formula for impedance

Exploring the Formula for Impedance in AC Circuits

The foundation of the formula for impedance is resistance

Resistance: the most important part of the formula for impedance

There are few, if any, physical activities that do not involve some degree of resistance. In fact, some of our everyday actions rely heavily on resistance. For example, virtually all modes of transportation use physical resistance through braking to prevent collisions. Resistance is also an important factor in electrical circuits.

For current to flow through any medium—conductors or the air—resistance must be overcome. On circuit boards, when the electrical energy transmission is AC, as opposed to DC, resistance must be expanded to impedance where high-frequency effects become a factor. When designing circuit boards, understanding the formula for impedance is important, as it impacts many decisions for the success of your board build and operation.

What Is the Impedance Formula for AC Circuits?

Beginning with the familiar Ohm’s Law, which defines the relationship between the voltage across, current through, and resistance of an element for DC, and expanding to include AC signal propagation, gives the following.

V = I x R ⇒ v(t) = i(t) x Z (1)

where V is DC voltage

v(t) is AC voltage

I is DC current

i(t) is AC current

R is resistance

Z is impedance

Impedance is a complex parameter that can be expressed in polar form as the sum of the resistance and the capacitive and inductive reactances, as shown below.

Z = R + j(XL - XC) = R + j(ωL - 1/ωC) (2)

where ω (=2𝝿f) is the fundamental frequency of the transmitted signal.

For transmission lines, every element on the path and the line itself has an impedance. Between two elements, the line impedance can be viewed from both the sending and receiving end with respect to the transmitted signal direction. This gives rise to the notion of characteristic impedance, whose basic equation is given in Eq. (3).

Z0 = [(R + jωL)/(G + jωC)]0.5 (3)

where R is the resistance,

L is the inductance,

G (1/R) is the conductance

C is the capacitance

⍵ is the signal frequency

To accurately calculate the impedance for a transmission line is quite involved, as varying signal parameters must be considered, as well as line lengths and conductor numbers and types. For PCBs, transmission lines are typically classified as microstrips or striplines, and impedances are also complicated to solve for in closed form. However, there are formulas for calculating board impedance that can yield reasonable results, as shown below.

Formulas for Impedance for PCBs

For single-ended microstrips, the following impedance formula can be used.

Z0 = [87/(Dk +1.41)0.5 ]{ln[5.98H/(0.8W + T)]} (4)

where Dk is the dielectric constant of the material or copper

H is the trace or layer height

W is the trace width

T is the trace thickness

For differential microstrips, the formula becomes as follows.

Z0 = [174/(Dk +1.41)0.5 ]{ln[5.98H/(0.8W + T)]}[1-0.48exp(0.96xD/H)] (5)

For striplines, the equation below can be used to determine the impedance.

Z0 = [60/(Dk)0.5]{ln [1.9B/(0.8W + T)]} (6)

Understanding how to determine impedance is an important design skill, as impedance control can be a major issue that impacts PCBA performance. Therefore, you should choose the best method to ensure your board impedance is incorporated correctly.

How To Best Apply the Formula for Impedance During PCBA Development

As mentioned above, accurately calculating impedance can be complicated. And, although formulas for impedance can be used to assist in the design process, additional issues should be considered. For example, accepted standards, such as IPC-2141A Design Guide for High-Speed Controlled Impedance Circuit Boards, should be utilized when developing high-speed PCBAs. Additionally, it can be difficult to verify impedance values without performing tests, such as time domain reflectometry (TDR), which is available from some CMs. These and other reasons why it may be best to rely on your CM are listed below.

Advantages of CM Formula for Impedance Utilization

  • CM should be familiar with standard requirements for impedance control
  • CM procures board materials and can verify parametric values
  • CM can readily make any necessary adjustments for impedance mismatch
  • CM can run impedance tests to verify impedance values on fabricated board

 

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Tempo Automation is the leading turnkey PCBA manufacturer in the industry for building prototype and low-volume boards of any complexity quickly and accurately. As the manager of your complete board build process, we will ensure that your board exhibits the impedance control needed for your design.

And to help you get started on the best path, we furnish information for your DFM checks and enable you to easily view and download DRC files. If you’re an Altium Designer or Cadence Allegro user, you can simply add these files to your PCB design software. For Mentor Pads or other design packages, we furnish DRC information in other CAD formats and Excel.

If you are ready to have your design manufactured, try our quote tool to upload your CAD and BOM files. If you want more information on formula for impedance calculations or how to best incorporate them in your board development, contact us.

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