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# PCB Impedance Calculator: Trace Characteristics

### Key Takeaways

• A transmission line’s characteristic impedance defines the relationship between a trace’s structure and parameters.

• Impedance calculators may have to balance speed and accuracy, depending on the complexity of the characteristic impedance equation.

• The characteristic impedance of striplines differs considerably from microstrips due to the continuous dielectric constant of their environment. A PCB impedance calculator uses field solvers to accurately approximate impedance values.

Ohm’s Law provides the framework for solving network analysis problems; when the curtain gets pulled back, Ohm’s Law updates to become the relationship between voltage, current, and impedance, not resistance. Impedance captures the real and complex opposition to alternating current flow that resistance alone cannot adequately describe. Knowing the impedance is very important for a whole host of circuit applications to drive performance or tune a network. However, determining the impedance values for intricate stackups and layouts requires powerful modeling software. Designers use a PCB impedance calculator to accurately determine the impedance of trace structures for improved performance without significant post-design adjustments.

## The History and Motivation for a PCB Impedance Calculator

The most fundamental definition of transmission line characteristic impedance is The general equation for characteristic impedance.

with L and C the inductance and capacitance per unit line length, respectively. This equation requires tweaking depending on whether the trace is within a uniform medium. For a stripline buried within a board substrate of dielectric constant εr, this relationship becomes Stripline impedance.

where c is the speed of light in a vacuum. The electromagnetic waves of a microstrip encounter a different dielectric in air and within the substrate. Therefore, they require a slight variation of the stripline characteristic impedance Microstrip (outer layer) impedance.

where Cair represents the capacitance of the trace in the atmosphere instead of the substrate. Capacitance calculations would then use an applied voltage and a Coulombmeter to measure the charge and calculate the total capacitance. However, this method assumes constant current charge distribution, which is not the case in actuality.

Modern PCB impedance calculators use field solvers to create accurate models of complex impedance. Impedance control is necessary for various circuit functions, primarily matching for effective power transfer. Since there are many different fabrication methods for incorporating traces into the board, a comprehensive field solver must handle each case differently to improve the calculator’s predictive ability. The underlying physics equations that describe the characteristic impedance are at the heart of these.

One of the primary issues with any field solver is the need to expand equations to capture real-world behavior better; this expansion complicates the equations. While the field solver is much better at powering through equations than a human, the number of calculations can slow field solvers to a crawl. Here lies the challenge: impedance calculators must balance speed and accuracy. High-end calculators can rely on top-of-the-line systems to cover any computing speed deficiencies, but general impedance calculators must make a clear design tradeoff.

A great example is the surface microstrip which forms on the outer layers of the PCB. There is a dielectric discontinuity (among other material property differences) from the surrounding atmosphere and the substrate underneath. IPC offers a characteristic microstrip impedance as IPC’s microstrip impedance equation.

where εr is the dielectric constant and h, w, and t are the trace height above the nearest plane, width, and thickness.

Simpler equations like those used by IPC perform well where the variables are close to common trace manufacturing variables, i.e., width, height, thickness, and dielectric; however, they contain higher (albeit likely acceptable) percent error than their robust counterparts. When the variables diverge considerably, the error in these simple approximations jumps (the more complex equations' accuracy is steady). It’s worth noting that the benefit of these detailed microstrip equations considerably diminishes if variables remain within a tight distribution around the common values.

## A Look At Stripline Impedance

Compared to microstrips, stripline calculations have simplifications due to the consistent dielectric constant. These equations delineate between single-ended and double-ended lines, with the latter used in differential pair setups. Just like with microstrips, however, equations experience some loss of accuracy as the thickness of the trace grows:

### Single Track Stripline

For a theoretical zero-thickness stripline, the characteristic impedance is A theoretical zero-thickness stripline impedance equation.

where η0 is the intrinsic impedance of the medium, and Κ is the complete elliptic wave function of the first kind as defined by the elliptic modulus k.

Practical models have to account for the thickness of the trace, which involves curve-fitting approximations to fundamental electromagnetic field equations. Cases where the trace is not centered in the medium or includes the more accurate trapezoidal shape of the trace caused by uneven etchant attack, will further frustrate the theoretical equation.

### Coupled Coplanar Striplines

Introducing a second stripline to the equation adds another layer of difficulty. Impedance calculations must account for the two distinct configurations of even (both traces at the same voltage as measured from ground) and odd (traces at the same voltage magnitude but opposite signs relative to ground) impedance. Differential signaling will produce a voltage effect between the traces similar to odd-mode impedance as defined by The relationship between differential and odd-mode impedance.

Once again, the equation for the theoretical odd-mode impedance is exact for a zero-thickness stripline. The thicker the stripline, the greater the contribution of approximations A stripline impedance equation with thickness approximations.