High Speeds and Fine Precision Knock PCB Traces Off Pedestal

Knowing how to measure PCB trace impedances can help you optimize circuit performance from DC to gigahertz.

Rick Nelson, Senior Technical Editor -- Test & Measurement World, 1/1/2000

The ideal conductor is history. If you don’t believe it, your designs could be toast. From DC to the gigahertz frequency ranges, PCB trace resistance and reactance can degrade product performance. Taking care to measure trace impedances accurately can help you determine why a circuit fails to perform optimally.

Problems begin with direct-current measurements. For DC measurements, you needn’t be concerned with reactance, but even the low levels of resistance that PCB traces exhibit can introduce errors. Consider the copper trace in Figure 1a, exhibiting a 0.45-mW /square resistance (Fig. 1b). That trace exhibits about 0.18-W resistance.¹ If it carries an analog signal to a 16-bit ADC having a 5-kW input impedance, it will contribute a per-unit error of 0.18/5k = 3.6x10^-5. That’s approximately equal to 2x2^-16, signifying that the trace resistance will contribute a 2-LSB error to the ADC’s output.

You can make a stab at measuring resistances of that magnitude using a standard DMM on its lowest resistance range. Just clip the DMM’s test leads together to measure their resistance; some DMMs will automatically autozero themselves.² A four-digit meter with a 50-W scale will provide 0.01-<W< p resolution, although accuracy at below 1-<W< values will suffer.<>

Figure 1. (a) The 10-cm-long PCB trace depicted here contributes 0.18 W to the circuit containing it. The resistance of PCB trace material is often expressed as ohms/square. The trace in part (a) consists of 400 0.25x0.25-mm squares set end to end; each square has a resistance of 0.18W/400 squares, or 0.45 mW/square. (b) You can combine the squares into any larger square and maintain the same resistance.

The Kelvin Double Bridge
The traditional method for accurately measuring such small resistances is to use the Kelvin double bridge³ (Fig. 2a). For R_A/R_C = R_B/R_D, the galvanometer G reads 0 V when the resistor under test, R_X, equals a standard calibrated resistance, R_S, times R_A/R_B. The resistances labeled R_X(lead) represent test-lead or PCB-trace resistances in series with R_X that you want to exclude from your measurement of R_X. R_L constitutes a low-resistance link from the ends of R_B and R_D opposite the galvanometer.

You first measure R_X with switch SW_L closed. Opening SW_L after balancing the bridge (that is, after adjusting R_S such that the galvanometer reads 0) serves as a check on the values of R_A through R_D. If the galvanometer deviates from 0 when you open the switch, then the R_A/R_C ratio no longer equals R_B/R_D, and you’ll need to recalibrate the bridge or your measurement won’t be accurate.

Note that the double bridge performs a four-wire measurement: Probes P_F1 and P_F2 serve as the forcing wires, delivering current to R_X; probes P_S1 and P_S2 serve as the sensing wires, measuring the voltage developed across R_X in response to the current applied to it. The keys are, first, to apply the sensing wires such that they will only measure the voltage across the resistance under test, and second, to minimize the current through them. For resistors R_A through R_B having 1-kW values, the Kelvin double bridge can measure resistances from 1 µW to 10W . If you substitute an AC source of frequency f for the battery shown in Figure 1, you can measure reactive impedances, where a capacitor’s impedance in ohms equals –j/(2pfC) and an inductor’s impedance in ohms equals j2pL.

Tinkering with the classical Kelvin double bridge is a tedious procedure—you have to fine tune the R_A through R_D values and find a calibrated R_S that corresponds to the desired value of your resistor under test. If you want to measure reactive elements (for instance, a PCB trace’s inductance or capacitance), you’ll need a calibrated frequency generator. Fortunately, modern digital milliohmmeters and impedance meters handle much of the grunt work for you. One approach that such instruments employ is the auto-balancing bridge⁴ (highlighted in gray in Fig. 2b) in conjunction with selectable RS values.

Figure 2. (a) The Kelvin double bridge compares an unknown resistance RX with a standard known resistance RS. (b) Modern milliohmmeters employ a variation known as the auto-balancing bridge. In-circuit measurements require a guard connection via probe PFG to ground.

The auto-balancing circuit maintains the Kelvin bridge’s four-wire measurement configuration. Forcing current flows from the voltage source to R_X through probe P_F1 into R_X. For the voltage polarity shown, the op amp develops a negative output voltage. To maintain the summing-junction voltage at 0, the output voltage reaches a negative level sufficient to sink the R_X current by way of probe P_F2 through R_X(lead), R_S(lead), and R_S. With knowledge of the calibrated R_S value, the voltage across R_X (which voltmeter VM₁ measures), and the voltage across R_S (which VM₂ measures), an auto-balancing instrument’s internal digital processor can calculate R_X. Such a technique can make measurements down to 1 mW.

If you are measuring impedances in-circuit, you’ll need to use a five- or six-wire measurement technique.⁵ Those wires include the two forcing and two sensing wires of the Kelvin configuration plus a guard connection (the fifth wire) and optionally a guard-sense wire (the sixth wire).

The portion of Figure 2b highlighted in yellow illustrates use of the guard configuration. R₁ and R₂ represent resistances in parallel with R_X; the dashed lines suggest that other components might be attached to the network comprising R_X, R₁, and R₂ as well. Any current flowing through R₂ into the op amp’s summing junction would disturb the auto-balancing bridge’s determination of R_X; for the configuration shown, the bridge would report the value of R_X in parallel with R₁ plus R₂ plus their lead resistances. Guarding minimizes current flow through R₂ by forcing the R₂ node opposite the summing junction to ground by means of probe P_FG—the fifth wire. Because the probe-to-ground connection won’t be completely resistance-free, you can add a sixth, guard-sensing, wire to measure the voltage at the R₂ guard connection, enabling your instrumentation to calculate and compensate for the error current flowing through R₂.

Transmission-Line Impedances
The configurations thus far discussed work at DC or low frequencies where Kirchhoff’s current law applies instantaneously: For every electron forced into one end of the Figure 1 trace, a displaced electron immediately pops out the other end. At high frequencies, this assumption falls apart. For instance, suppose you apply a 1-ps risetime edge to one end of the 10-cm trace. You’ll have to wait at least 333 ps, while your signal propagates along the trace near the speed of light, before anything happens at the other end. Your trace has turned from a simple low-resistance conductor into a transmission line.

To describe the trace voltage and current as your signal propagates along the trace, you can’t directly rely on Ohm’s and Kirchhoff’s laws. You can, however, investigate an infinitesimal section of the trace and nearby conductors, typically a ground plane separated from the trace by a layer of PCB substrate (Fig. 3), a configuration known as a microstrip line. Your PCB trace will have a resistance and inductance per unit length (R' and L'), and your board will exhibit a trace-to-ground-plane conductance and capacitance (G'and C', respectively). An infinitesimal length dx will have resistance, inductance, capacitance, and conductance values of R'dx, L'dx, C'dx, and G'dx, respectively. For this infinitesimal length, you can invoke Ohm’s law and Kirchhoff’s voltage and current laws to derive relationships between the voltage v(x,t) and current i(x,t) at position x.

Figure 3. This PCB trace, substrate, and ground plane constitute a microstrip line. You can apply Ohm’s and Kirchhoff’s laws to an infinitesimal segment of the line to calculate its characteristic impedance.

You can look in any introductory textbook⁶ on electromagnetics or transmission-line theory to see the algebraic details of the derivation. The result follows:

Z₀ is the PCB characteristic impedance, which for low resistance and conductance values (which characterize a nearly lossless line) reduces to . Characteristic impedance is a critical spec for high-speed circuitry on PCBs; traces for which characteristic impedance values are critical are called controlled-impedance traces.

You can use a field-solving controlled-impedance calculator program⁷ to calculate the characteristic impedance of microstrip lines and other PCB transmission-line configurations based on physical dimensions and conductor and dielectric properties. Measuring characteristic impedance to see how well it compares with the calculated value is more complicated than simply hooking up an ohmmeter. The impedance you will see depends not only on the line under test but also on what’s connected to it.

If you measure the input impedance Z_IN of an infinitely long microstrip line having characteristic impedance Z₀, you’ll find that Z_IN = Z₀. Similarly, if you measure a finite microstrip line having characteristic impedance Z₀ and that’s terminated with a matching load impedance Z_L = Z₀, you’ll also find that Z_IN = Z₀. For other cases—finite transmission lines terminated with impedances that don’t match Z₀—the measurement becomes more complex. When an applied incident signal encounters the impedance mismatch, a portion of its energy reflects back toward the source. The ratio of reflected voltage E_R to the applied, or incident, voltage E_I is called the reflection coefficient⁸:

If you know the reflection coefficient and one of the impedance values, you can use this equation to calculate the other impedance.

The instrument typically used to measure the reflection coefficient is the time-domain reflectometer (TDR)—essentially a time-domain oscilloscope coupled with a source for generating an incident voltage that it can apply to an impedance under test (Fig. 4a). Figure 4b through 4e show TDR scope waveforms you could expect to see for various load impedances. For the open-circuit case in Figure 4b, all incident energy reflects back toward the source. At time t₁, the time required for the incident wave to propagate down the line and the incident wave to return, the reflected voltage E_R = E_I adds to the incident voltage, and the TDR scope trace rises to 2E_I. For a short-circuit case (Fig. 4c), the reflected voltage is equal in magnitude but opposite in polarity to the incident voltage, and the reflection coefficient is –1.

Figure 4. (a) A time-domain reflectometer incorporates an oscilloscope and a signal generator; it can apply an incident voltage to a microstrip line terminated with a load impedance and monitor the reflected voltage. The displays indicate (b) an infinite load and (c) a short-circuit load as well as load impedances (d) slightly higher and (e) slightly lower than the microstrip line’s characteristic impedance.

Often, you’ll be comparing a microstrip line’s impedance with that of a known reference element. To test 28-W Rambus memory-board impedances, for example, you can compare them with a 25-W reference made up of two standard 50-W impedances in parallel.⁹ In such cases, you’ll see much lower reflection coefficients, such as shown in Figures 4d and 4e.

You don’t need a dedicated TDR to measure reflection coefficients—any signal source and oscilloscope that meets the speed requirements dictated by your application will do. TDRs do have advantages, however. Many will automatically calculate and display the reflection coefficient, so you won’t have to peer at a scope screen while crunching numbers on a pocket calculator.

In addition, TDRs make it easy to locate an impedance mismatch. Assuming you know the velocity of propagation for a signal traveling along a transmission line—which you can determine by performing a TDR measurement on a known length of the same type of line—you can calculate a mismatched impedance’s location based on the delay between generation of the incident voltage and the appearance of the reflected voltage. (This fact is more useful for measuring long cables than for microstrip lines.) Most TDRs can perform these calculations for you, displaying distance as well as time on their x axis.

Variations in trace width and plating thickness can create havoc on high-speed boards like Rambus boards. Verifying characteristic-impedance values on controlled-impedance boards during the manufacturing process is critical to ensure proper operation. Because controlled-impedance traces on multilayer controlled-impedance boards are often inaccessible, manufacturers typically fabricate test coupons along with their production boards. Test coupons¹⁰ are small PCBs having the same layer and trace construction as production boards. Test coupons can be extracted periodically from the manufacturing line to ensure they provide an adequate representation of any process variations that might degrade production boards. They are trimmed to provide access to transmission-line elements for test. T&MW

FOOTNOTES

1. Bryant, James, “Ask The Applications Engineer-10,” Analog Dialogue 30th Anniversary Reader Bonus, Analog Devices, 1995–1999, nwd2www1.analog.com/publications/magazines/Dialogue/Anniversary/10.html.

2. “Troubleshooting with TX-DMM True RMS Multimeters, #4,” Application Note 3MW-11873-0, Tektronix, Beaverton, OR, 1998, www.tek.com/Measurement/App_Notes/tx3/trnum4/eng/.

3. Fink, Donald, and Donald Christiansen, Electronic Engineers’ Handbook, 2nd ed., McGraw-Hill, New York, 1985, ISBN 0-07-020981-2, pp. 17–25.

4. “New Technologies for Accurate Impedance Measurement," Product Note 4294-2, Hewlett-Packard, Palo Alto, CA, 1999.

5. Bateson, John, In-Circuit Testing, Van Nostrand Reinhold Co., New York, 1985, ISBN 0-442-21284-4, p. 113.

6. One such book is Ulaby, Fawwaz T., Fundamentals of Applied Electromagnetics, 1999 Edition, Prentice Hall, Upper Saddle River, NJ, 1999, pp. 39–43, ISBN 0-13-011554-1.

7. Nelson, Rick, “Program Evaluates PCB-Trace Differential Impedance,” TestLit Review, Test & Measurement World, May 1999 Part 2, p. 9.

8. “Time Domain Reflectometry Theory," Application Note 1304-2, Hewlett-Packard, Palo Alto, CA, 1988.

9. “Printed Circuit Board (PCB) Test Methodology,” Intel, Hillsboro, OR, 1999, developer.intel.com/design/chipsets/memory/pcbtest.htm.

10. “Testing Controlled Impedance Boards with Test Coupons,” Application Note 124, Polar Instruments, Guernsey, UK, 1999, www.polar.co.uk/support/cits/AP124.html

You can reach Rick Nelson at rnelson@cahners.com

The Basic Measurement Series is an occasional series of articles that review basic techniqes and principles.