## Characteristic Impedance and Propagation Delay of a Transmission Line

A transmission line is any two conductors with some length separated by a dielectric material. One conductor is the signal path and the other is its return path. As the leading edge of a signal propagates down a transmission line, the electric field strength between two oppositely charged conductors creates a voltage between them. Likewise, the current passing through them produces a corresponding magnetic field. A uniform transmission line terminated in its characteristic impedance will have a constant ratio of voltage to current at a given frequency at every point on the line.

To ensure good signal integrity, it is important to maintain a constant impedance at every point along the way. Any change in the characteristic impedance results in reflections which manifests itself into noise on the signal. In any printed circuit board design, it is almost impossible to maintain a constant impedance of the transmission path from transmitter to receiver. Things like vias, non-homogeneous dielectric, thickness variation and other component paracitics all contribute to impedance mismatch. In high-speed designs, uncontrolled impedance can significantly reduce voltage and timing margins to the point where the circuit may be marginal or worst inoperable. The best you can do is to try to minimize each impedance discontinuity when they occur.

**Lossy Transmission Line Circuit Model:**

The circuit model for a lossy transmission line assumes an infinite series of two-port components as illustrated. The series resistor represents the distributed resistance with the units as ohms (Ω) per unit length. The series inductor represents the distributed loop inductance with the units as henries (H) per unit length. Separating the two conductors is the dielectric material represented by conductance G in siemens (S) per unit length. Finally, the shunt capacitor represents the distributed capacitance between the two conductors with units of farads (F) per unit length.

A 2D field solver is the best tool to extract these parameters from a given transmission line geometry. It assumes, however, that the same geometry is maintained through its entire length. Many spice like simulators need these RLGC parameters for their lossy transmission line models.

Given the RLGC parameters, the characteristic impedance can be calculated by the following equation:

Where:

*Zo* is the intrinsic characteristic impedance of the transmission line.

*Ro* is the intrinsic series resistance per unit length of the transmission line.

*Lo* is the intrinsic loop inductance per unit length of the transmission line.

*Go* is the intrinsic conductance per unit length of the transmission line.

*Co* is the intrinsic capacitance per unit length of the transmission line.

**Lossless Transmission Line:**

For the lossless transmission line model, *Ro* and *Go* are assumed to be zero. As a result, the equation reduces to simply:

**Propagation Delay:**

Propagation delay, as it relates to transmission lines, is the length of time it takes for the signal to propagate through the conductor from on point to another. Given the inductance and capacitance per unit length, the propagation delay of the signal can be determined by the following equation:

Where:

*tpd* is the propagation delay in seconds/unit length.

*Lo* is the intrinsic loop inductance per unit length of the transmission line.

*Co* is the intrinsic capacitance per unit length of the transmission line.

Relative permittivity is also known as relative dielectric constant . The number is a measure of an insulator material’s ability to transmit an electric field compared to a vacuum, which is 1. For simplicity, it is usually referred to it as just the dielectric constant, *Dk*.

Electromagnetic signals propagate at the speed of light through free space. When these signals are surrounded by insulating material other than air or a vacuum, the propagation delay increases proportionally. You can determine the propagation delay with a known *Dk* by the following equation:

Where:

*Dk* is the dielectric constant of the material.

*c* is the speed of light in free space = *2.998E8 m/s or 1.180E10 in/s.*

Nice and very helpful

WA2EINAugust 16, 2019 at 6:23 am

Hi Bert, I’m wondering what would it be the propagation delay when you have two different prepregs below your copper foil?, in other words to get the Dk of the dielectric should we add the Dk of both materials?.

e.g.: suppose we have the following stackup: copper-foil, prepreg1, prepreg2, core’s foil, dielectric-core, core’s foil, prepreg2, prepreg1, copper-foil; Where DK_prepreg1 = 3.12 and DK_prepreg2 = 4.58, on this case in order to calculate the propagation delay in a microstrip, the resultant Dk should be = (Dk_prepreg1 + Dk_prepreg2) or how should I calculate the Dk in order to be used with the propagation delay formula?. And one last thing, should we use Ereff instead of just the given Dk?(I’m asking because I have no Idea if this’s correct or not given the microstrip structure I will use with this stackup). Thanks Bert.

romano088April 15, 2022 at 10:56 pm

Dear romano088 ,

Thank you for your question. The short answer is the two dielectric constants need to be combined. But they do not simply add together. A crude ballpark approximation is 3.85 when they are averaged together. A better approximation is explained here.

For a microstrip geometry, without soldermask, Dkeff is the mixture of air Dk above the trace and laminate Dk below the trace. This is what you called Ereff and it is what you eventually use to calculate propagation delay of microstrip.

Recalling from first principles that a capacitor is simply a dielectric between two conductive plates. By definition, Dkeff is the ratio of the actual structure’s capacitance to the capacitance when the dielectric is replaced by air.

Cactual/Cair =Dkeff/Dkair ………. (1)

From eq. (1) we see Cactual is directly proportional to Dkeff.

When there are two prepregs, with different Dks between two copper plates, the effective total capacitance of the dielectric is the capacitance due to each prepreg in series. If we call this Cmix, then total capacitance in series is:

Cmix = 1/(1/C1 + 1/C2) ………. (2)

If C1 = Dk1*er0*A/h1 and C2 = Dk2*er0*A/h2; where: er0 is permittivity of free space (~8.85pf/m) A is the area of copper plates and h1, h2 are the thicknesses of each prepreg, then the total capacitance is:

Cmix = 1/(1/Dk1*er0*A/h1 + 1/Dk2*er0*A/h2) ………. (3)

Example:

Assume h1 = 152.4um; h2 = 127um; and A =1 sq.m; Dk1 = 3.12; Dk2 = 4.58;

Then from Eq. (3):

Cmix = 1/(1/3.12*8.85E-12*1/152.4E-6+ 1/4.58*8.85E-12*1/127E-6) = 1.156E-7F

Now if we replace the two prepreg thickness with air (Dk of 1) in eq. (3), then the capacitance between plates, Cair is:

Cair = 1/(1/1*8.85E-12*1/152.4E-6+ 1/1*8.85E-12*1/127E-6) = 3.168E-8F

Then from eq (1):

Dkmix = Dkair * Cmix/Cair = 1*(1.156E-7/3.168E-8) = 3.65

So this is the Dk to use for the mixed prepregs. Once you have the Dk of mixed prepregs, a good field solver, is best to calculate Dkeff for microstrip. Once you have Dkeff of microstrip geometry, then you can calculate the propagation delay.

But the best option is to use an accurate electromagnetic 2D field solver that allows for mixed dielectric to calculate the impedance and propagation delay of the transmission line geometry.

I hope this helps.

-Bert

Bert SimonovichApril 16, 2022 at 2:11 pm