Skip to main content

Differential Pairs in PCB Transmission Lines

Author Profile img: Atar Mittal

By Atar Mittal

February 3, 2020 | 9 Comments

Contents

webinar image
On-demand webinar

How Good is My Shield? An Introduction to Transfer Impedance and Shielding Effectiveness

by Karen Burnham

This article, written by Atar Mittal, the General Manager of our Design and PCB Assembly Divisions, will go over the characteristics and parameters of differential pairs in PCB transmission lines.

How to design differential pairs in PCB transmission lines

What is a single-ended line?

In our PCB transmission line article, we established that a single-ended transmission line can be modeled as follows:

Single ended transmission line
Single-ended transmission line

The relation between the voltage and current at any point on the line is given by:

relation-between-the-voltage-and-current.webp
The relation between the voltage and current.
 

Where ‘Z0’ is the characteristic (or instantaneous) impedance of the line. For a lossless or almost lossless line, we saw that ‘Z0’ is given by:

equation-for-characteristic-impedance-of-the-line.webp
Equation for characteristic impedance of a signal line.

Where ‘L0’ and ‘C0’ are respectively the inductance and capacitance per unit length (pul) of the line.

 

PCB Transmission Line eBook - Cover Image

PCB Transmission Line eBook

5 Chapters - 20 Pages - 25 Minute Read
What's Inside:
  • What is a PCB transmission line
  • Signal speed and propagation delay
  • Critical length, controlled impedance and rise/fall time
  • Analyzing a PCB transmission line

 

Differential pair line

A pair of lines can be modeled as follows:

Differential pair line
Differential pair line

We will assume here that both of the lines of the pair are identical and uniform. And that they have the same separation from each other along the entire length of the line. These are precisely the characteristics of a pair of lines to be designated as a differential pair.

When we have a pair of lines close to each other, it is fair to say that the presence of a current in line 2 will induce some voltage in line 1, and a current in line 1 will induce some voltage in line 2. Thus, the voltage ‘V1’ at line 1 will not only depend on the current ‘I1’ in line 1 (through the impedance ‘Z0’ of line 1). It will also depend on the current ‘I2’ in line 2 through coupling or mutual impedance ‘Zm’ between lines 1 and 2. The following equation can express this situation:

equation-for-mutual-impedance-Zm-between-lines.webp
Equation for mutual impedance between differential lines.

Where ‘Zse’ is the characteristic impedance of line 1 and ‘Zm’ is the mutual or coupling impedance between line 1 and line 2.

Coupling between the two lines

Similarly, for line 2 (of the differential pair), being identical to line 1, we can write the following equation:

The mutual impedance ‘Zm’ arises because of the coupling between the two lines:

equation-for-second-diffrential-line.webp
Voltage equation for the second differential line.

And the most important coupling agents are ‘Lm’, the mutual inductance pul, and ‘Cm’, the mutual capacitance pul between lines 1 and 2.

The closer the two lines are to each other, the greater is the coupling between them. If the separation ‘S’ between the lines is reduced, the values of all three parameters – ‘Lm’, ‘Cm’, and ‘Zm’ – increase.

The equations (3) and (4) are true for any point on line 1 and the corresponding point on line 2. And for a uniform differential pair, the ‘Zse’ and ‘Zm’ have the same value at every location along the differential pair.

Coupling coefficient

Since ‘Zm’ provides the magnitude of the signal voltage coupled from one line to another, as compared to the signal contributed through its own ‘Zse’, we may define the ratio ‘Zm/Zse’ as the coupling coefficient between the two lines of a differential pair:

coupling-coefficient-between-the-two-lines-of-a-differential-pair.webp
Coupling coefficient between two lines of a differential pair.

 

tool-image

PCB DESIGN TOOL

Conductor Spacing and Voltage Calculator

Calc TRY TOOL

 

What are differential and common-mode signals?

Odd and even modes

Let ‘V1’ and ‘V2’ be the signal voltages and ‘I1’ and ‘I2’ be the signal currents in the two lines of a differential pair characterized by impedances ‘Zse’ and ‘Zm’. These six quantities are related through equations (3) and (4).

The difference in signal voltage ‘V1’ and ‘V2’ is called the differential signal ‘Vdiff’. Half of it is also called the odd mode signal:

equation-for-odd-mode-signal.webp
Odd mode signal voltage equation.

The average value of ‘V1’ and ‘V2’ is the common mode signal ‘Vcom’. It is also called the even mode signal:

equation-for-even-mode-signal.webp
Even mode signal voltage equation.

From 5 and 6, we can express ‘V1’ and ‘V2’ in terms of ‘Vdiff’ and ‘Vcom’ as follows:

common-mode-signal-and-differential-mode-signal.webp
V1 and V2 are expressed in odd and even mode signal voltages.

These equations indicate the universal fact that any two arbitrary signal values ‘V1’ and ‘V2’ can always be expressed as and therefore analyzed in terms of a common (or even) mode signal and a differential (or odd) mode signal.

Furthermore, the equations (7a) and (7b) also allow us to think that ‘Vcom’ or ‘Veven’ part of the signals in ‘V1’ and ‘V2’ are a kind of ‘’bias’’ on top of which the differential mode (or odd mode) signals ‘+Vodd’ and ‘-Vodd’ ride to result in ‘V1’ and ‘V2’. This viewpoint is the most important aspect of differential signaling analysis.

Propagating time: Varying signals

At this point, before we proceed further in our conceptual analysis, let’s keep in mind that the use of transmission lines – single-ended or differential – is to propagate time-varying signals – usually high-speed digital signals or high-frequency analog signals – from one place to another. It is time-varying signals that constitute information. Static voltages and currents do not have any information.

Therefore, looking at equations (7a) and (7b) above, we need to emphasize that ‘Vcom’ (or ‘Veven’) is only a de-bias voltage on the two lines of a differential pair. The main signal is the differential signal (‘V1 – V2’). Half of which is added to line 1, usually called the positive line. It is identified by the suffix ‘+’ or ‘P’ in the name of the signal on it. The other half is subtracted from line 2, usually called the negative line and identified by the suffix ‘-‘ or ‘N’ in the name of the signal on it, to constitute ‘V1’ and ‘V2’ at the signal transmitter end.

At the destination

At the destination, the two lines go to the inputs of a differential receiver, which detects the difference (‘V1 – V2’) in the amplitude of the signals on the two lines as the true signal. Thereby, in the process, rejecting any common mode signal, deliberate AC bias, and/or common mode noise.

This ability of rejecting the common mode signal and thus any common mode noise in differential signaling makes it far superior to single-ended signaling, where there is no way to separate the noise from the actual signal.

Having stated this, we may be tempted to think that we need to analyze deeper only the differential or odd mode and ignore the common mode. But let’s not forget that as signals propagate on the lines, they are susceptible to all kinds of noise that gets superimposed on them. This may affect signal integrity adversely. Therefore, while differential pair lines’ response to the differential (ie. odd) part of the signal is our main concern, we must also analyze the differential pair’s response to the common (or even) mode signal.

How to analyze differential and odd-mode signals?

We will now analyze the differential pair when we have sent only odd-mode signals, without any common-mode part.

In this case, since ‘Vcom = Veven = 0’, we have from (7a) and (7b):

analyzing-differential-and-odd-mode-signals.webp
Equation establishing the relation between V1 and V2.

Since the lines are identical, we will have ‘I2 = -I’ so that ‘I1 + I2 = 0’. Thus, there will be zero current in the return path.

Equation (3) or (4) now gives us:

v1-equation.webp
V1 is expressed in terms of impedance and current.

We, as usual, define the ratio of ‘Vodd/Iodd’ as the odd-mode impedance of the line:

equation-for-odd-mode-impedance.webp
Equation for odd-mode impedance.

Now ‘Zm = K.Zse’ where ‘K’ is the coefficient of coupling.

odd-mode-impedance-in-terms-of-coefficient-of coupling.webp
Odd mode impedance in terms of the coefficient of coupling.

From this, it is clear that odd mode impedance is less than the single-ended impedance ‘Zse’ of the single line and greater the ‘Zm’ (or coupling between the two lines of the pair) is, the lesser will ‘Zodd’ be from ‘Z0’.

 

High-Speed PCB Design Guide - Cover Image

High-Speed PCB Design Guide

8 Chapters - 115 Pages - 150 Minute Read
What's Inside:
  • Explanations of signal integrity issues
  • Understanding transmission lines and controlled impedance
  • Selection process of high-speed PCB materials
  • High-speed layout guidelines

 

What is differential impedance?

The differential impedance is the impedance seen by a purely differential (ie, odd mode) signal over a differential pair. As:

equation-for-differential-impedance.webp
Equation for differential impedance.        

Thus, the differential impedance is twice the odd-mode impedance. Or the odd mode impedance is half of the differential impedance.

Most often, the only specified requirement of a differential pair is its differential impedance. Typical values for most common differential signal types are 90-ohm differential, 100-ohm differential, or 120-ohm differential. In some cases, we can also use 75-ohm differential impedance.

 

tool-image

PCB DESIGN TOOL

Impedance Calculator

Calc TRY TOOL

 

Differential pairs: even or common mode?

Let’s now take the case when both the lines of the differential pair are excited by a common signal:

equating-common-or-even-mode-with-v1-and-v2.webp
Equating the common or even mode with V1 and V2.

Since ‘V1 – V2 = 0’, there is no differential mode signal. As ‘V1 = V2’, ‘I1 = I2’, and the equations (3) and (4) become:

 

expressing-common-and-even-more-voltages-in-impedance-and-current.webp
Expressing common and even mode voltages in impedance and current.

Therefore, we can define the even mode impedance ‘Zeven’ as the ratio of the even mode voltage to the even mode current in each line:

expressing-even-mode-impedance-in-even-mode-voltage-and-current.webp
Expressing even-mode impedance in even-mode voltage and current.

As:

expressing-Z-even-in-terms-of-characteristic -line-impedance.webp
Expressing Z even in terms of characteristic line impedance.

And the common mode signal current in the differential pair is ‘I1 + I2 = 2I1’ so that the common mode impedance of the differential pair is:

rewriting-common-mode-impedance-in-terms-of-characteristic-impedance.webp
Rewriting common mode impedance in terms of characteristic impedance.

Thus, the common mode impedance of the differential pair is half of the even mode impedance of one line. It basically equals two even-mode impedances in parallel.

From equation (10), it is clear that ‘Zeven’ is greater than ‘Zse’. The greater the coupling between the two lines of the pair is, the larger ‘Zeven’ will be compared to ‘Zse’.

Teledyne Lecroy by Sierra Circuits banner

Recap: Odd and even mode impedances

Odd and even mode impedances of a differential pair of lines are fundamental characteristics of the pair. We can evaluate all other impedances from their values, as illustrated below:

Since:

even-and-odd-mode-impedance-interms-of-charecteristic-impedance.webp
Even and odd mode impedance in terms of characteristic impedance.

 

redefining-k.webp
Redefining k.

To these, we can add:

equations-for-differential-and-common-mode-impedance.webp
Equations for differential and common-mode impedance.

Detailed analysis of a differential pair in terms of line inductances and capacitances:

The analysis of a lossless single-ended transmission line was done using the following circuit model for an infinitesimally small line of length ‘delta x’:

Differential pair in terms of line inductances and capacitances
Differential pair in terms of line inductances and capacitances

Here, ‘L’ and ‘C’ are respectively the inductance and capacitance per unit length of the line. After analysis, we had derived that the characteristic (or instantaneous) impedance of the line at a point was given by:

 
equation-for-instantaneous-impedance.webp
Equation for instantaneous impedance.

And the propagation delay ‘Pd’ was given by:

 
propagation-delay-equation.webp
Equation for propagation delay.

We can now apply the above model and results to the analysis of the lines of a differential pair in terms of inductances and capacitances involved. And we are assuming that lines as such that the conductor resistances (‘Rs’) and dielectric conductances (‘Gs’) can be neglected for the impedance and propagation delay purposes. This will be the case at practical frequencies of interest.

The following diagram gives the circuit model of an infinitesimally small length of a differential pair line:

Circuit model of an infinitesimally small length of a differential pair line
Circuit model of an infinitesimally small length of a differential pair line

Here ‘L0’ and ‘C0’ are respective by the inductance and capacitance of each line per unit length. ‘Lm’ is the mutual inductance per unit length between line 1 and line 2. ‘Cm’ is the capacitance per unit length between line 1 and line 2.

Case 1: Odd mode (purely differential signal)

Here ‘V2 = -V1’ and ‘I2 = -I1’. Thus, no current flows in the return path. ‘Cm delta x’ can be considered as two capacitors, each of value ‘2Cm delta x’, in series, whose center point is at zero potential. (That is because of the potential division between two equal capacitors.) Thus, the equivalent circuit of the above in odd mode becomes:

Odd mode purely differential signal
Odd mode (purely differential signal)

Let’s look at line 1 (the line 2 case is exactly similar).

The inductive voltage in line 1 will consist of two parts. One due to ‘I’, flowing through ‘L0 delta x’. The other one due to ‘I2 = -I1’, flowing through ‘Lm delta x’. These can be equivalently stated as due to ‘I’, flowing through (L0–Lm) delta x. Thus, the effective inductance per unit length of line 1 in odd mode, denoted by ‘Lodd’, will be given by:

effective-impedance-per-unit-length-in-odd-mode.webp
Effective impedance per unit length in odd mode.

And the effective capacitance between line 1 and zero potential line is ‘(C0 + 2Cm) delta x’. Thus, the odd mode line capacitance per unit length is:

effective-capacitance-in-odd-mode.webp
Effective capacitance in odd mode.

By definition and similar to results derived in the case of the single-ended line, the odd mode characteristic impedance is given by:

characteristic-impedance-in-odd-mode.webp
Characteristic impedance in odd mode.

And the propagation delay per unit length for odd-mode signals is given by:

propagation-delay-in-odd-mode.webp
Propagation delay in odd mode.

This is the propagation delay per unit length that will take place for the purely differential part of the signal. The point to also remember here is that the odd mode or purely differential signal’s electromagnetic waves exist largely in and around the space between the two lines. And they are relatively less affected by the reference or ground planes.

Case 2: Even mode (purely common mode signal)

In this case, ‘V2 = V1’ and as the two lines of the pair are identical, ‘I1 = I2’. Since ‘V1 = V2’, the capacitance ‘Cm’ between the two lines will not affect the currents in the two lines. It can therefore be ignored, leading to the following equivalent circuit:

Even mode purely common mode signal
Even mode (purely common mode signal)

Let’s look at line 1 (the line 2 case is identical).

The inductive voltage in line 1 will be contributed by ‘I1’, flowing in ‘L0 delta x’. And by ‘I2 = I1’’, flowing in ‘Lm delta x’. This is equivalent to saying ‘I1’ flowing through ‘(L0 + Lm) delta x’. Therefore, the effective inductance per unit length of either line in even mode will be:

effective-impedance-per-unit-length-in-even-mode.webp
Effective impedance per unit length in even mode.

And the effective capacitance per unit length of either line in even mode will be:

effective-capacitance-per-unit-length-in-even-mode.webp
Effective capacitance per unit length in even mode.

Therefore, the even mode impedance of either line will be given by:

characteristic-impedance-in-even-mode.webp
Characteristic impedance in even mode.

And the propagation delay per unit length for even-mode signals is given by:

propagation-delay-in-even-mode.webp
Propagation delay in even mode.

Equations (13c) and (14c) clearly indicate that ‘Zeven’ is greater than ‘Zodd’. From theory, it is also known that ‘Lm’ is less than ‘L0’ and ‘Cm’ is less than ‘C0’.

‘Zse’, ‘Zm’, and ‘K’ in terms of ‘L0’, ‘C0’, ‘Lm’, ‘Cm’

Using equations (12), (13), and (14), we have:

Zse-Zm-K-interms-of-Lo-Co-Lm-and-Cm.webp
Zse, Zm, K in terms of Lo, Co, Lm, and Cm.

Some words about ‘Zse’

‘Zse’ is the single-ended impedance of either of the two lines in the presence of the other line. It is not exactly the same as the characteristic impedance. The presence of the second line somewhat reduces the impedance. The more coupled (or closer) the two lines are, ‘Zse’ will become a little less.

It is important to note that ‘Zse’ does not change much if the coupling between the two lines is not high. Or if we can make the separation between the two lines greater than the maximum conductor width or dielectric height between the signal layer and the closest ground/reference plane.

i.e., ‘Zse’ is relatively the same if the separation is greater than the maximum signal trace width, dielectric height between the signal layer and the nearest reference plane.

Further, ‘Zse’ is not approximate equal to ’L0/C0’ but is nearer to ’L0/(C0+Cm)’. This may appear surprising at first glance, but that is the case.

It is not out of place to define a few new terms:

defining-zo-pdo-kl-kc.webp
Defining Zo, PDo, KL, and Kc.

‘KL’ and ‘KC’ can be called the inductive and capacitive coupling coefficients, respectively. Using these, we can write equation (15) as:

zodd-and-zeven-in-terms-of-kl-and-kc.webp
Zodd and Zeven in terms of KL and Kc.

Since ‘KL’ and ‘KC’ are less than 1, and for most practical cases will be significantly less than 1, we can retain only the first-order terms so that:

redefining-zodd-and-zeven.webp
Redefining Zodd and Zeven.

If we compare Equation (18a) with (8a) and (18b) with (10a), it is easy to conclude that:

equating-zse-and-zo.webp
Equating Zse and Zo.

And:

expressing-k-in-terms-of-kl-and-kc.webp
Expressing K in terms of KL and KC.

It is worth mentioning here that in most practical cases of stripline differential pairs, the inductive coupling coefficient ‘KL’ and the capacitive coupling coefficient ‘KC’ are nearly equal if the dielectric constants of the PCB materials above and below the signal layer are nearly equal.

Two crosstalk-related parameters

Here, we would also like to define two more parameters, NEXT and FEXT:

defining-next-and-fext.webp
Defining NEXT and FEXT.

NEXT is called the nearest crosstalk coefficient. FEXT is called the far-end crosstalk coefficient. These parameters are important in crosstalk analysis when of two nearby lines, one is the main signal line and the other is the quiet line on which we wish to determine the crosstalk voltage induced due to signal voltage on the main line. We will develop this point in the next article about crosstalk analysis in detail.

 

For more design information, check with our DESIGN SERVICE team.

 

Differential Pair eBook - Cover Image

Differential Pair eBook

3 Chapters - 18 Pages - 30 Minute Read
What's Inside:
  • Differential and common mode signals
  • Differential impedance
  • Even and common modes
  • The physical parameters

 

About the author: Atar Mittal is the Director and General Manager of the design and assembly division at Sierra Circuits. He is responsible for the design and development of strategies and process automation tools for complex printed circuit boards and assemblies. Atar is also currently engaged in the development of productivity tools for electronics designers that would have a tremendous impact on shortening the development time.

post a question

Start the discussion at sierraconnect.protoexpress.com

Subscribe
Notify of
guest
9 Comments
Oldest
Newest Most Voted
Inline Feedbacks
View all comments

Talk to a Sierra Circuits PCB Expert today

24 hours a day, 7 days a week.

Call us: +1 (800) 763-7503
Book a Meeting with a Sales Rep
Email us: through our Customer Care form