, and hence, indirectly on time. Therefore, the calculation of the sensitivity [10]: dU dp = ∂U ∂x dx dp (6) involves the calculation of the derivative dx/dp of the circuit solution x w.r.t. the parameter p. ∂U/∂x is a mapping opera- tor that obtains the sensitivity for the QoI from the derivative dx dp . dx dp is obtained by taking the derivative of (1) w.r.t. p. dF dp = AC dẋ dp + JG dx dp + ( dAC dp ẋ + dAG dp x ) = 0. (7) This approach is referred to as direct sensitivity analysis (DSA) [5]. Equation (7) needs to be solved individually for every design parameter. As a result, DSA is expensive for the sensitivity analysis w.r.t. many design parameters. 3.2 Transient adjoint sensitivity analysis Transient adjoint sensitivity analysis determines sensitivities basedon the systemsolution x and a test functionλ,which are both continuously differentiable. To obtain the adjoint sensi- tivity analysis, differential equation (7) is multiplied with λ as a test function and integrated over time: ∫ tm 0 λT(t, tm) ( AC dẋ dp + JG dx dp ) dt = − ∫ tm 0 λT(t, tm) ( dAC dp ẋ + dAG dp x ) dt . (8) Integration by parts eliminates the time derivative of dx/dp [5]: ∫ tm 0 ( −λ̇TAC + λT(t, tm)JG ) dx dp dt = − ∫ tm 0 λT(t, tm) ( dAC dp ẋ + dAG dp x ) dt − [ λT(t, tm)AC dx dp ]tm 0 . (9) Without loss of generality λ is chosen as the solution of the adjoint system, to eliminate the boundary terms and accom- modate the left hand side term: AT Cλ̇ − JTGλ = ∂U ∂x , λ(t = tm, tm) = 0. (10) The adjoint solution is calculated backward in time to ensure the reverse initial condition λ(t = tm, tm) = 0 for arbitrary tm [10]. The term at t = 0 is eliminated due to the initial condition x(t = 0) = 0 in Eq. (1). Substituting Eq. (10) into Eq. (9) and multiplication by -1 yields the integration for the sensitivity of the QoI U . ∫ tm 0 dU dp dt = ∫ tm 0 λT(t, tm) ( dAC dp ẋ + dAG dp x ) dt (11) The generalized version of Leibniz integral rule is applied to the right hand side term [20], to obtain the sensitivity for the QoI at a specific time instant tm: dU dp (tm) = ∫ tm 0 ∂ ∂tm λT(t, tm) ( dAC dp ẋ + dAG dp x ) dt . (12) There is an adjoint solution for each time instant tm, as com- puted by solving equation (10). Thus, the task of computing the solution to Eq. (10) has to be redone for each instant in time where the sensitivity is to be considered. 3.3 Harmonic balance-based direct sensitivity analysis Analogously to the transient case, DSA is derived by sym- bolic differentiation of the reverse initial condition from Eq. (4) for the HB method. dF(x) dp = dA dp x + ( A − ∂ inl ∂x ) dx dp = dA dp x + J dx dp = 0 (13) From Eq. (13), we can solve the system for the sensitivity dx/dp: dA dp x + J dx dp = 0 ⇒ dx dp = −J−1 dA dp x (14) 123 7834 Electrical Engineering (2024) 106:7831–7838 The Jacobian is determined in each Newton iteration when used for the harmonic balance solver. If the goal is to perform sensitivity analysis based on an existing circuit solution, the Jacobian is given as the converged solution from the nonlin- ear problem [12]. If many design parameters p are analyzed, the numerical efficiency of HB-based direct sensitivity decreases, because Eq. (14) contains a matrix multiplication with the dense matrix J−1. This dense matrix multiplication can be avoided by introducing an HB-based adjoint sensitivity analysis. 3.4 Harmonic balance-based adjoint sensitivity analysis The adjoint system for the harmonic balance case (13) is given as JHλ = ∂U ∂x . (15) As the Jacobian does not depend on the adjoint solution λ, the adjoint problem is linear. The sensitivity of the QoIU (p) w.r.t. the design parameter p follows as: dU (p) dp = − ( ∂U ∂x )H J−1 dA dp x = −λH dA dp x. (16) In frequency domain, the adjoint solution can be reused for all considered frequencies, which eliminates the issues of the transient analysis but increases the number of DoFs for large frequency spaces. Compared to the HB-based direct sensitiv- ity analysis, HB-based adjoint sensitivity analysis does not contain any dense matrix multiplications, improving perfor- mance. The procedure is outlined as the green dashed box in Fig. 1. The issue that remains is the bad performance for strongly nonlinear systems. This is particularly problematic when the Jacobian has to be approximated along many convergence steps of the HB iteration (Eq. (4)). A hybrid time–frequency domain method, that eliminates the issue of many Newton iteration steps, is proposed in the following section. 4 Transient forward harmonic adjoint sensitivity analysis The HB method is well suited for simulating steady-state operation. If strong nonlinearities occur, the harmonics become coupled across a wide range of frequencies. Espe- cially in the context of power electronics, short rise and fall times are present, which requires us to consider a large num- ber of harmonics [3]. A procedure consisting of a forward HB solve combined with a harmonic adjoint method for calculating the sensi- tivities was proposed in [12] and has been recapitulated in Sect. 3.3. Obtaining the forward solution for x with the har- monic balance method is not compulsory for the sensitivity analysis. This motivates the following procedure. The circuit solution is calculated by a transient circuit solver. The adjoint solution is obtained by a harmonic analysis. This procedure extends the HB-based adjoint sensitivity analysis (in green) as depicted by the red dotted box in Fig. 1. The TFHA obtains the circuit solution x through the uti- lization of an efficient transient solver of choice. Once an approximation to the steady state is found, the Fourier trans- form x of one period of x is used to calculate the Jacobian J . The approximated Jacobian matrix is then used to obtain the adjoint solution in frequency domain, analogously to the way presented in Sect. 3.4. Subsequently, the adjoint solution is combined with the symbolic derivative of system matrix dA/dp to calculate the sensitivities. The harmonic adjoint simulation is implemented in Python using the SciPy sparse solver. Depending on the application, the sensitivity can be illustrated as a frequency spectrum or be transformed back to time. The TFHA is exposed to information loss which results in a residual error, due to a finite number of harmonics. The Euclidean distance of the solution with fewer harmon- ics against the solution with more harmonics quantifies the residual. This is based on the idea of the Zienkiewicz-Zhu error estimator [21]. The relative error is approximated by the quotient of the absolute error divided by the norm of the fine solution: Fig. 1 Workflow for the harmonic balance-based adjoint sensitivity analysis and the TFHA 123 Electrical Engineering (2024) 106:7831–7838 7835 Erel ≈ ∥∥∥( dU dp ) fine − ( dU dp ) coarse ∥∥∥ 2∥∥∥( dU dp ) fine ∥∥∥ 2 . (17) Without prior considerations of the spectral circuit behav- ior, the sensitivity is determined by iteratively increasing the number of harmonics. The quantification of the error (Eq. (17)) is finally used as a termination condition to assess the converged solution. 5 Results The TFHA is illustrated for three different examples: A half-wave rectifier as a nonlinear, academic example and two different power electronics circuits serving as industrial examples. 5.1 Half-wave rectifier A half-wave rectifier (Fig. 2) consists of only one nonlinear and two linear components. The nominal device values of the half-wave rectifier are: • Vin = 5V sin(ωt) • R = 20� • C = 200 nF Figure 3 displays both the time domain and the fre- quency domain sensitivity of the output voltage w.r.t. the resistance R. Due to the weak nonlinearities within the circuit, the sen- sitivity solution converges fastly even for a small number of harmonics. Fig. 2 Schematic of a functional half-wave rectifying circuit 5.2 Boost converter Aboost converter is a type of DC-DC converter circuit which is used in many power electronic applications. In a circuit model, parasitic effects can be modeled as lumped circuit elements such as resistances, inductances and capacitances. The circuit with parasitic lumped elements is shown in Fig. 4. The boost converter is simulated with the device values given in Table 1. The sensitivity of the drain voltage Vdrain w.r.t. the output resistance R1 is shown in Fig. 5. A significant overshoot is observed for the sensitivity and the drain voltage throughout the switching process. This is attributed to the broadband nature of the system. The over- shoot can heavily influence the sensitivities especially close to the switching of the transistor. Hence, this is a relevant test for accuracy and performance of the algorithm. The results in Fig. 5 show that the algorithm correctly approximates the overshoot while limiting the Gibbs ripples when a frequency domain solution with an insufficient number of harmonics is calculated. The spectral TFHA solution gives a good indica- tion of this behavior (Fig. 6). The spectrum does not decrease steadily for higher har- monics, but rather shows a side lobe, with higher peaks at around 0.8MHz. 5.3 Active filter circuit An active filter circuit that injects a reversed polarity noise signal serves as an examplewith a larger number of elements. This circuit is nonlinear. Figure 7 shows a reduced schematic of the entire active filter setup as used in the analysis. The complete circuit consists of 240 linear and nonlinear ele- ments and has 152 degrees of freedom. The circuit contains several specific transistor and diode models. The Xyce [22] circuit simulator is used to assemble the Jacobian in fre- quency domain. The sensitivity of several QoIs is analyzed for several design parameters of the circuit. The first QoI is the output potential of the OPAMP V(OPout). Due to the filter proper- ties of the circuit, the quantity OPout is smoother than other quantities in the circuit (Fig. 7). The sensitivity results for Fig. 3 Sensitivity of the output voltage w.r.t. the resistance R for the half-wave rectifying circuit in time and frequency domain. The termination frequency is indicated by the green vertical line at 35 harmonics for a residual of under 1% 123 7836 Electrical Engineering (2024) 106:7831–7838 Fig. 4 Boost converter circuit model with parasitics. The elements L1, R1, C and M model the functional behavior of the boost converter, the elements R2, L2, R3, L3, R4 and L4 are parasitic elements whichmodel the EMC effects on the circuit board the QoI V(OPout) w.r.t. the capacitances C1 and C2 as well as the resistances R1 and R2 are shown in Fig. 8. The capacitances show a large influence on the amplitude of V(OPout). The resistances strongly influence the pre-filter properties on the input of the OPAMP. As a result, variations of the resistances lead to sensitivities which vary quickly in time. It is conclusive that the resistances have the most influence on the flank steepness of the OPAMP output. The sensitivity analysis is executed for all design parame- ters in the circuit to benchmark the performance of theTFHA. The circuit solution is found with the Xyce transient cir- cuit solver in around 12s. In comparison, a full HB iteration would take over 4h in Xyce with direct solver and for 50 harmonics that are necessary to achieve the required accu- racy of 10−2. The adjoint solution is calculated in 20s using the implementation of Eq. (15) in Python. The evaluation of the sensitivity with Eq. (16) takes less than 2s for each design parameter. This is a significant speedup compared to existing methods for the given example. Direct sensitiv- ity analysis with Xyce takes 12s for each design parameter. Performing the simulation with first order finite differences equally takes around 12s for each design parameter. As a result, a speedup of 85% can be observed for the considered number of design parameters compared to the finite differ- ence method. The sensitivities for the ten most influential design parameters are listed in Table 2. Note here that the design parameters not visible in Fig. 7 are part of the load impedance Z load. The input node of the active filter has more oscillations at high frequencies. As Fig. 9 shows, the V(OPin) sensitiv- ity w.r.t. C2 requires more spectral components for a high frequency approximation. To illustrate this, the sensitivity of V(OPin) w.r.t. element C2 is calculated with the TFHA and Fig. 5 Time-resolved sensitivity of the gate voltage Vdrain w.r.t. the resistance R1 related to the drain and gate voltages of the boost converter MOSFET Fig. 6 Sensitivity spectrum dVdrain/dR1 of the boost converter circuit showing a significant spectral content at around 0.8MHz that must not be neglected. The termination frequency is indicated by the green vertical line at 225 harmonics for a residual of under 1% Fig. 7 Functional schematic for the nonlinear active filter circuit Fig. 8 Sensitivity results for V(OPout) of the nonlinear active filter circuit w.r.t. capacitor C2. Results compared to Xyce DSA reference Table 1 Device values used for the simulation of the boost converter circuit Label R1 C L1 R2 L2 R3 L3 Value 10� 100 µF 100 µH 71.36 µ� 10.12 nH 71.35 µ� 10.11 nH Label R4 L4 Vin Vgate Value 149.62 µ� 19.48 nH 12V 20V (pulsed: 10 ns tswitch, 100 µs tcycle) 123 Electrical Engineering (2024) 106:7831–7838 7837 Table 2 List of the ten most influential design parameters (not normalized) Sensitivity Value (not normalized) Sensitivity Value (not normalized) dV (OPout)/dL4 6.265MVH−1 dV (OPout)/dC3 2.295MVF−1 dV (OPout)/dC1 6.098MVF−1 dV (OPout)/dR11 1.873MV�−1 dV (OPout)/dR3 6.008MV�−1 dV (OPout)/dC5 0.792MVF−1 dV (OPout)/dC2 2.713MVF−1 dV (OPout)/dR2 0.662MV�−1 dV (OPout)/dC6 2.299MVF−1 dV (OPout)/dR1 0.615MV�−1 Fig. 9 Sensitivity dV (OPin)/dC2 of the filter circuit computed by the TFHA compared to the transient direct Xyce [23] sensitivity solution. Values normalized to nominal device value of capacitor C2 compared to the sensitivity solution of the direct sensitivity solver of Xyce [23]. The comparison is shown in Fig. 9. As expected, TFHA can accurately determine the sensitivity in smooth intervals but is unable to resolve sharp peaks. 6 Conclusions This paper recapitulated transient adjoint sensitivity analysis methods as well as HB-based adjoint sensitivity analysis for nonlinear circuits. Transient forward harmonic adjoint sen- sitivity analysis (TFHA) which combines transient solvers with harmonic adjoint sensitivity analysiswas introduced as a novel method that improves the efficiency of sensitivity com- putation with regard to a large number of parameters. Results calculated by the TFHA have been presented and compared to results calculated by transient or harmonic adjoint sensi- tivity analysis. TFHA is advantageous in applications where purely tran- sient sensitivity analysis is costly, but the HB method itself might need many Newton iterations to converge. This can be the case in fastly oscillating or strongly nonlinear sys- tems [3], such as power electronic applications. The TFHA requires one steady-state time domain solution obtained by a solver of choice and only one solution for the adjoint sys- tem and therefore one inversion of the Jacobian matrix. The TFHA performs well in weakly nonlinear systems where many design parameters are considered. Additionally, TFHA is particularly performant for sensitivity analysis on time- dependent QoIs, since it requires only one adjoint solution in such scenarios. Supplementary Information The online version contains supplemen- tary material available at https://doi.org/10.1007/s00202-024-02463- z. Author Contributions J. Sarpe wrote the main manuscript text and prepared the figures. A. Klaedtke provided the models for the "Boost Converter" and the "Active Filter Circuit". All authors wrote the intro- duction and the conclusions. All authors reviewed the manuscript. Funding Open Access funding enabled and organized by Projekt DEAL. Availability of data andmaterials Not applicable. Declarations Conflict of interest The authors declare no conflict of interest. Ethical approval Not applicable. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adap- tation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indi- cate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. 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(SAND2016-9437, 1562422), 2016-94371562422. https://doi.org/10.2172/1562422 Publisher’s Note Springer Nature remains neutral with regard to juris- dictional claims in published maps and institutional affiliations. 123 https://doi.org/10.1109/TCAD.1986.1270223 https://doi.org/10.1109/TCAD.1986.1270223 https://doi.org/10.1109/TMTT.1983.1131587 https://doi.org/10.1109/TMTT.1983.1131587 https://doi.org/10.1109/TMTT.2003.820905 https://doi.org/10.1109/TMTT.2003.820905 https://doi.org/10.1016/S0378-4754(00)00270-6 https://doi.org/10.1016/j.ress.2007.04.002 https://doi.org/10.1016/j.ress.2007.04.002 https://doi.org/10.1063/1.525186 https://doi.org/10.1109/TCSI.2009.2015720 https://doi.org/10.1109/TCSI.2009.2015720 https://doi.org/10.1137/S1064827501380630 https://doi.org/10.1137/S1064827501380630 https://doi.org/10.1007/978-3-540-71980-9_18 https://doi.org/10.1109/22.17397 https://doi.org/10.1115/GT2020-16208 https://doi.org/10.1115/GT2020-16208 https://doi.org/10.1364/oe.22.010831 https://doi.org/10.1364/oe.22.010831 https://doi.org/10.1007/s00202-023-01797-4 https://doi.org/10.1007/s00202-023-01797-4 https://doi.org/10.1007/s00158-017-1858-2 https://doi.org/10.1007/s00158-017-1858-2 https://doi.org/10.1109/TCS.1975.1084079 https://doi.org/10.2307/2319163 https://doi.org/10.1002/nme.1620330702 https://doi.org/10.1002/nme.1620330702 https://doi.org/10.2172/1562422 Transient forward harmonic adjoint sensitivity analysis Abstract 1 Introduction 2 Nonlinear circuit analysis 2.1 Modified nodal analysis 2.2 Harmonic balance method 3 Sensitivity analysis 3.1 Transient direct sensitivity analysis 3.2 Transient adjoint sensitivity analysis 3.3 Harmonic balance-based direct sensitivity analysis 3.4 Harmonic balance-based adjoint sensitivity analysis 4 Transient forward harmonic adjoint sensitivity analysis 5 Results 5.1 Half-wave rectifier 5.2 Boost converter 5.3 Active filter circuit 6 Conclusions References