Today's higher population densities on pc boards, faster switching speeds, and greater switched power contribute to the increased probability of interference between circuits, modules, and systems. This means emissions requirements are much tighter. Design engineers now recognize the need for more rational and effective bypass and decoupling strategies as they combat the dark side of electromagnetism.

The old "rabbit's foot" approach of sprinkling capacitors around the board proves much less effective under the new conditions, possibly even aggravating an existing problem. All capacitors, leads, and board traces have associated series inductance which, at the higher switching rates, is neglected at one's peril. These formerly negligible reactances can interact with the capacitance to produce not only zeroes, but also poles that could yield unpleasant or mystifying results.

We approached this issue in a previous article *("Tuned Decoupling Tames Noise In Switching Circuits," *Electronic Design,* July 6, 1998, p. 42)* by exploring the use of capacitor-based decoupling networks tuned via pc-board trace inductance to specific frequencies that we desired to suppress. The article detailed the advantages of the technique as well as the caveats of employing it. Here, we revisit tuned decouplers and look at the use of discrete inductors to improve and extend the decouplers' effectiveness.

Effective decoupling requires that the decoupler provide the ac content of the switched current while the main power source supplies the average current, and therefore, all of the energy. In this way, high-frequency currents stay off of the power and ground buses and remain confined to small (decoupling) loops near the target, thereby minimizing emissions. Therefore, the primary analytical tool is a model based on an ac-current-source representation of the switching component, or network *(Fig. 1)*. In this model, i_{O} is the ac current through the switched unit, i_{S} is the part of i_{O} that flows through the inductance L_{S} of the power-bus trace, and Z is the impedance of the decoupling network. The response i_{S}(f)/i_{O}(f) is obtained from this model.

An empirical comparison for the analytical inferences was possible with a hardware implementation *(Fig. 2)*. Top-side (component) traces were sized for desired inductance value, according to the formulas of Rostek *("Avoid Wiring-Inductance Problems," *electronic design,* Dec. 6, 1974, p. 62)*. The entire bottom side of the board was used for the ground plane.

The power bus was connected to a 10-V dc power source and was bulk-decoupled by two 47-µF tantalum capacitors at the connection point. The 2N7000 FET switch was driven by a 5-V square wave at a 1-MHz rate with a 50% duty cycle. A magnetic field probe monitored radiation off of the power-bus trace, and the obtained information was displayed on a Hewlett-Packard spectrum analyzer.

For additional insight, a Spice model of the hardware was used to study the various decoupling configurations. In all Spice runs, 100 cycles of the driving signal were sufficient to attain steady-state values. Three circuit representations were studied, and in general, the three stood in agreement.

The three basic decoupling networks included a trace-tuned, capacitor-based parallel branch or branches *(Fig. 3)*; a pi filter with series discrete inductor and trace-tuned, capacitor-based parallel branches *(Fig. 4)*; and an LC filter with discrete series inductor and trace-tuned, capacitor-based parallel branches *(Fig. 5)*.

The capacitors C1 and C2, shown schematically in these figures, were MuRata Erie radial-leaded ceramic devices, made with X7R dielectric. The reactances represented by L1 and L2 are trace and/or lead inductances. The reactance represented by L_{X} was implemented with Coilcraft 90-13 axial-leaded inductors.

Any 1-µF ceramic capacitor has an inherent antiresonance (i.e., minimum impedance) at 5 MHz, indicating an intrinsic inductance of about 1 nH. Adding 24 nH of series trace/lead inductance yields the antiresonance at about 1 MHz. Similar considerations hold for a 0.033-µF capacitor, which has an inherent antiresonance at 27.5 MHz.

For convenience, the investigation assumed that the spectral region of concern extended from the fundamental (1 MHz) to the fifth harmonic (5 MHz). You can find the empirical results for important points along the decoupler development path in Figures 6, 7, 8, 9 and 10. The pole frequencies were computed from the ac model. Additional plots of hardware results can be accessed on the Web at *www.elecdesign.com*.

Tuned decoupling networks can very effectively suppress emissions produced by switching circuits. A single-branch tuned decoupler acts to quench emissions by producing a zero at the frequency to which it's tuned. Additionally, it can reduce emissions at higher frequencies. Compare Figure 7 and Figure 11 with Figure 6 (the no-decoupling case). When a single frequency dominates the emission pattern, a single-branch decoupler tuned to the offending frequency obtains a good quench.

Reacting With Trace Inductances

The designer must realize that the single-branch decoupler will interact with power-bus trace inductance and ground-return trace inductance (if the latter isn't negligible) to produce a resonance (pole) at a frequency less than that of the decoupler's zero. So, sufficient energy at frequencies in the neighborhood of this newly created pole could result in a new emissions issue.

A dual-branch decoupler, wherein each branch is tuned to a separate frequency, quenches emissions at both tuned frequencies by producing zeroes at these frequencies. It can also reduce emissions at frequencies that are greater than the higher of the two tuned frequencies. For examples, see Figure 8 and Figure 12.

Like the single-branch de-coupler, the dual-branch version produces a pole at a frequency below that of the lower of the two zeroes. Plus, it produces a pole at a frequency that's intermediate between the frequencies of the two zeroes. Energy at frequencies in the neighborhood of either new pole could then become a complicating factor in the EMC picture. These same statements can be generalized for n-branch decouplers (n > 2) for situations in which it's desired to quench emissions at n number of different frequencies.

A series inductor placed between the two branches of a dual-branch decoupler creates a pi filter. Depending on the value of the series inductance and/or juxtaposition of the parallel branches, this filter can either move the pole out of the frequency range of concern (to a frequency that's less than the decoupler's low-frequency zero), or move the pole to a different frequency within the range of concern. If the L1-C1 branch in Figure 4 is the decoupler's low-frequency-zero branch, the pole will be moved out of the range of concern (assuming a sufficiently large series inductance). For an illustration of this, check out Figures 9 and 13. Also, compare Figure 8 and Figure 12.

On the other hand, if the L1-C1 branch is that of the high-frequency-zero, the pole will move to a different frequency within the range of concern *(see item #8 in the table)*. In either case, the pi filter also suppresses emissions at frequencies greater than that of the decoupler's high-frequency-zero.

The LC filter in Figure 5 can be a very good multifrequency decoupler *(Fig. 10, again)*. For complete effectiveness, the tuned parallel branch (the "C" part of the filter) should be tuned to a frequency that's less than or equal to the lowest frequency in the range of concern. An example of the consequences of failing to meet this criterion is shown in Figure 15.

Both the pi filter and the LC filter efficiently suppress multiple frequencies. In each case, the price is the insertion of a discrete inductor in series with the power trace. This inductor must be physically large enough to carry the average current and some of the ac current without going into magnetic saturation.

The decouplers might interact with parasitic reactances or among themselves (as we have seen in the dual-branch case) to produce new resonances and potentially complicate the suppression effort. For instance, a single-branch decoupler may interact with parasitic reactances, or with the decouplers of nearby switching circuits to produce poles at intermediate frequencies.

The response equations, derived from the ac model, have proven themselves as good predictors of decoupler performance *(see the table)*. They can be used to analyze the effectiveness of a given decoupler configuration. Furthermore, they have the capability to serve as a decoupler design tool.

Let's pause for a brief review of the analytically derived response formulas of the various decoupler configurations. For a single-branch decoupler *(Fig. 3a, again)*, we have:

where:

is a zero of the response and:

is a pole. In our specific example, the decoupler's f_{1} is equal to 1.0066 MHz, which is slightly off antiresonance at f = 1 MHz.

For the dual-branch decoupler (*Fig. 3b, again)* in which the branches are tuned to different frequencies, f_{1} and f_{2}, we have Equation 4. The f_{1} is given by Equation 2, where:

and:

is the frequency at which i_{S}/i_{O} = 1, and the pole frequencies f_{P1} and f_{P2} are obtained from the roots of the denominator of Equation 4.

For the pi filter shown in Figure 4, we use Equation 7. The f_{1} and f_{2} are given by Equations 2 and 5, respectively, and f_{S} is given by Equation 6. The pole frequencies are the roots of the denominator of Equation 7.

For the LC filter in Figure 5, the transfer function is given by:

where f_{1} is obtained from Equation 2, and the pole frequency is:

Here's an instructive example of the progressive decoupling of a circuit switching 200 mA at 1 MHz. For convenience, we assumed the frequency range of concern extended from the fundamental (1 MHz) to the fifth harmonic (5 MHz). Because the switching signal has a 50% duty cycle, the spectrum of interest is concentrated at 1, 3, and 5 MHz. Any content at 2 or 4 MHz would be attributable to the nonzero rise and/or fall times of the switching signal.

The table summarizes the response results and pole frequency results for 10 of the 13 decoupler configurations. Response ratios and pole frequencies for the model prediction were obtained directly from the appropriate equation. The response ratio for the test results were computed from the test data, as shown in the following example for item #4. The computation is based on the following relationship:

At f = 1 MHz, we have Ä(dB) = 0 − 16 = −16. Therefore, at 1 MHz, Equation 10 yields i_{S}/i_{O }= 0.025.

Note the generally good agreement in results between the circuit representations for a given decoupler *(see the table, again)*. Only for items #2 (1 MHz) and #11 (3 MHz) do the responses differ by a factor of 10 or more. For configurations that have a more isolated power source from the current switch than is the case for item #2 (heavier filtering at 1 MHz), the hardware circuit more nearly approaches the circuit of the ideal current source used in the ac model. Agreement between analytical and empirical results is close in these cases of greater isolation.

It isn't clear why the result is very different for the third harmonic in item #11. Interestingly, the Spice result is close to the test result for Item #2 (as expected), and the Spice result is close to the model prediction for item #11. Evidently, in the case of item #11, the resonance in the hardware located near 1 MHz is considerably narrower than the models predict.

*No Decoupling:* The results of operating the circuit without decoupling are shown in Figure 6. For convenience, we designated the response at each frequency of interest equal to zero dB.

*Single-Branch Decoupling:* The results of decoupling the circuit with a single branch, tuned to 1.0066 MHz, are shown in items #2 and #3 of the table. The degree of filtering increases as one moves from item #2 to item #3, as does the portion of i_{O} provided by the decoupler.

Ideally, a single branch decoupler, tuned to a specific frequency, will supply all of the current for that particular frequency with minimal filtering. In reality, the decoupler in this example is tuned somewhat off 1 MHz. Hence, the theoretical impedance of the decoupler is nonzero at 1 MHz. In addition, even at antiresonance, there's a residual impedance in the decoupler, making for a nonzero component of i_{S} at the target frequency. For these reasons, the heavier the filtering, the greater the amount of i_{O} provided by the decoupler.

With the exception of its overly optimistic result for the fundamental of item #2, the ac model offers an accurate prediction of the actual hardware outcome for single-branch decoupling.

*Dual-Branch Decoupling:* The results of dual-branch decoupling the switching circuit appear in items #4 and #5 of the table. One of the duals is tuned to 1.0066 MHz and the other to 5.06 MHz. Comparing the test results of #4 to those of #2, we find two things.

First, the decoupler of item #4 provides much greater suppression of the fundamental than does #2. It also provides greater suppression of the fifth harmonic than do either of the item #2 or #3 circuits. This is as expected, because its second parallel branch is tuned to 5.06 MHz. But a resonance is now created at about 4 MHz, and this resonance is broad enough to cause the third harmonic to spike. The ac model helps us to see that this should be expected.

Second, the heavier filtering of the decoupler of item #5 suppresses the response at all three of the odd frequencies, compared to the item #4 decoupler. It matches the best results of items #2 and #3 decouplers at 1 MHz and 3 MHz, and it significantly improves on those results at 5 MHz. The resonance still exists at about 4 MHz, as predicted by the ac model and corroborated by Spice. This resonance could present a problem if the even-harmonic content of the current increases (via a change in rise/fall time or in duty cycle).

*Pi Filter:* One means of moving the resonance caused by the dual-branch decoupler out of the frequency range of concern is by constructing a pi filter, obtained by separating the dual branches of Figure 3b with an appropriate-size inductor. This construction can be seen in the circuit of Figure 4. In addition to being of at least a certain minimum inductance value, the inductor must be chosen so that it won't saturate at the expected current levels.

With L1 = 25 nH, C1 = 1 µF, L2 = 30 nH, C2 = 0.033 µF, and L_{X} = 1 µH, the item #6 decoupler shifts the 4-MHz resonance of the item #4 circuit into the 800- to 900-kHz range. The responses at 3 MHz and at 5 MHz are almost quenched. But, the response at 1 MHz undergoes a substantial increase, compared to the item #4 results, because it then resides in the neighborhood of the resonance. All of these outcomes can be predicted, at least directionally, from the ac model.

The heavier filtering of the item #7 decoupler virtually quenches the responses at all three odd frequencies. It moves the resonances to still lower regions of the kilohertz range.

The decoupler of item #8 exchanges the parallel branches of the item #6 decoupler about L_{X}. This results in the 1-MHz and 5-MHz responses being quenched while the 3-MHz response increases slightly from its item #6 value. Consider the resonance that appears for the circuit of item #8. This frequency was found by figuring out the roots of the denominator in Equation 7. We obtain a resonance at around 3.7575 MHz.

This range-of-concern resonance is quite narrow. Its influence appears in the slight increase in the response at 3 MHz for item #8 compared to that of item #6 at 3 MHz. Given the suppression advantage that the lightly filtered item #8 configuration enjoys over the lightly filtered item #6 configuration, it may be worth the risk of accepting a weak resonance in the frequency range of concern by choosing the item #8 decoupler not only over the item #6 circuit, but also over the heavily filtered item #7 decoupler. This would achieve a lower component count/cost. Keep in mind, though, that when the higher-frequency branch is placed first to the voltage source, a resonance, however weak, remains in the frequency range of concern.

As a final consideration for the pi decoupler, we investigated the question of a minimum value for L_{X}, which would be the smallest inductance capable of moving the resonance out of the frequency range of concern. Using the decoupler from Figure 4 (with L1 = 25 nH, C1 = 1 µF, L2 = 30 nH, and C2 = 0.033 µF), the ac model found the minimum L_{X} value, 368 nH, which is the L_{X }value that zeroes the denominator of Equation 7 at f = 1 MHz. The Spice method found 340 nH for minimum L_{X}. Furthermore, inserting L_{X} = 333 nH into our hardware model yielded the result of Figure 14.

*LC Filter:* The generic LC filter is shown in Figure 5. Both the lightly filtered (item #9) and the heavily filtered (item #10) versions provide good suppression. Also, both produce resonances well below the frequency range of concern. In fact, the lightly filtered item #9 decoupler results in suppression matched only by the higher-parts-counts configurations of items #7, #8, and #10. On a cost-effective basis, the configuration of item #9 is the decoupler of choice for multifrequency suppression.

The results obtained with the decoupler of item #11 illustrate the importance of careful tuning (in this case, tuning the parallel branch to a frequency equal to or less than the lowest frequency value of concern). The resonance at 864 kHz is sufficiently strong and wide to cause a spike, rather than a suppression, in the response for the 1-MHz component of i_{S}.

One may legitimately ask why we spent so much time studying the pi configuration if the simple LC filter is so effective and cost-effective as a multifrequency decouple. Well, several reasons come immediately to mind. First, the pi filter is sometimes touted as the configuration of choice. So, it's essential to know its strengths and its weaknesses.

Consider the case of a single-branch tuned decoupler too, as in Figure 3a. If a parasitic reactive branch occurs in parallel with that decoupler, and that parasitic branch is antiresonant at a frequency other than where the decoupler produces a zero, a situation like the one in Figure 3b would result. A pole would develop at an intermediate frequency. Plus, increased emissions could occur if the new resonance multiplies one of the harmonics of i_{O}.

A similar situation would result if the second parallel branch, rather than being parasitic, was the tuned decoupler of a nearby circuit that was switching at a rate other than that of the first circuit. In such instances, one might want to consider introducing a series inductance (L_{X}), observing all of the points required to make the introduction effective.

Another potential cause of those "unintended consequences" so familiar to EMC workers is an LC decoupler on a nearby circuit. This could combine with the first decoupler to form a network like that shown in Figure 4, with the possibility of the various undesirable outcomes described in the *Pi Filter* section.

Whether it's with or without discrete inductors, tuned decoupling can be a useful way to suppress electromagnetic emissions from switched circuits. It's important, though, to carefully choose and design decouplers in order to obtain the optimum effect in a given situation.