Series Resistor Analysis

It may not be widely recognized but hardware design engineers have encounters with signal integrity or SI issues practically on a daily basis.

On one of recent projects I have encountered a line termination application that is slightly different from usual techniques. The usual ones, like pull-ups and Thevenin equivalent circuits at the end of interface line or series resistors next to driver, are well understood. But the Micron Technology Application Note TN-46-14 has the following line termination recommendation for DDR memory data line termination: "For bidirectional I/O signals, such as DQ (Data line – M.G.), minimize ringing, overshoot, and undershoots by placing the R_S halfway between source and sink devices." That configuration is shown in Figure 1.

Figure 1. Resistor in the middle of the line

So… let us look why and how this approach works. The purpose of the series termination resistor is to reduce the value of the signal amplitude and rely upon reflection from the end of the line to bring the signal back into the range appropriate for receiver. The lower amplitude is beneficial for limiting reflections, crosstalk and emissions. The series resistor in the middle of the line is supposed to imitate the performance of a typical application when series resistor is placed close to the driver and, therefore, we should compare these two cases.

Figure 2. The typical case of series resistor is very close to the driver

The series resistor immediately after the driver, as shown in Figure 2, increases the driver's output resistance value and thus reduces the signal amplitude at point A calculated from the voltage divider generated by resistances R_out+R_s and Z:
V_A = V · Z/( Z + R_out + R_s) (1)

In the case of the resistor in the middle of the line, as shown in Figure 1, the point C is in the same position (in front of transmission line segment before receiver) as the point A in Figure 2. Therefore, we should compare the signal amplitudes at these points.
Let us consider the middle of the line case. The circuit configuration as seen from the point A is similar to one in the Figure 2 but without the series resistor R_s. Therefore:
V_A = V · Z/(Z + R_out) (2)

After the point A the signal with amplitude V_A traverses the transmission line to the point B (see Figure 3). It reflects from the point B, where Reflection Coefficient is:

r = [(Z+R_s)-Z]/[(Z+R_s)+Z]

Therefore, the signal amplitude at point B is:
V_B = V_A · [1 + r] = V_A · {1 + [(Z+R_s)-Z]/[(Z+R_s)+Z]} = V_A · 2(Z+R_s)/(2Z+R_s) (3)

Figure 3. Transmission line discontinuity

The signal amplitude at point C may be calculated from the divider as it is shown in Figure 4:
V_C = V_B · Z/(Z + R_s) (4)

Figure 4. Voltage divider created by series resistance and transmission line impedance

Combining the expressions for V_A, V_B and V_C we obtain the expression for V_C as a function of the input value V:
V_C = V · Z/( Z + R_out + R_s/2 + R_outR_s/2Z) (5)

As it could be seen comparing the expressions (5) and (1) the R_s
member of denominator in expression (1) is replaced by the (R_s/2 + R_outR_s/2Z ) in (5). That difference may show us the effect of moving the series resistor from the position near the driver to the middle of the transmission line.

The effect of the change clearly depends on the value of driver's output impedance. Consider that R_out of a high-speed transmitter circuit is relatively low. As a rule it is much lower than the value of line characteristic impedance that typically ranges between 45 Ohms and 65 Ohms. Therefore, depending on the value of R_out in respect to Z the range of the signal amplitude at point C varies between: V_C = V · Z/( Z + R_out + R_s/2) for R_out
approaching 0, which is slightly larger than V_A in (1), and V_C = V · Z/( Z + R_out + R_s) for R_out = Z, which is the same as V_A in (1).

The analysis shown above is beneficial in several ways:

• it shows that the series resistor placed in the middle of the transmission line does perform its duty of signal amplitude reduction
• it provides engineer with methodology to calculate the series resistor value
• it also shows the generic approach toward analysis of the high-speed interfaces with series termination resistors