# Equivalent series resistance vs. parallel leakage conductance

I am studying a book called RF Circuit Design by Reinhold Ludwig and Gene Bogdanov (2nd Edition).

In the first chapter on capacitors, they introduce the loss tangent and equivalent series resistance, with $$\ESR=\frac{tan\Delta}{\omega C}\$$.

This seems to make sense to me, with the loss angle $$\\Delta\$$ being the angle between the real and complex components of the impedance of a capacitor in series with a resistance.

However, just before that, they also define the loss tangent relative to the leakage conductance $$\G_e=\omega Ctan\Delta\$$. This leads to an interesting connection between series resistance and parallel conductance: $$\ESR=\frac{G_e}{\omega^2 C^2}\$$.

After thinking about this quite a bit, I suspect what they've done is calculate the impedance of the capacitor and leakage conductance in parallel, and make an approximation for very small conductance:

$$Z=\frac{1}{G_e+j\omega C} \approx \frac{G_e}{\omega^2 C^2}-\frac{j}{\omega C}$$

with the assumption $$\\omega C >> G_e\$$.

Firstly, am I understanding this correctly? If so, what this tells me is that parallel resistance can be modelled as an equivalent series resistance. Is this a standard way of calculating ESR? All other references I have seen on equivalent circuits for capacitors include two separate resistors, one in series and one in parallel, equating ESR with the resistor in series. Is it normal to lump the parallel resistance in with the ESR like this book seems to be doing?

• Let's set $R_e=\frac1{G_e}$. Then $R_e$, in parallel with $C$, is $Z=\frac{R_e}{s\,R_e\,C + 1}$. Your condition is the same as saying: $s\,R_e\,C\gg 1$. So then $\frac{R_e}{s\,R_e\,C + 1}\approx \frac{R_e}{s\,R_e\,C}=\frac1{s\,C}$. Which says, given your condition, that the leakage is ignorable. It does not say that it can be treated as an in-series resistance. All you've done is to define it away. But that is not the same thing as showing equivalence. I suspect the authors wouldn't introduce the idea of leakage conductance/loss angle if it were ignorable.
– jonk
Commented Apr 27, 2021 at 18:02
• @jonk $Z=\frac{1}{G_e + j\omega C} = \frac{G_e-j\omega C}{G_e^2 + \omega ^2 C^2} = (\frac{1}{1+\frac{G_e^2}{\omega^2 C^2}})(\frac{G_e}{\omega^2 C^2} - \frac{j}{\omega C})$ Commented Apr 27, 2021 at 19:04
• The approximation of $\omega C >> G_e$ means we approximate the first term in the parentheses as 1 Commented Apr 27, 2021 at 19:06

This relationship does apply not to all caps.

• The leakage is usually defined as a current max or Rp at Vmax at some fixed temperature. (Not $$\G_e\$$).
• We know that Plastic caps are closest to ideal with very high $$\\tau_{leakage}=R_p*C\$$ and very low $$\\tau_{conduction}=ESR*C\$$
• for all caps there is a SRF with Zmin(f) series inductance XL(f) that matches the impedance of C, Xc(f) for a series resonant frequency (SRF) after which Zc(f) rises with frequency.

Note : SRF is often called Self-Resonant Frequency for parallel tank ccts. But for RF some cap suppliers prefer to call this PRF for parallel. Don’t let this confuse you, it is a label than has been “grand-fathered”.

So the two time constants may have some ratio within only the same P/N family of dielectric parts. Plastic or Metal Film vs Ceramic vs NPO ceramic vs Aluminum Electrolytic (e-caps) vs Double Electric Layer Ultracaps and batteries all have different ratios and values of the two different time constants.

You may make a spreadsheet of different dielectrics to see how this varies.

I have done this for e-caps and found that in most cases, standard (cheap) e-caps have an ESR * C time constant of >>100 us, while “Low ESR” e-caps have this time constant 1 to 10 us. ESR does reduce with rising voltage ratings somewhat but not as significant as the dielectric/terminal interface related ESR which affects cost. Also, I have noticed, they specify Dissipation Factor, DF and no ESR, then it has always been a “standard cap” when used with storing rectified line frequency pulses thus specified at 120 Hz (industry std.) so Beware for SMPS caps.

• for same value of say 1uF ceramic caps tend to be slightly lower ESR than low ESR caps but in the <100 pF range NP0/C0G caps are much better with lower density and smaller max value.
• Plastic caps with a much lower dielectric constant have a very low leakage and lower ESR so the two time constants are far apart making them more ideal, yet far more expensive in large values as well as size.
• loss tangent is certainly used for RF especially above 300 MHz and becomes critical for FR4 at >=1GHz to choose a low loss tangent material and NP0 caps.