# Understanding losses is key to state-of-the-art interfacing

**Author's Intro**

The interface world is going through revolutionary times. That's quite a happy time for an engineer. After all that is why we are in this profession, to be constantly challenged and excited. The new signal data rates have changed the interface arena completely. For number of years engineers were able to utilize their experience gained on one of the projects in the past. This is the time when signal integrity discipline has advanced so much that even experienced digital designers and system engineers need to go back to books, application notes and articles in the press.

In this article I address the issues that have become critical since data rates have reached into Gigabits range. The effects that previously were familiar only to the communication engineers dealing with high-speed transmissions over long cables are influencing the data transmissions on every circuit module and backplane: Inter Symbol Interference (ISI), losses in copper and dielectric, signal jitter and so on. In order to deal with these effects one has to know their nature.

In addition, the interface technology moves so rapidly that it requires an effort to stay informed. The IC manufacturers' engineers have come up with a number of methods that may save you money and extend life of old components and materials. One should be familiar with all the options in order to be able to address the interface circuitry design intelligently. It is also imperative to know the tools that are available today and their capabilities.

**CONTENTS - Part 1**

Examining Interface Line Losses Dielectric Losses Skin Effect Some Practical Considerations

**CONTENTS - Part 1**

Inter Symbol Interference (ISI) Eye Pattern Diagram Dealing with Line Losses Signal Losses Compensation Methodology Line Equalization Methods Signal Compensation Multilevel Signaling Summary and Conclusions References

**Examining Interface Line Losses**

It is interesting to trace the history of the interface line on a PCB. Some of us still remember the times of wire-wrapped boards or backplanes wired by solid wires and so on. At that time we have not thought much of interconnection between components. The simple wire interconnects were quite sufficient for the kilobits per second data rates that these interconnects served. With increase of the data rates the wire interconnects were dressed in various parameters and limitations. Terminology and technology used in RF and microwave applications have become useful and were introduced into the digital design applications. This introduction was limited to some special cases, when active components of characteristic line impedance, like copper resistance and dielectric conductance, were much lower in values than reactive components. That simplified the analysis and calculations. Lines like that are called "lossless" transmission lines. The active components on these lines may be neglected. The term "lossless" indicates that there is no power loss on lines like that.

With further increase of the data rates the frequencies of signal's harmonic content were growing as well. RF and microwave techniques and methodology have become again a template for expansion of the signal integrity theory for digital signaling. In addition to more significant effect of small values of capacitance and inductance of vias, connectors and so forth, in addition to increased effect of reflections and crosstalk, the high data rate signals suffer from reduction in amplitude and phase reduction as they propagate along the line. As in RF and microwave applications the issue of losses on transmission lines becomes dominant as frequencies reached over 500MHz level.

Let us consider a transmission line, which is represented by an infinite chain of L, C, R and G cells as it is shown in

**Figure 1. Transmission Line Model**

Consider a transmission line in the Figure 1 above. It is represented by a lumped component model, which consists of unit length cells represented by L, C, R and G unit length parameters of the transmission line. We may write two equations that represent the Kirchhoff's voltage and current laws for an individual cell.

In these equations the values of voltage and current on the left side are represented by differences of the output and input values. The voltage and current parameters on the right sides of the equations are the values at the input of the cell, or at an arbitrary point z of the line. With admission of Dz to approach zero, the Kirchhoff's equations are transformed into a set of partial differential equations called General Transmission Line Equations. These equations are also known as telegraphist's equations.

In order to simplify the process a general representation of voltage and current parameters is replaced by time-harmonic function. In addition, the signal propagation is considered in one direction only, z-direction. That yields differential equations in respect to phasors V(z) and and I(z). A simple transformation will lead to quadratic differential equations.

For time-harmonic signals:

From the first equation:

plugging I(z) into second equation:

In a similar fashion we may obtain:

The solution of these equations yields the following results:

For finite line:

For infinite line:

If we plug the solution for the infinitely long line into one of the General Transmission Line Equations an easy transformation will yield the formula for the characteristic impedance of the lossy line:

For the lossless line, where R = 0 and G = 0, from the generic impedance formula we can easily obtain:

From the nature of transmission line characteristic impedance one may observe and intuitively understand the nature of the losses in a transmission line. In case of the lossless transmission line the members of the impedance expression are reactive, therefore, the signal traveling along the line converts its energy from magnetic field in inductor into electric field in capacitor. This process proceeds without any losses of power or degradation in signal's amplitude.

When trace resistance and dielectric conductance are not zero, the signal looses its energy in the form of heat on active components and, of course, its amplitude would degrade.

Let us now consider the propagation constant in greater details, since as we see from the expressions of the voltage and current on transmission line the propagation constant defines the amplitude and phase of the signals. We may separate the components responsible for the amplitude and phase values if we record the propagation constant as: = + j , where: - attenuation constant (Np/m) -- defines amplitude function - phase constant (rad/m) -- defines phase variations. Then:

As any complex number the expression of the propagation constant may be represented as a sum of real and imaginary parts. Then the expression for the signal voltage on the infinitely long line may be shown as:

This expression shows that the amplitude of the signal propagating on lossy transmission line is changing exponentially with distance.

It is obvious that for the lossless line, when R = 0 and G = 0, then:

The solutions of the General Transmission Line Equations are representing the signals propagating on transmission line. In case of the finite line or line with discontinuities there are incident and reflected waves propagating in opposite directions. In the voltage and current expressions the incident and reflected waves are represented by -- and + superscripts respectively. For the infinite (semi-infinite) line or properly terminated line there is only incident signal. The General Transmission Line Equations solutions show that propagation constant parameter defines the amplitude and phase of the signal at each point on the transmission line.

The Low-Loss Line is a special case of the transmission line when: R 0 and G 0, however, R < l,="" g="">< c.="" this="" case="" is="" of="" interest="" for="" designers="" of="" high-speed="" interface="" applications="" because="" the="" case="" of="" low-loss="" line="" covers="" majority="" of="" practical="" pcb="" traces.="" for="" this="" case="" one="" may="" find="" in="" a="" number="" of="" sources="" (for="" example:="" 1)="" with="" the="" following="" formula="" that="" defines="" the="" attenuation="" and="" phase="" constants.="">

**Dielectric Losses**

It is obvious that in order to determine the attenuation and phase constants one needs to determine the values of R and G. Let me start with dielectric losses. I will confine the discussion to a certain level of details in order to ensure the clarity, however, if someone is interested in even more details and full understanding of the subject he or she should pick up a field and wave text book (for example 2). Or one may skip the explanation completely and move directly to the results.

I would like to start with consideration of one of Maxwell's equations:

This equation is Ampere's circular law (in differential form) and it states that magnetic field around the closed loop is generated by current passing through the loop and also by the changing electric field. In this equation the factor in front of the electric field intensity E contains the dielectric permittivity constant and conductivity of the dielectric material. In case of conductive dielectric material the conductivity parameter is not zero. In order to construct a uniform format and simplify the equation for the practical use a parameter of complex permittivity was introduced:

Where:

In order to summarize the discussion above, the complex dielectric permittivity includes the conductive property of dielectric in addition to original function of dielectric constant -- coefficient of dielectric polarization in the electric field. The whole point of this discussion is to show that conductive element of dielectric is real and as we know a current on a resistive element dissipates heat and loses power. That loss of signal power is the source of signal degradation.

As a next step in our discussion I would like to evaluate the magnitude of the dielectric conductance. I would like to start with introduction of a new parameter. This parameter is called a loss tangent or dissipation factor. This parameter measures the magnitude of the conduction current relative to the displacement current. The dielectric polarization is expressed by means of displacement current. The loss tangent or loss factor, as it often called, actually represents the ohmic loss in the dielectric. The reason we are considering this parameter is that the loss tangent parameter is routinely available from the manufacturers of dielectric materials. Using the loss tangent with dielectric permittivity information one may determine the value of dielectric conductivity for a specific frequency, as it is shown below:

For example:

In the formula of signal attenuation for Low-Loss Line the attenuation is calculated using the parameter of conductance and not the conductivity. The formula below shows how to obtain the value of conductance:

Where: A - area of the unit length of the dielectric under conductor of the signal line, and h - the distance in dielectric between signal and return lines.

It is easy to notice that expression for capacitance uses the same dimension values of the dielectric:

From these two expressions one may deduce another formula for conductance below:

One should consider, however, the details of trace construction. For example for a microstrip construction the value of the dielectric permittivity , should be calculated with consideration of dielectric change from PCB material to air. In case of FR4 PCB the value of eff is about 3.5 instead of 4.5. Another example may be the case of a stripline. In this case the capacitance of the trace to reference plane consists of two parallel capacitors. One -- between bottom surface of the line and plane below, another -- between the top surface and the plane above.

For 1 m long stripline with 0.005" wide trace and 0.005" dielectrics to ground layers, the value of conductance at 1GHz may be estimated as follows:

That gives us resistance of dielectric between the line and ground at about 100 Ohms.

Now let us look into another aspect of the transmission line in PCB that will cause signal degradation or loss.

**Skin Effect**

Before I go into formulas I would like to wave my hands and present a description of the skin effect that is easily understood.

The skin effect is a phenomenon of current occupying just the outer shell of the conductor instead of the whole cross section area. As we know the current usually selects the path of least resistance. For a DC current in a conductor any unit area of a cross section is the same as any other unit area of this cross section. The only resistance that DC current sees is the ohmic resistance. In case of AC current, of course, another parameter of the conductor should be considered -- inductance. The inductance presents impedance to the alternating current. With frequency increase of the alternating current the impedance is increasing. Inductance, on other hand, is a function of the conductor's geometry. With reduction of conductor's width the inductance is increasing because the narrower inductor is embraced by larger amount of magnetic flux.

In reality the current does not have a specific ring where it is uniformly distributed. The current density is decreasing closer to the center of the conductor. The voltage of the signal is the highest at the conductor's surface and decreasing toward the center. In fact, for a good conductor the amplitude of high frequency electromagnetic wave is attenuated very rapidly inside the conductor. From the practical point of view it is nice to define a region that may be used for comparison of different conditions. This region was defined as follows:

Once the skin depth parameter is calculated the resistance of the conductor is easy to determine. The copper resistivity r is a known value and resistivity is not changing with frequency. The factor that varies with frequency is the volume of the conductor that is used for the current conduction. Below are the formulas for line resistance calculations in high-frequency applications:

**Figure 2. Illustration of the skin depth in a PCB microstrip**

It is interesting to notice that the total resistance of the interface line conductor includes the sum of DC and AC resistances. The DC value is much smaller than the AC part of the resistance, as it is shown below, however, it still should be considered for the accurate assessment of the interface performance since the signals do have DC components.

The example of calculations below illustrates the values for a more-less typical line, 0.005" wide. The length of the line considered is unit long, or in this case it is 1 m. For copper:

Consider:

**Some Practical Considerations**

The discussion above provides some formulas and insight into the losses that interface line presents to a high-speed signal. These formulas also provide an insight how the dielectric losses and conductor's skin effect are affected by frequency. The attenuation due to dielectric conductance is directly proportional to frequency. The attenuation due to conductor's skin effect is a function of square root of frequency. In the examples above we have considered the 0.005" wide trace, 1m long with 50 Ohms characteristic impedance. The value of combined, AC and DC, resistance is about 70 Ohm, which is comparable to the 100 Ohm value of the resistance calculated for the dielectric body between the line and the ground. As you can see the values of these parameters are at about the same level and, therefore the their attenuation effects are of about comparable values. Of course with increase of the frequency the increase will be larger for the dielectric attenuation constant. The dielectric attenuation is proportional to frequency and conductive attenuation is proportional to the square root of frequency. Therefore, at some frequencies the effect of dielectric loss becomes dominant. That frequency for the typical PCB traces is around 1 GHz.

In order to model the losses accurately one should use appropriate simulation tools. Not all tools have the capability to address the signal losses. In addition, in order to obtain the correct model the correct parameters of the line and dielectric should be used. These parameters may be obtained from the vendor. Quite often the best approach includes development of a prototype, measurement of the line parameters and then building the models based on the these measurement to use in simulation tools. The measurements of line parameters could be tricky and require some expertise. For example, the measurement of loss tangent is not a straight forward procedure. It is describes it in more details in reference 1.

Another aspect of the conductive losses, which was not considered by the formulas and not considered by the simulators as well, is the quality of the conductor or its surface roughness. A rough conductor may cause additional up to 50% increase in resistance since the depth of the skin at high frequencies becomes comparable with the dimensions of the picks and valleys of the copper surface.

**This is the end of Part 1**

post a commentregarding this story.