Topology optimization of conductors in electrical circuit

Abstract

This study proposes a topology optimization method for realizing a free-form design of conductors in electrical circuits. Conductors in a circuit must connect components, such as voltage sources, resistors, capacitors, and inductors, according to the given circuit diagram. The shape of conductors has a strong effect on the high-frequency performance of a circuit due to parasitic circuit elements such as parasitic inductance and capacitance. In this study, we apply topology optimization to the design of such conductors to minimize parasitic effects with maximum flexibility of shape manipulation. However, when the distribution of conductors is repeatedly updated in topology optimization, disconnections and connections of conductors that cause open and short circuits, respectively, may occur. To prevent this, a method that uses fictitious electric current and electric field calculations is proposed. Disallowed disconnections are prevented by limiting the maximum value of the fictitious current density in conductors where a current is induced. This concept is based on the fact that an electric current becomes concentrated in a thin conductor before disconnection occurs. Disallowed connections are prevented by limiting the maximum value of the fictitious electric field strength around conductors where a voltage is applied. This is based on the fact that the electric field in a parallel plate capacitor is inversely proportional to the distance between the plates. These limitations are aggregated as a single constraint using the Kreisselmeier-Steinhauser function in the formulation of optimization problems. This constraint prevents only disallowed disconnections and connections, but does not prevent allowed topology changes. The effectiveness of the constraint is confirmed using simple examples, and an actual design problem involving conductors in electromagnetic interference filters is used to verify that the proposed constraint can be utilized for conductor optimization.

Appendices

Appendix A: Calibration term in the KS function

The second terms in (14) and (15), which were not included in a previous study (Kreisselmeier and Steinhauser 1979), are used to calibrate the effect of the section of the integral domain. Here, the calibration term is derived.

The KS function in integral form can be written as follows:

$$ KS (f_{\text{KS}}) = \frac{1}{\rho_{\text{KS}}} \ln {\int}_{{\Omega}_{\text{KS}}}\exp\left( \rho_{\text{KS}} f_{\text{KS}} \right){\mathrm{d}}V, $$

(37)

where f_KS is the target function that the maximum value is extracted using the KS function, Ω_KS is the integral domain, and ρ_KS is a parameter. If f_KS takes a value of f_high in the domain Ω_high and takes a value of f_low in the domain Ω_low = Ω_KS∖Ω_high, then the KS function can be written as follows:

$$\begin{array}{@{}rcl@{}} KS (f_{\text{KS}}) &=& \frac{1}{\rho_{\text{KS}}} \ln \left( {\int}_{{\Omega}_{\text{high}}}\exp\left( \rho_{\text{KS}} f_{\text{high}} \right){\mathrm{d}}V \right) \\ &&+ \frac{1}{\rho_{\text{KS}}} \ln \left( {\int}_{{\Omega}_{\text{low}}}\exp\left( \rho_{\text{KS}} f_{\text{low}} \right){\mathrm{d}}V \right) \end{array} $$

(38)

The second term is negligible if f_high ≫ f_low is satisfied and hence the equation can be written as follows:

$$\begin{array}{@{}rcl@{}} KS (f_{\text{KS}}) &\approx& \frac{1}{\rho_{\text{KS}}} \ln \left( {\int}_{{\Omega}_{\text{high}}}\exp \left( \rho_{\text{KS}} f_{\text{high}} \right){\mathrm{d}}V \right) \end{array} $$

(39)

$$\begin{array}{@{}rcl@{}} &=& \frac{1}{\rho_{\text{KS}}} \ln \left( \exp\left( \rho_{\text{KS}} f_{\text{high}} \right) {\int}_{{\Omega}_{\text{high}}}1{\mathrm{d}}V \right) \end{array} $$

(40)

$$\begin{array}{@{}rcl@{}} &=& f_{\text{high}} + \frac{1}{\rho_{\text{KS}}}\ln\left( {\int}_{{\Omega}_{\text{high}}}1{\mathrm{d}}V \right). \end{array} $$

(41)

The presence of the second term in (41) implies that there is a contribution from the integral domain Ω_high. If the section of Ω_high is defined as the product of the scalar K_KS and the section of Ω_KS, then the equation can be written as follows:

$$ KS (f_{\text{KS}}) = f_{\text{high}} + \frac{1}{\rho_{\text{KS}}}\ln\left( K_{\text{KS}} {\int}_{{\Omega}_{\text{KS}}}1{\mathrm{d}}V \right). $$

(42)

Thus, to calibrate the term due to the integral area, the second terms in (14) and (15) must be considered.

Appendix B: Meaning and setting of KS function parameters

There are four parameters in the KS functions in (14)–(16), namely ρ₁, ρ₂, K_ec, and K_ef. Among them, ρ₁ and ρ₂ determine the extraction ability of the KS functions. Lower values of these parameters make the calculation more stable but decrease the accuracy of extraction of the maximum value. Typical values of ρ₁ and ρ₂ are between 5 and 200 (Wrenn 1989). K_ec and K_ef determine the values of the calibration terms in the KS functions, as explained in Appendix A; lower (higher) values of these parameters increase (decrease) the KS functions. These parameters are the ratio of the section of the analytical domain Ω_ec or Ω_ef to the section where the function $|\boldsymbol {J}_{i}|/J_{\min }$ or $|\boldsymbol {E}_{j}|/E_{\min }$ takes a sufficiently large value (i.e., close to the maximum value). These parameters cannot be determined exactly before optimization. For this reason, K_ec and K_ef are both set to rough values at first (e.g., 0.1). Then, if the constraint does not work appropriately, the parameters are adjusted by checking the values of the KS functions; the parameter is decreased (increased) if the constraint is too weak (severe).

Setting the parameters for the KS functions in Section 4 was conducted as follows. For volume minimization and maximization, ρ₁, ρ₂, K_ec, and K_ef were set to 5, 5, 0.1, and 0.1, respectively. Here, to prevent the convergence issue of min-max problems, ρ₁ and ρ₂ were both set to small values. This setting produced good optimization results, as shown in Sections 4.1 and 4.2. Optimization of filter 1 was initially carried out with the same parameters as those in the previous example, that is, ρ₁ = 5, ρ₂ = 5, K_ec = 0.1, and K_ef = 0.1. However, with these settings, the constraint in (25) was always violated during optimization (i.e., g > 1). Table 1 shows the parameters and values for the KS functions in the initial structure. Although the maximum value for each KS function in (14) and (15) was 0.79, g was 1.02. This is because of the poor ability of the KS function in (16) to extract the maximum value due to the small ρ₂. To enhance this ability, ρ₂ was set to 50 and optimization was conducted again. It can be seen from Table 1 that the maximum value among the KS functions was extracted more correctly with these settings because g = 0.79. However, although constant violation of the constraint was prevented, disconnections occurred during optimization because the constraint was too weak. To strengthen the constraint, K_ec and K_ef were set to 0.05, half of the previous values, and optimization was conducted again. With these settings, it can be seen from Table 1 that the values of the KS functions were increased by 0.14, which is equal to $-1/5 \ln (1/2)$; this means that the constraint became severe. Using these parameters, optimization of all three EMI filters was successfully achieved, as described in Section 4.2.

Table 1 Parameters and values for KS functions in the initial structure

Appendix C: Derivation of (35)

S-parameter S₂₁ is defined as follows (Frickey 1994):

$$ S_{21} = \left. \frac{b_{2}}{a_{1}} \right|_{a_{2}= 0} , $$

(43)

where a₁ and a₂ are the incident power waves at ports 1 and 2, respectively, and b₂ is the reflected power wave at port 2. Note that the dimension of the power wave are the square root of the power rather than the power itself (Kurokawa 1965). These power waves are defined as follows:

$$ a_{1} = \sqrt{\text{Re}(Z_{1})} I_{1i}, a_{2} = \sqrt{\text{Re}(Z_{2})} I_{2i}, b_{2} = \sqrt{\text{Re}(Z_{2})} I_{2r}, $$

(44)

where Z₁ and Z₂ are the characteristic impedance of ports 1 and 2, respectively; I_1i and I_2i are the incident current at ports 1 and 2, respectively; I_2r is the reflected current at port 2; Re(Z) is the real part of the complex number Z. I_1i, I_2i, and I_2r are defined as follows:

$$ I_{1i} = \frac{V_{1i}}{Z_{1}}, I_{2i} = \frac{V_{2i}}{Z_{2}}, I_{2r} = \frac{V_{2r}}{Z_{2}}, $$

(45)

where V_1i and V_2i are the incident voltages at ports 1 and 2, respectively, and V_2r is the reflected voltage at port 2. By substituting (44) and (45) into (43), the following equation is obtained as follows:

$$ S_{21} = \left. \sqrt{\frac{\text{Re}(Z_{2})}{\text{Re}(Z_{1})}} \frac{Z_{1}}{Z_{2}} \frac{V_{2r}}{V_{1i}} \right|_{V_{2i}= 0} . $$

(46)

As described in Section 4.2, the impedances of the two ports are both 50 Ω, and the amplitudes of the voltage sources at ports 1 and 2 are 1 and 0 V, respectively; that is, Z₁ = Z₂ = 50, V_1i = 1, and V_2i = 0. By substituting these equations and

$$ V_{2r} = V_{\text{em,p,}2} - V_{2i} = \frac{1}{w_{2}}{\int}_{{\Gamma}_{\text{em,p,}2}} (\mathbf{E} \cdot \mathbf{a}_{\mathrm{p},2})\mathrm{d}V - V_{2i} $$

(47)

into (46), (35) is obtained.