Inductance extraction of Press-Pack IGBT by Considering Displacement Current

Taking the displacement current generated by charges accumulation into consideration, this paper presents a inductance extraction method for press-pack IGBT where conductor might not form a closed loop in simulation. Charge accumulating on ends of conductor is considered and then Ampere’s circuital law (ACL) could be meet. Vector potential A also could be affected by displacement current, which leads to a different formulation of partial inductance. A simplified model based on practical structure are used in numerical experiment for comparison between existing method and proposed method.


Introduction
The press-pack IGBT has become much more attractive than ever due to its high voltage and high current ratings. In the switching process, the maximum current overshoot of paralleled chips inside PPI decides the application limit of the device, which is directly affected by stray parameters of the package module.
The inductance in package is important for current distribution between chips, which mainly exists in the conductors of package. The magnetic coupling effect between conductor segments that don't form a closed loop needs to be considered because the return current path might be distant. Therefore, Ruehli [1] proposed the definition of partial inductance based on flux linkage method and gave a series of formulas to establish the relationship between incomplete loops and closed loops. In recent years, Paul [2] and Holloway [3] made some detailed analysis and summary. Afterwards, the partial element equivalent circuit (PEEC) method that based on the partial inductance theory is widely applied [4][5][6].
To obtain partial inductance, magnetic vector potential A generated by each conductor segment needs to be calculated at first. Kalhor's study [7] showed that two charges accumulating with time need to be placed at both ends of a conductor, which generate displacement current in space, to meet Ampere's circuital law (ACL) in incomplete loop. Kalhor [8] pointed out that Biot-Savart's Law is still applicable in this situation, but effect on A is not analyzed. According to (1), differential form of ACL is applied in the calculation of A, which means total current that includes displacement current is used.
Hence, this paper presents an inductance extraction method of partial inductance calculation by considering both conduction current and displacement current. In section II, A is derived when charges accumulating at both ends of a conductor. Section III gives a method of partial inductance calculation based on the result of A. Section IV makes a numerical experiment for a simplified structure and show difference of results between methods before and after considering displacement current.    q j q q A q j q q I (2) As mentioned above, ACL is satisfied in this model. So (4) is obtained by substitute (3) (4) where V c represents the whole area of source, c r and r are space vectors pointing to source point and field point,

A Generated by Displacement Current
generated by M q M q and N q N q , and can be calculated by (5).
According to Gauss' law, D D generated by a charge in space is given in (6) where M r and N r are space vectors pointing to M and N, respectively. Thus, sum A A in (4) can be also spited into two items C

A C A and D A D
A , corresponding to current density.

Therefore, D
A D A can be calculated as followed.

A A A A A A A A A A A
Similarly, when conductor segments form a closed loop, all effect of displacement can be compensated based on current continuity equation. That means the introduced item D A D A will not influence any results of closed loops by existing method, so it can be also applied in a closed loop. But D A D A may modify result of partial inductance for conductor segments.

Partial Inductance Calculation
Existing method of partial inductance in [4] assumed that conductor segment S1 as a portion of half-infinite loop in Fig.  3 Considering displacement current, because direction of D A is different with current, last three parts of integral in (18) is not all of zero. So partial inductance of conductor segments is redefined as followed.

A and D1
A are generated by segment S1.
pij Lc represents mutual partial inductance between segment Si and Sj. When i=j, pij Lc means self partial inductance of segment Si. Therefore, inductance of segment S1 can be calculated by p11 C 1 D 1 S1 S1 As mentioned above, D1 A is irrotational, i.e., the second integral is merely related to locations of head and end points in path S1, leading to a convenient calculation. Similarly, mutual partial inductance between two segments Si and Sj is defined by where Cij

A and Dij
A are along path Sj and are generated by segment Si.

Numerical Experiment for Comparison
A simplified structure based on practical module, as shown in Fig. 4, is used for numerical experiment and comparison of proposed method and existing method.
Inductance due to pillars is important part of stray inductance of package and has obvious influence on current distribution [9], hence we compare the partial inductance of pillars using both proposed method and existing method. The calculated model is shown in Fig.5. We make one simplification that the notches of pedestals are neglected, so all of pedestals are cuboid. Self-inductance and mutual-inductance of pillars are calculated and shown followed.  show that displacement current obviously affect calculation of partial inductance in practical situation. The influence caused by inductance extraction is analyzed by simulation of IGBT current calculation. The electrical structures of press-pack IGBT in Fig.4  Simulation and experiment are made for comparing above two methods. Result of current is shown followed. Results show that proposed method could improve the calculation accuracy of simulation. The influence will be obvious whereas IGBT module become larger.

Conclusion
Precious inductance extraction of conductor is important in analysis of electrical performance. This paper proposes a method of inductance extraction for press-pack IGBT where conductor segments might not form a closed loop. By considering displacement current, the conductor segment model could meet ACL. Vector potential A also could be affected by displacement current, which leads to a different formulation of partial inductance. A simplified model based on practical structure are used in numerical experiment, and obtains different results of existing method and proposed method.