6.2.1 Inductance–Resistance (Lp,R)PEEC Model for Bar Conductor

Since the PEEC equivalent circuit does not form a closed loop, the inductance of the model will be an open-loop inductance according to Section 5.4. We refer the reader to Section 5.6 for an example that consists of three partial inductances. Most likely, this model is a part of a larger structure.

It is important to use this term so that we are aware of the fact that the circuit is not closed, as is also the case for a partial inductance. Fortunately, PEEC elements have electrical nodes to connect parts together such as building blocks. Some of the parts of a PEEC circuit without capacitive elements consist of partial or open-loop inductances.

The example in Fig. 6.2 is a building block that consists of a simple thin metal sheet with a finite resistivity, for example, copper. We see from Fig. 6.2 how the metal panel is subdivided into cells. For this example, we divide the current into and components while we assume that the thickness is small enough that we do not have to subdivide it. We need to recognize that the subdivision of the current into rectangular Manhattan components is part of the WRM approximation approach.

Hence, this reduces the unknown currents to complex or, in the time domain response of the currents in the cells. As usual, the cells carved out of the metal panel are electrically isolated from each other with infinitesimal gaps while the connections are at the nodes only. For clarity, we show in Fig. 6.2 only a few cells where the current vectors are shown. We should mention that the connections at the nodes have to obey the continuity equation that is given by Kirchhoff's current law (KCL) in Section 6.3.1.

We observe that only the cells in the same direction are inductively coupled since the dot product in (6.5) is zero for perpendicular conductors. Hence, the coupling for each element is between the parallel partial inductances only, which is similar to (6.12), or for example

6.15

There are nine partial inductances in the -direction and eight coupled inductances in the -direction and nine coupled partial inductances in the -direction. We note that the longitudinal cells along all sides are half-width cells.

We see that using a half cell at the edge does in general improve the solution accuracy. Also, assume that we want to connect two of the models in Fig. 6.2 to each other. Then, we simply connect the neighboring nodes together. We also observe the neighboring edge half cells merge into full cells. We consider this as the building block approach where large problems can be viewed as being built from smaller subsystems.

The second part of the process is related to the circuit equations. We note that the circuit corresponding to applying the coupling equation (6.15) to each physical cell is in Fig. 6.2a. The resultant complete (Lp,R)PEEC equivalent circuit in Fig. 6.2b needs to be solved.

Before we consider the general case, we want to observe that the (Lp,R)PEEC equivalent circuit in Fig. 6.2b does not include a ground connection. We can declare any one of the 12 nodes as the ground. Then, the circuit analysis techniques in Chapter 2 can be used to find all voltages and currents in the solved system if a current is injected at any node. However, this may not be an issue in a PEEC model that also includes capacitances with a ground node corresponding to . However, care must be taken on a case-by-casesituation.

6.3.1 Continuity Equation and KCL

The solution of a PEEC circuit also needs to satisfy the continuity equation (3.3) for the charge conservation as shown in Chapter 3.

In the time domain, it seems to be a simple approach to use the continuity equation (6.16) to replace the capacitive charge in the solution with currents in the overall solution as

6.16

This result is what we call a current-only electric field integral Equation (cEFIE) formulation that is widely used for a dynamic electromagnetic solution.

However, this approach leads to an overall behavior that is singular for very low frequencies . This is a problem for electronics type problems. For example, a remotely located antenna does not couple at . Then, such a solution is appropriate. From a circuit's point of view, this is similar to a high pass filter where the low frequencies are irrelevant. Hence, this approach is mostly suitable for problems that do not require a solution.

In the PEEC approach, since the capacitance and inductances are separate, a to daylight full-spectrum solution is possible by using the continuity equation (6.16). The continuity equation needs to be satisfied at each PEEC node as shown in Fig. 6.3. We can see four inductive cells joining a capacitance surface area.

First, we apply a Gaussian surface over the area that corresponds to the node at the center of the volume covered with the capacitive cell on both top and bottom. It is clear that, if we apply Gauss surfaces to each cell in Fig. 6.3, we will wrap up the entire volume of the center part of the conductors. Multiple applications of the continuity equation will cover the entire area.

Surface charge is present on both surfaces of the conductive cells of the finite thickness conducting cells shown in Fig. 6.3. The top and bottom surfaces have capacitive charge. In the following section, it is shown that the charge is on the surface only.

The Gaussian volume over which we integrate the continuity equation also includes the side surface where the current flows. This results in

6.17

where one of the Gaussian volumes and the surface includes all sides as indicated in Fig. 6.3, where the charge is on top and bottom and thecurrent flows through the internal surfaces .

By comparing this result with Fig. 6.3, we see that the normal direction current flow occurs through the four partial inductance cross sections of the conductors. Since they are interior to the structure, the charge density is zero. We notice that the integrals over the side cross sections are

6.18

where is the total current flowing into the four partial inductance cells.

On the other hand, the conductor top and bottom surfaces are charged with surface charge densities . Integrating over both top and bottom surface will yield

6.19

Finally, we add up all the contributions for all surfaces to yield

6.20

It is clear that this is exactly the KCL equation at the node. Hence, we recognize that applying the KCL at each node is the same as applying the continuity equation. Simply stated, the solution of the circuit equations by the techniques of Chapter 2 with the above meshing will satisfy the continuity equation.

6.3.2 Relaxation Time for Charge to Surface

An important issue to validate is if the charge is indeed on the conductor surfaces even for the highest frequencies involved. The time domain equivalent is that the charges must appear on the surfaces faster than the fastest transients in the time domain responses. We start with the Maxwell's equation (3.1c) and the displacement flux density definition .

6.21

where the last step is valid for piecewise continuous dielectrics. On the other hand, we can insert the conducting current into the continuity equation (6.16) to obtain

6.22

Replacing with (6.21) yields the differential equation

6.23

The solution of this simple differential equation is given by

6.24

Hence, the time constant is . As an example, we consider the copper conductor for microelectronics. Its conductivity and the permittivity . Thereby, the corresponding time constant . It means that the charge density inside a conductor will disappear in the volume much faster than any transient signal. Hence, we can safely assume the charge density is zero inside conductors. The charges only reside on surfaces. Then in the interior of conductors the continuity equation (3.55) becomes

6.25

while on the surfaces of conductors, using the surface divergence, we have

6.26

where is the outward normal to the surface .

6.3.3 Physical Aspect of the Capacitance Model

We considered most aspects of the building or PEEC models with the exception of the capacitive part. Also, stand-alone capacitance computations have been treated in Chapter 4. Here, we consider the capacitance term for PEEC models.

We again start with the total electric field equation (3.58d)

6.27

where the capacitance part is represented by the gradient . The electric field generated by the scalar potential

6.28

The scalar potential is related to the charge distribution by equation (3.46) as

6.29

We can consider each direction separately by denoting since they will turn into different capacitive elements. Hence, the terms to be added are of the form

6.30

We use the same symmetrical WRM discretization as we apply for the other circuit elements. As in Chapter 4, we are averaging potentials over the capacitive cells. Hence, we have to apply the symmetric integration to obtain voltages , which is defined as

6.31

If we apply this to (6.30), we get

6.32

where is a volume. The charge is confined to the surface of the conductors such that we can eliminate the conductor cross-section in the derivation. Next, we approximate the derivative using the central finite difference scheme

6.33

It means that the centers of potential cells are half a step shifted with respect to the center of the inductance cell. To clearly illustrate this, Fig. 6.4 taken from Chapter 8, Fig. 8.2 is shown here.

We see that the potential or capacitive cells are centered at the nodes, whereas the inductive cells are connected between the nodes. We also observe that some capacitive cells are corner quarter cells or side half cells in addition to the full cells.

To consistently apply the averaging in the derivation, we also average the evaluation point over the cell

6.34

where represents the positive half shift in position. Bringing (6.33) into (6.32), and applying the averaging process to (6.32), results in

6.35

where the charge density over the cell is replaced by the total charge on that cell. is the free space Green's function from Section 3.4.2

6.36

where the time delay due to the finite value of the speed of light in the vacuum has been neglected, for example, for the QS case. From this derivation, we can also define the partial potential coefficient as

6.37

Potential coefficients are similar to partial inductances in terms of the fact that each cell couples to all other capacitive surface cells in the system. Hence, it corresponds to a dense coupling network and thereby a dense linear algebra matrix in the numerical model. The following section considers how the capacitance models derived in this section can be implemented into the PEEC circuits.

6.3.4 Equivalent Circuits for PEEC Capacitances

Models that include only capacitance and their computations are introduced in detail in Section 4.5.2. Here, we concentrate on the aspect that pertains to the complete PEEC model construction. Equations (4.39a)–(4.39c) are used for the capacitive circuit model. Figure 4.16 is repeated for the three-cell system for convenience in Fig. 6.5. The partial coefficient of potential equations for this example are

6.38

Chapter 4 considers how to obtain a capacitance equivalent circuit as shown in Fig. 6.6.

A more efficient model is desirable for general PEEC models, while this model is appropriate for some small problems. As was considered in Section 2.7.3, we want to connect the capacitance model at the nodes with the total current . We start with (6.38), or . To obtain an efficient formulation, we subdivide into a diagonal part

6.39

and a part with the off-diagonal elements

6.40

Hence, all we did so far is to split the matrix into two parts

6.41

where is easily invertible since it is a diagonal matrix

6.42

If we left-multiply (6.41) by , we get

6.43

where is the identity matrix. With the definition of as

6.44

we can simplify (6.43) by differentiating both sides with respect to time to the important form

6.45

Since , equation (6.45) leads to

6.46

where includes the current of all the elements connected to the capacitor nodes. We note that this equation corresponds to the equivalent circuit in Fig. 6.7 where the current controlled current sources (CCCS) are given by

6.47

where the coefficients are represented in the matrix (6.44). We should note that we can include the delay or retardation between the capacitive cells by including them in the current source models.

6.3.5 (Pp,R)PEEC Resistive Capacitive Inductor-Less Models

A (Pp,R)PEEC model without partial inductances is the only combination of partial elements we did not treat so far. Examples using these models are relevant for very high resistive circuits. Because the inductive voltage drops are relatively small compared to voltage drops over high resistive components, the inductive voltage drops are of minor importance to the overall calculation accuracy. Then, partial inductances could be ignored in the PEEC model. Due to the continuous miniaturization of integrated chips, the internal resistive loss is increasing when the cross section of copper wires is decreasing. Hence, inductances could be ignored at low frequencies for some interconnect parasitic modeling. To construct a (Pp,R)PEEC model, the process is straightforward since all partial self- and mutual inductances can be ignored.

It is clear that we have two choices to include resistances in the MNA equations. Due to the absence of the partial inductance part, the circuit matrix can be restructured, depending on which resistor stamp is used. Conventionally, we use the Table B.3 combined with the partial inductance as outlined in Table B.7 in Appendix B to reduce the number of unknowns. For the models without , we can also use Table B.2 stamp to further reduce unknowns.

6.3.6 Delayed (Lp,Pp,R,)PEEC Models

So far, Quasi Static QS models are considered in most of our derivation. Fundamentally, the QS models are only valid when the overall dimensions of the system such as the length of the signal transmission lines is small compared to the wavelength in the spectrum corresponding to the highest frequency applied by the sources. If the highest frequency corresponds to the wavelength that is comparable to or smaller than the overall object's size, full-wave (FW) models become necessary. FW means that the frequency dependency is completely considered in the modeling process for all propagation modes. The finite speed of wave propagating is considered.

Conventionally, the largest physical size of all PEEC model cells for the subdivided system requires that for all cells , where is the working wavelength of the highest frequency component. In the time domain, the delay or retardation time is between any two elements, which are where is the speed of light in vacuum. In the frequency domain or phasor format, the corresponding concept is the phase delay . Here, the vectors and are the points for which the delay factor is calculated.

It is important to add additional insights to this statement. Conventional TEM-mode transmission line models are physically long with a constant two-dimensional cross section. For this case, QS capacitances and inductances are appropriate 2D transmission line parameters. However, if the cross-sectional dimensions become comparable to the highest frequency wavelength, then retardation becomes relevant for the cross-section.

6.3.7 Simple Full-Wave (Lp,Pp,R,)PEEC Models Implementation

Using zero-volt voltage sources is a common approach to compute current unknowns in an MNA SPICE-type circuit. We use the approach from Ref. [7] to set up the PEEC equations in a conventional SPICE solver using the SPICE input language current using NMA stamps in Appendix B. We also use this approach for analytical studies for small circuits. However, for the implementation of large circuit solvers, we use the approach presented in the following section and in Section 2.7.3.

In Fig. 6.8, we show an elementary equivalent PEEC circuit for a single PEEC loop using the voltage sources. Today, we use the efficient formulation in (6.54) for an efficient solver implementation. However, for some experiments and analytical formulations, we can use the approach based on Ref. [7] where the circuit equations are formulated without using the matrix KCL given in Section 2.7.1. From the previous section, we can determine the coupling capacitive current sources in the frequency domain as

6.48

The total capacitive currents and are coupled using zero-volt and zero-resistance voltage source stamps. This is a common way to measure the current using MNA stamps in Appendix B. Hence, the delay term is included in the above potential coupling formulation. The rest of the circuit matrix is done using conventional MNA stamps in Appendix B.

Next, we give an example for setting up the MNA equation by using conventional KCL and Kirchhoff's voltage law (KVL) equations in the frequency domain. For a FW formulation, the model has to include models for the retardation. Starting with the KCL for nodes 1 and 2, we have

6.49

The KCL at the capacitor 1 is given by

6.50

and for capacitor 2

6.51

It is seen that the zero-volt voltage source enables the separation of KCL equations by adding and to the unknowns. Note that is the delay between the two capacitive cells. We also have the following two KVL equations due to the added voltage sources

6.52

Finally, we can set up the MNA matrix from the above equations as

6.53

This leads to the solution of the circuit equations in the frequency domain for one basic PEEC loop. We again note that this type of PEEC model can be implemented using the conventional SPICE input language for (QS)PEEC models.

6.4.1 Full-Wave Models

The FW circuit model has also been described in Chapter 2. We have a so-called PEEC descriptor type model of the circuits in the form of (6.54) where we also add voltage sources

6.54

The key difference is that is a diagonal matrix such that the inverse does not add additional costs to the solution. represents the off-diagonal part of the partial coefficient of potential (6.44) matrix that does not have to be inverted.

6.4.2 Quasistatic PEEC Models

First, we take the simple case of a QS model with a capacitance model and, of course without the current source model introduced in Chapter 2. In this section, we again call the input current , which is our usual notation. Figure 6.9 shows a conventional (QS)PEEC model where the capacitances can be represented by a two-terminal capacitance matrix as is done in (4.7). It is evident from this that a QS model consists of entirely conventional circuit elements, including partial mutual inductances as shown in Fig. 6.9. This capacitance model is equivalent to the full wave (FW) capacitance model (6.54) if the delays are set to zero.

6.55

where we can define the input source part as

6.56

where is the vector of inputs. Since this formulation requires the inverse , it is computationally not very efficient. Also, the model uses passive circuit elements and only inductive couplings, while the capacitances are shown to be equivalent to capacitance model in Fig. 6.9.

We want to clearly specify the variables involved in the transfer function for the model. It is sufficient to consider a single node from Fig. 6.9 for this purpose. An input current source and an input voltage source is shown in Fig. 6.10. The existence of the sources at each node is specified by the selector matrix .

An output vector for node 1 consists of the potential and/or the inductive branch current as part of the . The presence or the absence of sources in the circuit is represented by the matrices and . It should be evident that the input and output selector matrices are the same also for (FW)PEEC models with delays.

6.4.3 Input and Output Selectors

In a matrix form, we need to select the appropriate inputs and outputs for the problem at hand. For this, we chose the conventional form of the MNA equations that works for both the above QS and the FW equations

6.57a

6.57b

where can consist of output currents, voltages, or both and is a vector of inputs as defined above. The matrix is the output selector matrix.

The definition of the different matrices for the delay case is clear by comparing (6.54) with (6.57) where

6.58

6.59

and the right-hand side is

6.60

The solution vector in the frequency domain is

6.61

that completes the description in the frequency domain.

The translation into the time domain is also rather straightforward by replacing the Laplace variable with . Since the matrices including and are independent of time or , the only difference is the system itself which is in time

6.62a

6.62b

Note the similarity between the frequency and the time domain formulations.

As the last step in this section, we want to point out that the transfer function can be of an impedance, admittance, or a mixed type since both the inputs and outputs can be of different types. For this reason, we use the transfer function notation

6.63

where finally the general transfer function is

6.64

Hence, the above equations represent the general system to be solved.

6.4.4 Power/Energy Type Circuit Model

For completeness, we derive a power/energy formulation for Quasi-static PEEC models. This is based on similar derivations in Ref. [8] and other circuit-theory texts. By denoting with the complex current vector, the general matrix KCL (2.37) can be written as

6.65

for the equivalent circuit shown in Fig. 6.9. Here, is the current for the connected capacitances, represents the Current Controlled Current Source (CCCS) current, denotes the currents flowing in the inductances while are the currents of the external current sources. Note that this is not related to the potential external current defined in this section.

The corresponding voltages for all four contributions, according to Section 2.7.2, are given by

6.66a

6.66b

6.66c

We use the short circuit capacitances instead of the two terminal capacitances shown in Fig. 6.9. Then, the (QS)PEEC–MNA given by (2.11) is somewhat simpler since we do not have to use the delayed capacitance model for the QS case. This leads to (6.55), or

6.67

Next, we form the energy type products, where is the conjugate complex of the current vector . Premultiplying (6.65) with yields

6.68

However, we noticed before in (2.38) that , which we use again for the identity matrix and , and . Using this fact and by comparing this result with (6.66), we can simplify the result to

6.69

where the last term includes external currents.

Considering any single port to observe the behavior of the system since final result (6.68) includes the entire system. Hence, the right-hand side can be reduced to . It is a standard approach to compute the input impedance as

6.70

If the direction of the current is taken into account, the input impedance can be written as

6.71

As a first observation, it is clear that this function has to be positive on the positive imaginary axis . The first term on the right-hand side relates to the capacitance model, whereas the last term represents the resistance and partial inductance model. The advantage of this formulation is that we can separately check the inductance/resistor and the coefficient of potential or capacitance models. It is clear that the two models have to be positive independent of each other for all frequencies.

6.4.5 Resistances, Inductance, and Capacitive Terms

So far, we did not consider the resistors in series with the inductors. We start with the inductance part of the circuit description of the

6.72

where the last step on the right-hand side is due to the matrix voltage law (6.66c). Taking the transpose of the last two parts of (6.72) and substituting it into the Lp,R part of (6.71), we get

6.73

Two observations further simplify this result. First, we can subdivide the equation into an inductive and a resistive part. Second, both the and the matrices are symmetrical. This leads to

6.74

It is well known that a properly computed partial inductance matrix is symmetrical and positive definite. Also, we use the definition of positive definiteness [9].

It now is clear that in (6.74) in the right half of the -plane where . Again, we only consider a (QS)PEEC model.

The capacitive term can again be treated separately from the input impedance matrix (6.73) as

6.75

where the last part has been added for the (QS)PEEC model. We have a simpler capacitance model with one current for the connection to the nodes, which we call . For the QS case considered, we take the first equation of (6.55) as

6.76

We can simplify this by using the matrix KCL (Section 2.7.1) to

6.77

since the capacitance model is connected directly to the nodes where is not included since it is a unity matrix. Inserting this for in (6.75), we get for the QS case that

6.78

It is clear from the positive definite Definition 6.1 that this part is also positive in the right half plane where . We also know that the capacitive model can be replaced by a set of conventional coupled capacitances as is shown in Fig. 6.9. If the capacitances are computed accurately, we know that the capacitive part is passive. More details on these issues are considered in Section 4.4.1.

Hence, we can conclude the following:

Based on the above derivation (see also Refs [10] and [11]), it is shown that the transfer function is passive. More details on passivity issues are given in Chapter 13.