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# Quiet Power: How to Read the ESR Curve

*This column originally appeared in the November 2012 issue of The PCB Design Magazine.*

The most widely used power distribution component is undoubtedly the bypass capacitor. After the nominal capacitance, its next most important parameter is its effective series resistance, or ESR. To use bypass capacitors properly, designers must need to understand exactly what ESR means and how to read the ESR curve in measured or simulated plots.

A real-world capacitor, whether it is a discrete capacitor or the capacitance of a power-ground plane pair, is never ideal. It always has parasitic resistance and inductance. The inductance is associated with the shape and size of the capacitor electrodes as well as the shape and size of external connections—the pads, vias and planes closing the current loop around the capacitor. The size of the current loop, which determines the inductance, depends only partly on the capacitor itself. It is a widely recognized and accepted fact that the loop inductance of a capacitor also depends on how we connect it to our board.

In contrast, we tend to assume that the ESR of the capacitor depends only on the capacitor, and not on its external connections. While this may be true for many applications, there are cases (to be shown in a future column) where even the ESR of a capacitor depends on the external connection geometry. To get to that interesting case, we need first to understand the contributors to ESR.

The resistance of a capacitor comes from two sources: conductor losses and dielectric losses. For instance, in a multilayer ceramic capacitor, the capacitor plates and the connecting terminals have finite resistance and they make up the series losses. The dielectric layers, especially if we have high dielectric constant materials, have a finite loss tangent and will create a parallel loss conductance. When we use a series C-R-L equivalent circuit for the capacitor, the series and parallel losses combine into a single series resistance, called effective series resistance, or ESR. This is shown in the equivalent circuits of Figure 1.

Figure 1: Equivalent circuit of a capacitor with separate conductive and dielectric losses on the left and combined effective series losses on the right. ESR is the sum of the Rs(f) conductive and Rps(f) dielectric losses.

The conductive loss follows the frequency dependence of skin loss: Starting from a DC resistance value, the resistance rises with the square root of frequency. In some cases, it may be important that different contributors of conductive losses, such as terminal and capacitor-plate losses have different DC resistances and knee frequencies. Figure 2 shows a simple example: the Rs(f) conductive resistance with 10 mOhm DC resistance and 1 MHz knee frequency.

Figure 2: Typical frequency dependence of conductive losses in a ceramic capacitor.

The parallel losses are described by a DC leakage current and a dielectric loss tangent, D(f). For AC modeling, we can usually ignore the DC leakage. In a simplified model we can also assume that the dielectric loss tangent is frequency-independent; therefore, the parallel conductance is:

If we want a causal model, the slight frequency dependence of C and Df has to be taken into account. Figure 3 shows the conductance (G) and the capacitive susceptance (B) for a 1 uF capacitor with 1% Df using the causal wideband Debye model. Figure 4 shows the series equivalent of the same lossy capacitance.

Figure 3: Parallel conductance and capacitive susceptance as a function of frequency.

Figure 4: Series equivalent of the parallel conductance and series capacitive reactance for the capacitor from Figure 3.

We notice that the two contributors of the equivalent series resistance vary with frequency the opposite way: the resistance of conductor losses increases, and the resistance of the dielectric losses decreases with increasing frequency. The sum of the two creates a typical basin-like plot (Figure 5). With reasonably low Df values and at low frequencies, the ratio of the magnitude of capacitive reactance |X(f)| and ESR(f) is constant and equals the (approximately frequency independent) loss tangent. We can see that in Figure 5, in the 100 Hz to 10 kHz frequency range, the red ESR curve is approximately 100 xsmaller than the green |X(f)| curve, which corresponds to a 1% Df loss.

Figure 5: Capacitive reactance and effective series resistance (ESR) of a 1 uF capacitor with 1% Df and 10 mOhm DC series resistance.

You can try different input numbers, plot the result and see all the expressions behind these calculations in the bypass capacitor model, available at www.electrical-integrity.com.

*Dr. Istvan Novak is a distinguished engineer at Oracle, working on signal and power integrity designs of mid-range servers and new technology developments. Novak received* *his M.S. degree from the Technical University of Budapest, Hungary and his Ph.D. degree from the Hungarian Academy of Sciences in 1976 and 1989, respectively. With 25 patents to his name, Novak is co-author of "Frequency-Domain Characterization of Power Distribution Networks." To contact Istvan, click **here**.*

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