# PCB Parasitic Resistance

This section emphasizes better Printed Circuit Board (PCB) designs. For this reason, detailed analyses of leakage currents were not included. The content below shows you how to estimate parasitic resistances on a PCB. This will help adjust designs sensitive to leakage currents.

## PCB Resistivities

Surface resistivity (ρS, in units of MΩ, or equivalent) describes the local, physical behavior of a PCB surface under a voltage gradient field. The aggregate behavior over all such points creates an equivalent, Parasitic Surface Resistance (RPS) between any two points. Bulk resistivity (ρV, in units of MΩ cm, or equivalent) describes the local, physical behavior inside a PCB’s dielectric under a voltage gradient field. The aggregate behavior over all such points creates an equivalent, Parasitic Bulk Resistance (RPB) between any two points. Typical resistivities for FR4, based on several manufacturer’s data, are:

- ρS ≈ 1 × 105 MΩ
- ρV ≈ 1 × 106 MΩ cm

Different operating conditions and manufacturing flows will produce different values; sometimes by two or three orders of magnitude in either direction.

## Measuring Resistivities

Figure 1 shows two parallel, serpentine traces on the same PCB surface. This geometry is useful for measuring ρS on a given PCB process. Make the traces as long as possible, and as close together as possible, to maximize the surface leakage current. For example, a 6 in × 6 in PCB (60 mils thick) could have two serpentine traces 10 mils wide and 10 mils apart. If the overlap area is 5 in × 5 in, the equivalent length would be 1250 in. With ρS = 1 × 10^{5} MΩ and ρV ≈ 1 × 10^{6} MΩ cm, we estimate RPS ≈ 0.8 MΩ andRPB ≈ 2 GΩ.

Figure 2 shows two parallel planes on opposite PCB surfaces. This geometry is useful for measuring ρV on a given PCB process. Make the planes as large as possible, to maximize the bulk leakage current. For instance, a 6 in × 6 in PCB (60 mils thick) could have two planes 5 in long and 5 in wide. With ρS = 1 × 10^{5} MΩ and ρV = 1 × 10^{6} MΩ cm, we can estimate RPS ≈ ∞ and RPB ≈ 0.9 GΩ.

Be sure to measure ρS and ρV under various conditions seen by your circuit. These values will also change between different manufacturers and processes.

## Numerical Solutions

### Software

When optimizing a complicated PCB geometry it may pay to use a Partial Differential Equation (PDE) solver. Searching the internet for “partial differential equation software,” or the equivalent should bring up several commercially available software packages.

### Using Spice

It is possible to use SPICE to implement a network of resistors representing heat flow between adjacent points. Select an array of equally spaced points. Resistors connect adjacent points and represent the local resistance to current flow, in that direction.

To simulate a particular parasitic resistance, force a voltage at its input and measure the current at its output. The voltage-to-current ratio is that resistance.

Figure 3 shows a typical array point, in a two-dimensional (2D) array. The central point is at voltage V0. The four adjacent points are connected by the four resistors R_{1}, R_{2}, R_{3}, and R_{4}.

The resistors represent the local resistivity between adjacent points. Use very low-valued resistors for points connected by metal. For instance, R_{1} is:

### Equation 1

R_{1} = ρS Δx / Δy, for surface calculation

R_{1} = ρV Δx / (Δy Δz), for bulk calculations

Where:

ρS = Local Surface Resistivity (MΩ)

ρV = Local Bulk Resistivity (MΩ cm)

Δx = grid spacing in the x-direction

Δy = grid spacing in the y-direction

Δz = z-dimension (common to all objects)

This approach is easily extended to three dimensions (3D).

### Using a Spreadsheet

It is possible to simulate in a spreadsheet by using an iterative approach. Set up the resistive array like the last section. The iteration equation at V_{0} is:

### Equation 2

V_{0} = (V_{1}/R_{1} + V_{2}/R_{2} + V_{3}/R_{3} + V_{4}/R_{4})

/ (1/R_{1} + 1/R_{2} + 1/R_{3} + 1/R_{4})

(V_{1} + V_{2} + V_{3} + V_{4}) / 4, all Rs equal

The convergence can be slow, so this approach should be used for simple problems. Once the voltages are determined, it is a simple matter to calculate the sum of currents into (or out of) a node of interest.

This approach is easily extended to three dimensions (3D).