2.23 Measuring Characteristic Impedance

A time domain reflectometer can be used to measure the characteristic impedance of a transmission line. The line is driven with a step function through a known matching resistor. If the transmitted voltage is reduced by a factor of 2, the characteristic impedance is equal to the resistor value. The voltage is sensed after the initial rise time. Obviously, measurements made on actual traces allows for very little experimentation.

An analog method can be used to measure the characteristic impedance of many transmission line geometries. The method makes use of the fact that characteristic impedance involves the ratio of conductor dimensions. For example, two 5-mil wide traces that are 2 mil thick that are spaced 5 mil apart has the same characteristic impedance as two bars of metal 0.5 in wide and 1/16 in thick, which are spaced 0.5 in apart. The bars of metal can be laid out on the surface of a test bench. The capacitance of this model can be measured by using simple test equipment. The characteristic impedance is related to the capacitance by Equations 2.3 and 2.4. The capacitance of the model is the capacitance of the traces multiplied by the model scale factor. In the example above, this factor is 100.

The conductors used in a model must be long enough to provide a useful capacitance. In the example above, a length of 1 m will yield a capacitance of about 50 pF. To measure this capacitance, a function generator that supplies triangle waves can drive the capacitance through a 100-ohm series resistor. The voltage can be set to change 20 V in 0.5 μs. Here dV/dt equals 40 V/μs. The current in the capacitance is CdV/dt or ± 200 μA. The voltage across the resistor in this example is ± 20 mV. The characteristic impedance is 1/cC, where c is the velocity of the wave in meters per microseconds and C is the capacitance in microfarads per meter. In this example, Z = 1/300 · 50 · 10⁻⁶ = 66.6 ohm.

For traces over a ground plane, the electric field concentrates in the dielectric. This means that most of the energy travels in the dielectric. For side-by-side traces over a dielectric, the electric field divides between the air and the dielectric. Note that the energy density in the dielectric is reduced by the relative dielectric constant. This means that more of the energy travels in air, and this energy travels faster. A measure of the capacitance does not relate to how the wave action divides or to the fact that the wave moves slower in the dielectric. Note that the time domain reflectometer method of measurement does not consider the way the energy divides between air and the dielectric.

This modeling method can also be used to measure leakage capacitance to a nearby trace. The measurement is made by connecting the nearby trace to common through a 100-ohm resistor and noting the voltage across the resistor. When the signal line is driven, some of the electric flux terminates on the nearby trace. This changing flux induces the current that is measured. The leakage capacitance determined in the model must be reduced by the scale factor to obtain the capacitance on an actual circuit board. In the example above, the scale factor is 100. If the mutual capacitance of the model is 3 pF, the mutual capacitance to a centimeter of trace is 0.03 pF. It is easy to see that this modeling method makes it possible to measure very small capacitances.