The canonical decoupling network for an operational amplifier consists of a bulk capacitor, commonly $10\,\mu\mathrm{F}$, placed in parallel with a small ceramic, commonly $0.1\,\mu\mathrm{F}$, at each supply terminal. Under the ideal capacitor model, the aggregate capacitance is $10.1\,\mu\mathrm{F}$ and the small device contributes negligibly. Yet omission of the small capacitor is observed to degrade the power-supply rejection of precision analog front-ends and, in extreme cases, to precipitate instability. The apparent paradox is resolved by recognising that physical capacitors exhibit parasitic inductance and resistance, and that the two components in the standard pair occupy disjoint regions of the frequency axis.
The output voltage of an operational amplifier is synthesised from charge drawn from its positive and negative supply rails. Because the transistors of the output stage exhibit finite transconductance and finite output conductance, any perturbation of the rail potential is transferred to the output with a non-zero, frequency-dependent coupling coefficient.
This coupling is quantified by the power-supply rejection ratio (PSRR), defined as the ratio of a perturbation at the supply to the resulting perturbation at the output, referred to the input. Precision devices typically exhibit a low-frequency PSRR on the order of $100\,\mathrm{dB}$. Above a break frequency of a few hundred hertz, however, the rejection falls at approximately $20\,\mathrm{dB}$ per decade, reaching $20\,\mathrm{dB}$ in the vicinity of $1\,\mathrm{MHz}$ and vanishing, for practical purposes, above $10\,\mathrm{MHz}$. In the high-frequency band the amplifier therefore provides little attenuation between supply perturbations and the output signal.
The sources of such perturbations are intrinsic to the system. Switch-mode regulators inject a fundamental at the switching frequency, commonly $500\,\mathrm{kHz}$ to several megahertz, together with its harmonics. Digital logic elsewhere on the printed circuit board draws pulsed currents through the distributed inductance of the supply network, producing wideband transient voltages. The output stage of the amplifier itself demands transient current which the upstream regulator, whose control loop typically has a bandwidth of a few kilohertz, cannot supply within the relevant time scale.
The impedance of the ideal capacitor is
On logarithmic axes the magnitude response is a straight line of slope $-20\,\mathrm{dB}$ per decade, so that each decade of frequency yields a corresponding decade of impedance reduction. Under this model a $10\,\mu\mathrm{F}$ capacitor presents $1.6\,\Omega$ at $10\,\mathrm{kHz}$, $160\,\mathrm{m}\Omega$ at $100\,\mathrm{kHz}$, and $16\,\mathrm{m}\Omega$ at $1\,\mathrm{MHz}$. It would follow that a single large capacitor provides the lowest attainable bypass impedance at every frequency, and that augmenting it with a $0.1\,\mu\mathrm{F}$ capacitor in parallel, whose ideal impedance exceeds that of the bulk device by two orders of magnitude at all frequencies, can provide only a marginal improvement.
Were the ideal model exact, the aggregate capacitance would indeed be $10.1\,\mu\mathrm{F}$, the combination would present the impedance of a slightly enlarged single capacitor, and the smaller device would serve no identifiable purpose. Measured impedance spectra of physical capacitors demonstrate otherwise.
A physical capacitor departs from the ideal model in two systematic respects. Its terminals, internal electrodes, and interconnect contribute a non-zero loop area through which the charging and discharging currents must flow, and this loop possesses a finite self-inductance. Dissipative mechanisms in the dielectric, the electrodes, and the terminations contribute a non-zero series resistance. The component is accordingly represented by the lumped series connection of the nominal capacitance $C$, an equivalent series inductance $\mathrm{ESL}$, and an equivalent series resistance $\mathrm{ESR}$:
$R_p$ is the dielectric leakage resistance, typically $\mathrm{M\Omega}$ to $\mathrm{T\Omega}$. At bypass frequencies its admittance is negligible compared to $j\omega C$, so it is omitted from the $Z(j\omega)$ expression below.
The corresponding magnitude response exhibits a V-shaped profile on logarithmic axes. At sufficiently low frequencies the term $1/(\omega C)$ is dominant and the impedance decreases at $-20\,\mathrm{dB}$ per decade, in accordance with the ideal model. At sufficiently high frequencies the term $\omega\,\mathrm{ESL}$ is dominant, the impedance increases at $+20\,\mathrm{dB}$ per decade, and the component behaves as a series inductor. The transition between these two regimes occurs at the self-resonant frequency,
at which the inductive and capacitive reactances cancel exactly, and the impedance attains a minimum equal to the $\mathrm{ESR}$.
The quantity of central interest is the dependence of $f_{\text{SRF}}$ on $C$ when the device geometry is varied accordingly. An electrolytic or tantalum capacitor of $10\,\mu\mathrm{F}$ typically occupies a comparatively large case, possesses an extended internal current path, and exhibits an $\mathrm{ESL}$ of order $5$ to $10\,\mathrm{nH}$. The corresponding self-resonant frequency is
Above this frequency the device no longer behaves capacitively, and its impedance increases with frequency. A ceramic capacitor of $0.1\,\mu\mathrm{F}$ in an 0603 or 0402 surface-mount package presents a short internal current path and an $\mathrm{ESL}$ of approximately $0.5$ to $1\,\mathrm{nH}$. Its self-resonant frequency is therefore
The ceramic, notwithstanding its two-order-of-magnitude smaller capacitance, is self-resonant at a frequency more than twenty times higher than the electrolytic. This follows from the dependence of $f_{\text{SRF}}$ on the geometric mean of $\mathrm{ESL}$ and $C$: the reduction in capacitance by a factor of one hundred is partially offset by a concurrent reduction in parasitic inductance by a factor of approximately five.
The parallel connection of the two devices yields an admittance equal to the sum of the individual admittances, from which the composite impedance follows:
At any given frequency, the current distributes between the two branches in inverse proportion to their impedances, so that the composite response is governed by the branch of lower impedance. Below the self-resonance of the bulk device, the $10\,\mu\mathrm{F}$ dominates, since its capacitive reactance is substantially smaller than that of the $0.1\,\mu\mathrm{F}$. Above the bulk self-resonance, the impedance of the $10\,\mu\mathrm{F}$ device increases with frequency while that of the $0.1\,\mu\mathrm{F}$ continues to decrease toward its own self-resonance; at the frequency at which the two impedances are equal, the ceramic assumes the role of principal conductor for high-frequency noise currents. The composite impedance therefore follows the lower envelope of the two individual curves at every frequency, producing an extended interval of low impedance that neither component is capable of providing in isolation.
The composite does not behave as a $10.1\,\mu\mathrm{F}$ capacitor. The effective bypass capacitance at any frequency is determined by whichever component presents the lower impedance at that frequency, not by the arithmetic sum of the nominal values. The additive rule is valid exclusively in the low-frequency limit, in which both components are well described by the ideal model and the contribution of the smaller capacitor is demonstrably negligible. Above a few hundred kilohertz the bulk component contributes no useful bypass admittance, and the composite behaves as a $0.1\,\mu\mathrm{F}$ capacitor with $\mathrm{ESL} \approx 1\,\mathrm{nH}$ rather than as a $10.1\,\mu\mathrm{F}$ capacitor with any meaningful effective inductance.
A concluding remark is warranted. The nominal distinction between "$10\,\mu\mathrm{F}$" and "$0.1\,\mu\mathrm{F}$" corresponds, in practice, to a distinction between two classes of physical construction: a high-capacitance electrolytic or tantalum device of relatively large volume and long internal current path, and a low-capacitance ceramic device of small volume and short current path. Were it technologically feasible to realise a $10\,\mu\mathrm{F}$ capacitor in an 0402 ceramic package with an $\mathrm{ESL}$ of order $1\,\mathrm{nH}$, a single such component would satisfy the bypass requirements of the amplifier, and the classical bulk–ceramic combination would be rendered superfluous. Contemporary high-density ceramic technologies approach this regime, and correspondingly, a number of recent design practices dispense with the bulk component in favour of two or three ceramic capacitors of comparable value.
The qualitative argument of the preceding section admits a formal confirmation through the transfer function representation of the bypass network. Treated as a linear time-invariant two-terminal element, each capacitor branch is characterised by an impedance expressible as a rational function of the Laplace variable $s$.
The series connection of $\mathrm{ESR}_k$, $\mathrm{ESL}_k$, and $C_k$ yields, for the $k$-th branch,
The branch admittance $Y_k(s) = sC_k / D_k(s)$ therefore possesses a simple zero at the origin and a pair of poles at the roots of $D_k(s)$. Under the low-damping condition $\mathrm{ESR}_k^2 < 4\,\mathrm{ESL}_k/C_k$, these roots form a complex-conjugate pair situated in the left half-plane with natural frequency $\omega_{0,k} = 1/\sqrt{\mathrm{ESL}_k C_k}$, precisely the self-resonant frequency of the branch.
Since admittances are additive in parallel, the composite admittance is
and the composite impedance is therefore the rational function
The numerator is of degree four and the denominator of degree three; $Z_{\text{tot}}(s)$ is accordingly a proper rational function possessing four zeros and three poles, whose configuration in the $s$-plane admits the following interpretation.
The zeros of $Z_{\text{tot}}(s)$ are the roots of $D_1(s)$ and $D_2(s)$, coinciding with the self-resonant frequencies of the two individual branches. Each conjugate pair lies close to the imaginary axis at a distance $\alpha_k = \mathrm{ESR}_k / (2\,\mathrm{ESL}_k)$ determined by the dissipation of the corresponding branch. On the imaginary axis, these zeros manifest as the two local minima of $|Z_{\text{tot}}(j\omega)|$ observed in the plot of the preceding section.
The denominator factorises into a trivial pole at $s = 0$, corresponding to the open-circuit behaviour of capacitors at DC, and the roots of the quadratic
In the underdamped regime these roots constitute a complex-conjugate pair with natural frequency
The approximation is instructive: the resonance of the composite network is governed by the equivalent series inductance of the bulk device and the capacitance of the ceramic device. Physically, this is the frequency at which the post-resonance inductive reactance of the bulk capacitor and the pre-resonance capacitive reactance of the ceramic capacitor combine to form a parallel LC tank. For the nominal parameters of the present analysis this yields $f_{\text{AR}} \approx 6.5\,\mathrm{MHz}$, situated between the two self-resonances.
A conjugate pole pair of small real part produces a local maximum of $|Z_{\text{tot}}(j\omega)|$ at $\omega \approx \omega_{\text{AR}}$. The composite impedance is therefore not monotonic between the two self-resonant minima: a residual peak — the anti-resonance — is an unavoidable feature of any parallel arrangement of capacitors of dissimilar self-resonant frequency. The amplitude of the peak is bounded by $\mathrm{ESR}_1 + \mathrm{ESR}_2$, which supplies the dissipation that damps the parallel tank.