Differential Impedance

Nowadays it is not unusual to use differential signaling. Majority of high-speed interfaces are using the differential signaling. However, many engineers are not clear on the nature of one of the major parameters of differential lines – the differential characteristic impedance.

Let us now consider the specific aspects of signal propagation on differential transmission line. The signals on two wires of a differential line may have any of the infinite number of possible relationships. If the signal and signal-not lines are coupled the signals on both lines will affect each other and distort their waveforms. There are two modes of signals relationship, however, when signals reach their destinations undistorted: the differential or odd mode and common or even mode. In case of differential mode the signals are exactly 180 degree out of phase, and in case of common mode the signals are exactly in phase. If the branches of differential line are geometrically slightly different the differential signal will arrive distorted to destination. That means that geometry of lines defines the modes that will carry the signals to the end undistorted. For practical purposes we will consider symmetrical or balanced lines.

In case of the single ended lines we want to minimize the mutual components in order to reduce the effect of one line on another, or crosstalk. For differential application the mutual parameters are defined and should be controlled. We may start with general equations that govern performance of two lines with mutual capacitance and inductance.

The first two equations express voltage as a function of current with help of self and mutual inductances.

V1 = L1 * dI1/dt + Lm * dI2/dt (1a)
V2 = Lm * dI1/dt + L2 * dI2/dt (1b)

The second two equations represent current as a function of voltage with help of capacitances from the lines to reference plane and between the lines themselves.

I1 = C1 * dV1/dt + Cm * d(V1 - V2)/dt

I2 = C2 * dV2/dt + Cm * d(V2 - V1)/dt
or
I1 = (C1 + Cm) * dV1/dt - Cm * dV2/dt (2a)
I2 = - Cm * dV1/dt + (C2 + Cm) * dV2/dt (2b)

Where:
L1 and L2 are respective self-inductances of each branch of differential line
Lm is a mutual inductance between two branches of differential line
C1 and C2 are respective self-capacitances of each branch of differential line to ground
Cm is a mutual capacitance between two branches of differential line

Now we may address the Odd or Differential signal propagation mode, where I1 = -I2; V1 = -V2. From equations (1) and (2) for one branch of differential lines:

V1 = L1 * dI1/dt - Lm * dI1/dt => V1 = jw(L1-Lm)*I1
I1 = (C1 + Cm) * dV1/dt + Cm * dV1/dt => I1 = jw(C1+2Cm)*V1

From these two equations it is easy to obtain the relationship between the voltage and current, which represents the characteristic line impedance of one branch of differential line and that is called an Odd characteristic line impedance:

Zodd = V1/I1 = Ö[(L1 – Lm)/(C1 + 2Cm)]

Also considering that we are dealing with symmetrical lines and, therefore, L1 = L2 = L and C1 = C2 = C, than:

Zodd = Ö[(L – Lm)/(C + 2Cm)] (3)

In order to determine the differential characteristic impedance we may write equations for both branches of the differential line:

V1 = Zodd * I1
V2 = Zodd * I2

Considering that differential voltage between the signals on branches of differential line is the difference between voltages on individual branches, than:

Vdiff = V1 - V2 = Zodd * (I1 -I2), and since I1 = -I2 = I, than:
Vdiff = Zodd * 2 * I = 2*Zodd * I

Therefore:

Zdiff = 2*Zodd = 2*Ö[(L – Lm)/(C + 2Cm)] (4)

Comparing Zodd expression (3) to the characteristic line impedance for a single ended line Zse = Ö(L/C) it is easy to notice that Zodd < Zse. That may be assessed intuitively as well. If two lines are coupled a current will exist between these lines in addition to the current between the line and reference plane and since for the same signal voltage level the current is higher, therefore, the line impedance is the lower. If we move the branches of differential line far away from each other, so the Lm = 0 and Cm = 0, then Zodd = Zse = Ö(L/C) and Zdiff = 2* Zse = 2*Ö(L/C), which is also a valid way to construct a differential line in some circumstances.