Tutorial on Partial Element Equivalent Circuit (PEEC) method based analysis of EMC problems

Program

Background - History PEEC Introduction

EMC Europe 2014 tutorial on Partial Element Equivalent Circuit (PEEC) based analysis of EMC problems

Jonas Ekman

EISLAB

Lule˚ a University of Technology Jonas.Ekman@ltu.se

Giulio Antonini

UAq EMC Lab University of L’Aquila Giulio.Antonini@univaq.it

Sep 1, 2014

Program

Background - History PEEC Introduction

1. Program

Monday Sep 1st

14:00-14:30 Introduction and the

history of PEEC

Jonas Ekman (Lulea University of Technology)

14:35-15:05 Recent progress Giulio Antonini &

- Acceleration techniques Daniele Romano (University of L’Aquila)

15:10-15:40 Parameterized PEEC

models

Francesco Ferranti &

Giulio Antonini (Ghent University &

University of L’Aquila) COFFEE

16:10-16:40 Circuit and

electromagnetic co-simulation for active, non-linear circuit systems

Sohrab Safavi & Jonas Ekman (Lulea University of Technology)

16:45-17:15 Industry view on PEEC

for EMC simulations

Didier Cottet (ABB)

17:20-17:50 Application of PEEC

formulation to analysis of electrical networks in aircraft

Mauro Bandinelli &

Alessandro Mori (Ingegneria dei Sistemi (IDS))

Program

Background - History PEEC Introduction

2. Background - History

2.1. Edward Bennett Rosa

1861-1921, Chief Physicist at the National Bureau of Standards. ^{1 2}

”The actual self-inductance of any closed circuit will be the sum of the self- inductances of all the parts, plus the sum of the mutual inductances of each one of the component parts on all the other parts”

2.2. Frederick Warren Grover

1876-1973, Associate Physicist at the National Bureau of Standards, Prof.

Union College. ³

1E. B. Rosa & L. Cohen, ”Formulae and tables for the calculation of mutual and self- inductance”, Bulletin of the Bureau of Standards, National Institute of Standards & Tech- nology, vol. 5, no. 1. 1907. https://archive.org/stream/formulae5113219089393rosa

2E. B. Rosa & F. W. Grover, ”Formulas and tables for the calculation of mutual and self-inductance”, Bulletin of the Bureau of Standards, National Institute of Standards &

Technology, vol. 8, no. 1. 1907. http://www.g3ynh.info/zdocs/refs/NBS/Sci169noerr.pdf

3F. W. Grover, ”Inductance calculations”, Courier Dover Publications, 2004.

Program

Background - History PEEC Introduction

2.3. Albert E. Ruehli

1937-, IBM T.J. Watson Research Center, NY, USA.

“A new comprehensive theory of inductance is offered, which is called the theory of partial inductance since the calculations are based on the inductance of loop segments.” ⁴

“Specifically, the use of partial capacitances (defined later) in conjunction with partial inductances leads to a three-dimensional technique, and partial-element equivalent circuits result in a complete characterization of three-dimensional in- terconnection structures.” ⁵

“So-called partial element equivalent circuits are proposed, which are based on an integral equation description of the geometry rather than a differential equation description.” ⁶

”It was incomprehensible to Grover that we wanted to compute inductances for such small objects as integrated circuits (ICs).”

4A. E. Ruehli, ”Inductance Calculations in a Complex Integrated Circuit Environment”, IBM Journal of Research and Development, vol. 16, no. 5, 1972.

5A. E. Ruehli & P. A. Brennan, ”Efficient Capacitance Calculations for Three-Dimensional Multiconductor Systems”, IEEE Trans. on Microwave Theory and Techniques, vol. 21, no.

2, 1973.

6A. E. Ruehli, ”Electrical analysis of interconnections in a solid-state circuit environment”, IEEE Int. Solid-State Circuit Conf., pp. 64-65, 1972.

Program

Background - History PEEC Introduction

2.4. Giulio Antonini

1969-, Prof. at University of L’Aquila.

• Acceleration using wavelet theory, fast multipole method, waveform re- laxation, reluctance modeling

• Material modeling including dielectrics and magnetics

• Stability and passivity enhancements and studies

• Skin-effect models

Program

Background - History PEEC Introduction

3. PEEC Introduction

3.1. Basic theory

The theoretical derivation starts from the expression of the total electric field in free space as

E~^T(~r, t) = ~Eⁱ(~r, t) −∂ ~A(~r, t)

∂t − ∇φ(~r, t). (1)

If the observation point, ~r, is on the surface of a conductor, the total electric field can be written as ~E^T = ^{J (~~r,t)}_σ , in which ~J (~r, t) is the current density on a conductor. To transform into the electric field integral equation (EFIE), the definitions of the electromagnetic potentials, ~A and φ are used.

A(~~ r, t) =

K

X

k=1

µ Z

v_k

G(~r, ~r0) ~J (~r0, t_d)dv_k (2)

φ(~r, t) =

K

X

k=1

1

0

Z

v_k

G(~r, ~r0)q(~r0, td)dvk (3)

Combining results in the well known electric field integral equation.

Program

Background - History PEEC Introduction

The EFIE to be solved E~ⁱ(~r, t) =

J (~~ r, t)

σ + µ

Z

v0

G(~r, ~r0)∂ ~J (~r0, td)

∂t dv0 +∇

0

Z

v0

G(~r, ~r0)q(~r0, td)dv0 (4)

Interpreting the EFIE as KVL

It is possible to interpret each term in the above equation as KVL since

V = RI + sLpI + Q/C (5)

This results in interpreting:

• The first RHS term as Resistance,

• The second RHS term as Partial inductance

• The third RHS term as Coefficient of potential

Program

Background - History PEEC Introduction

The resulting equivalent circuit (QS)

C

Lpmm Rmm

a b

C

Figure 1: One way of visualizing the interpretation of the EFIE as a partial element equivalent circuit (quasi-static approximation).

Program

Background - History PEEC Introduction

The resulting equivalent circuit (FW)

P 1

I

P 1

I

Lpmm Rmm

I

I

a

I

Vmm^L

Figure 2: One way of visualizing the interpretation of the EFIE as a partial element equivalent circuit (full-wave).

Program

Background - History PEEC Introduction

3.2. Simple PEEC model for two wires in free space

I_Pⁱ

(a)

P1_ii I_Pⁱ P1_jj I_P^j

Lpmm Rmm

I_Cⁱ I_C^j

a I b

Vmm^L

P1_ii I_Pⁱ P1_jj I_P^j

Lpmm Rmm

I_Cⁱ I_C^j

a I b

Vmm^L

I_Pⁱ

(b)

Figure 3: PEEC model for two conducting wires (a) using controlled voltage- and current- sources to account for the electromagnetic couplings (b).

Program

Background - History PEEC Introduction

3.3. Non-orthogonal formulation

1. Developed in 2003⁷.

2. Enables the modeling of geometrically complex structures.

3. Consistent with the orthogonal formulation.

4. More complex partial elements. Compare:

Lpnonort = µ Z

a

Z

b

Z

c

Z

a⁰

Z

b⁰

Z

c⁰

(ˆa · ˆa⁰) ha⁰ (6) G[ ~r_g(a, b, c); ~r_g(a⁰, b⁰, c⁰)] da⁰db⁰dc⁰ h_a da db dc with

Lport= µ

2π[ln 2l/w

1 + t/w + 0.5 + 2

9l/w(1 + t/w)] (7)

7A. E. Ruehli, G. Antonini, J. Esch, J. Ekman, A. Mayo, A. Orlandi, ”Nonorthogonal PEEC Formulation for Time- and Frequency-Domain EM and Circuit Modeling,” IEEE Transactions on EMC, 45, pp. 167-176, 2, 2003.

Program

Background - History PEEC Introduction

a

b c

(a)

Lp²²

Lp33

Lp⁴⁴ P33

1 I_p

P44

1 Ip

P11

1 I_p

P22

1 Ip

-⁺

Lp₁₁

-⁺

-⁺ -⁺

VL

V_L

V_L

VL

1

4

3 2

Ip1

I_p⁴

I_p³ I_p²

3

2

1

4 I3

I₁ I₂

I₄

f2

f1

f₄

f3

(b)

Figure 4: Nonorthogonal metal patch (a) and PEEC model (b).

Program

Background - History PEEC Introduction

a

b c

(a)

Lp²²

Lp33

Lp⁴⁴

C33

C₄₄ C₁₁

C22

Lp₁₁

C14

C₂₃

C34

C₁₂ C₁₃

C24

(b)

Figure 5: Nonorthogonal metal patch (a) and QS-PEEC model (b).

Program

Background - History PEEC Introduction

a

b c

(a)

Lp11 1 4 {Lp11_value} C11 1 0 {C11_value} Lp22 1 2 {Lp22_value} C22 2 0 {C22_value} Lp33 2 3 {Lp33_value} C33 3 0 {C33_value} Lp44 4 3 {Lp44_value} C44 4 0 {C44_value} K13 Lp11 Lp33q _{Lp13

value}²

{Lp11value}{Lp33value} C12 1 2 {C12value} K24 Lp22 Lp44q

{Lp24value}²

{Lp22value}{Lp44value} C13 1 3 {C13_value} C14 1 4 {C14value} C23 2 3 {C23value} C24 2 4 {C24value} C34 3 4 {C34value} (b)

Figure 6: Nonorthogonal metal patch (a) and SPICE .cir-file in (b).

Program

Background - History PEEC Introduction

3.4. Steps performed in a PEEC solver

Graphical User

Interface (GUI) Meshing Partial Element Post-processor

Calculation Matrix

Formulation Matrix Solution

3.4.1. GUI

Creating, editing, archiving simulation projects.

3.4.2. Meshing

Orthogonal vs. nonorthogonal. Electrically large or small. Model simplification.

3.4.3. Partial element calculation

Orthogonal vs. nonorthogonal. Closed formulas vs. numerical integration.

Program

Background - History PEEC Introduction Graphical User

Interface (GUI) Meshing Partial Element Post-processor

Calculation Matrix

Formulation Matrix Solution

3.4.4. Matrix formulation

Nodal analysis vs. Modified nodal analysis.

3.4.5. Matrix solution

Direct, iterative, and/or accelerated.

3.4.6. Post-processing

Visualize current and potential distribution, port parameter calculations.

Program

Background - History PEEC Introduction

3.5. Summary of approach

• Pros:

+ Same time- and frequency- domain model.

+ Direct inclusion of other SPICE circuit elements.

+ Solvable with SPICE-like solvers.

+ Engineering-type of approach.

• Cons:

- Large number of partial elements (speed-up, approximations).

- Complex materials and geometries?

• The PEEC method is equivalent to a Method of Moments solution (if the Moment Method is formulated in a special way).