Component insights: the capacitor

27 min readFeb 23, 2025

For me — and maybe for other engineers as well — the relationship between electrical components’ physics of operation, functionality in circuits, and mathematical representations (e.g., as modifiers of time series and transfer functions) didn’t click right away. The connection was never made explicitly in class. Hopefully, this article will make that connection explicit for one component: the capacitor.

Your understanding of capacitors before (left) and after (right) reading this article. Left figure by source. Right figure by author, reproduced in larger size at the end of this article.

· Physics
∘ General design
∘ A recap of the electric field inside a dielectric
∘ Derivation of fundamental relationships for a parallel-plate capacitor
· Time domain characteristics
∘ How does current flow through a capacitor?
∘ Why can’t the voltage across a capacitor change instantaneously?
· Frequency domain characteristics
∘ A recap of phasor representation
∘ The impedance of a capacitor
· Non-ideal behavior
∘ Complex permittivity
∘ Loss tangent
· Summary
· About the author
· Sources and further reading

Physics

General design

Capacitors are devices that consist of two conductors separated by an insulating material called a dielectric. When a voltage is applied to the two conductors, positive charge (+Q) builds up on one, and negative charge (-Q) builds up on the other. While capacitors can take many shapes (e.g., concentric metal spheres, coaxial metal cylinders, etc.), practical applications often use parallel-plate capacitors in which the conductors are plates of area A, separated by a thickness d.

The electric potential is constant over each conducting plate, as electric charges can flow freely through a conductor and differences in potential are removed almost immediately by charge redistribution. Thus, the potential difference V between the two plates can be expressed as follows, where E is the electric field between the plates and dl is the incremental distance traversed when going from the negative to the positive plate:

E can be found using Gauss’s law. If a surface is chosen with appropriate symmetry — i.e., such that the magnitude of E is uniform over the surface and the direction of E is the same as that of the surface normal (E points perpendicular to the surface) — then the magnitude of the electric fields due to the positively and negatively charged plates (E+ and E-, respectively) can each be found as:

Gaussian surfaces can be chosen such that Q_enc is +Q for E+ and -Q for E-. Since both electric fields are proportional to Q, the net E (an arithmetic combination of the two) is proportional to Q. It follows that V is proportional to Q as well. The constant of proportionality is called the capacitance C, and is an indicator of a capacitor’s ability to store electrical charge (at a given voltage):

A recap of the electric field inside a dielectric

Note that most of the information in this section is taken from Griffiths (2023, pp. 181–189). I highly recommend reading this section and/or Feynman et al. (1964) for a more detailed recap.

First, recall that a dielectric usually consists of polar molecules, and when the material is placed in an electric field, each permanent dipole experiences a torque that pushes it into alignment with the field. This is how the dielectric material becomes polarized: while the dipoles are oriented randomly in the absence of an external electric field, the presence of an external field aligns the dipoles in a “cooperative” manner so that they yield a net dipole moment. This amounts to a net polarization P, which is defined as the dipole moment per unit volume.

Recall that the effect of polarization is to produce accumulations of bound charge ρ_b within the dielectric and σ_b on the surface:

In calculating the net electric field between the conductors, we must consider both the original electric field responsible for P and the new field that results from P, i.e., the field of the bound charge. Considering any charge not due to polarization as “free charge” ρ_f (e.g., electrons in the conductor or ions in the dielectric material), the total charge density within the dielectric can be written as ρ = ρ_b + ρ_f. Gauss’s law can then be expressed as:

Now define the electric displacement D as

so that

We can summarize the three fields (E, P, and D) as follows:

Polarization field P: arises due to (polarization-induced) bound charge ρ_b
Electric displacement field D: arises due to free charge ρ_f
Total electric field E: arises due to the net charge ρ

By rewriting the definition of D, we get ε_0 E = D-P, which makes sense, as the polarization-induced electric field acts oppositely to the original, free charge-induced electric field and attenuates it, yielding a smaller net electric field.

For simplicity (and because the assumption generally holds), we will consider linear dielectrics, in which the polarization is proportional to the field via a constant of proportionality χ_e, the electric susceptibility:

Thus, for linear dielectrics, we can write

The last two expressions for D are equivalent, but one uses ε, the permittivity of the material (which includes a factor of ε_0), and the other uses ε_r, the relative permittivity or dielectric constant of the material (which excludes the factor of ε_0).

For a space entirely filled with a homogenous linear dielectric (where the curl of P is zero, an hence the curl of D is zero), D can be found from just the free charge, i.e., from E_vac, the field the same free charge distribution would produce in the absence of a dielectric:

Therefore, the presence of a homogenous linear dielectric reduces the electric field by a factor of one over the dielectric constant. Since the dielectric decreases the electric field, the voltage decreases as well, and from C=Q/V, the capacitance increases. A capacitor with a dielectric thus stores the same charge as one without a dielectric but at a lower voltage.

Physically, this is because for a given voltage, additional free charge is needed on each conductor to overcome the partial neutralization of the polarization-induced bound charge.

Illustration of why the addition of a dielectric between the capacitor plates increases the capacitance. Left: for a given V, the addition of a dielectric increases the amount of charge stored on the plates Q. Right: for a given Q, the addition of a dielectric lowers the voltage V across the capacitor. Figure by author.

Derivation of fundamental relationships for a parallel-plate capacitor

Consider a parallel-plate capacitor filled by a slab of linear dielectric material. The plates have area A, and the dielectric slab has thickness d (the distance between the two plates is d) and dielectric constant ε_r. The free charge densities on the top and bottom plates are +σ and -σ, respectively.

We will

Find the electric field E between the plates via the vacuum electric field E_vac
Find the electric field E between the plates via Gauss’s law, i.e., via finding the polarization bound charge (to cross-check our answer from 1)
Find the potential difference V between the plates
Find the capacitance C

Part 1: Find the electric field E between the plates via the vacuum electric field E_vac

Part 2: Find the electric field E between the plates via Gauss’s law (to cross-check our answer from 1)

Part 3: Find the potential difference V between the plates

Part 4: Find the capacitance C

Time domain characteristics

How does current flow through a capacitor?

Note that most of the information in this section is taken from Griffiths (2023, pp.340–342).

By differentiating both sides of Q = CV with respect to time, we arrive at the relation that the current through a capacitor is proportional to the rate of change of voltage:

Given that we treat perfect (lossless) dielectrics as incapable of conducting free current (permitting the movement of free charge), you might be confused about how current can flow across a capacitor, i.e., how we can satisfy Kirchhoff’s Current Law (KCL) — current into a volume equals current out of that volume at any instant —when analyzing circuits with capacitors. In this section, we’ll see that need to consider both the free current and the displacement current.

Consider the local conservation law for electrical charge, which says that the charge density at a point can only change if current flows into or out of the point:

The first term is the rate of change of the charge density ρ at a point. The second term is the divergence of the current density J (amount of charge crossing a unit area in unit time) at the same point.

Charged particles must also satisfy Gauss’s law:

By substituting Gauss’s law into the charge conservation law, we get:

Identical results can be obtained from the divergence of Ampere’s law with Maxwell’s modification:

The last line above establishes a local conservation of current. The ∂E/∂t term is a type of current density related to the time-variation of the electric field. From the relation between the curl of B (circulating magnetic field) and the “source current” in the first line above, we see that this term allows for the generation of a magnetic field even in regions where no actual free current (J) exists. It therefore provides us with a coupling between E and B necessary to support wave propagation (the changing electric field associated with the displacement current can induce a changing magnetic field, and vice versa, creating a self-propagating wave phenomenon).

Note that the current density J can be split into three parts:

J_f is the free current (also called conduction current) caused by moving charges
J_m = ∇×M is the magnetization current, i.e., the current needed to generate the magnetization M
J_p = ∂P/∂t is the polarization or bound current, where P is the electric polarization resulting from bound charges in/at the surface of the dielectric

Given this deconstruction of the current density, the current conservation equation can be rewritten as follows:

The ∂D/∂t term is referred to as the displacement current density, J_d. The total current can therefore be written as the sum of the free current I_f and the displacement current I_d:

Displacement current might be weird to think about. We usually think of electric current as the flow of charges (i.e., the free current), while this sort of current can flow through empty space without any charges present. We can detect it because, like conduction current, it generates magnetic fields.

In the case of a capacitor, if the dielectric has negligible polarization (e.g., the space between the conductors is filled by air), D simply equals ε_0 E, and the displacement current is just the rate of change of E between the plates of the capacitor. But if the dielectric experiences significant polarization, D = ε_0 E + P, and we must consider how the changing orientation of dipoles (and thus the movement of the bound charges comprising the dipoles) mediates the displacement current.

Consider the example in the figure below. When the top terminal of the source voltage is positively biased, positive charges are driven away from it (electrons are driven to it) and flow as free current (I_1) toward the top plate of the capacitor. As positive charges accumulate at the top plate, they give rise to an electric field E that penetrates the dielectric and is seen by positive charges on the bottom plate. The positive charges on the bottom plate are repelled by E and flow as free current until they make it to the negative terminal or ground. The displacement current that “flows” across the capacitor (I_2) doesn’t involve movement of charge across it; instead, it’s a sort of “capacitive coupling” between charges on the top and bottom plates, mediated by dipoles that permit polarization current to flow through the dielectric.

Therefore, the current in the circuit is continuous without the physical movement of charges between the capacitor plates. The changing electric field across the capacitor determines the relationship between charge accumulation at the plates, and the dipoles in the dielectric help carry the current by oscillating between their two extreme positions of alignment with the field.

Why can’t the voltage across a capacitor change instantaneously?

In your studies, you might’ve heard of several ways to think about the time-domain behavior of a capacitor in a circuit:

The voltage across a capacitor can’t change instantaneously
A capacitor acts like a charge buffer that resists changes in voltage
In a capacitive circuit, the current waveform precedes the voltage waveform(the “ICE” part of ELI the ICE man)

These are all related ideas.

Idea 1: The voltage across a capacitor can’t change instantaneously

From a mathematical point of view, if the voltage changes instantly from one value to another (i.e. discontinuously), the derivative is not finite. This implies that an infinite current would be required to instantly change the voltage. Since an infinite current is not physically realizable, the voltage cannot change instantaneously.

But how can we explain this physically? If we change the external electric field (via changing the applied voltage that the capacitor experiences), the charges on the capacitor plates will have to redistribute/accumulate to support the new voltage, and this is a process that takes time. Thus, current must flow before the capacitor can achieve a new voltage.

Note that the finite time it takes for dipoles to respond (reorient themselves) to the electric field may pose an additional constraint on how quickly the voltage across the capacitor can change. This is considered to be non-ideal behavior and is quantified via the relaxation time of the dielectric (discussed in the “Complex permittivity” section).

Idea 2: A capacitor acts like a charge buffer that resists changes in voltage

According to the “buffer analogy,” a capacitor acts like a charge buffer that needs some time to fill and some time to deliver its stored charge. How slow it charges and discharges depends on the capacity.

The reasoning from Idea 1 can explain the buffer analogy too: the capacitor exhibits a “voltage memory” because it takes time for charges on the capacitor plates to redistribute/accumulate. Before the voltage across the capacitor can achieve a new value, the flow of current needs to deliver the new distribution of charges on the capacitor plates.

Most of the time, this concept is introduced via RC circuits. When connected to a DC voltage source through a resistor, a capacitor exhibits exponential charging or discharging. The time constant τ=RC dictates the rate at which the capacitor charges or discharges.

In this way, you can consider the capacitor to “resist” changes in voltage. When the voltage of the source goes up quickly, the capacitor starts charging with time constant τ, forcing the voltage across the capacitor to go up slowly. When the voltage of the source goes down quickly, the capacitor starts discharging with time constant τ, forcing the voltage across the capacitor to go down slowly.

Idea 3: In a capacitive circuit, the current waveform precedes the voltage waveform(the “ICE” part of ELI the ICE man)

Given the equations for the free and displacement current (copied below for convenience), we know that the displacement current is responsible for the lag between applied voltage and current.

When you differentiate a sine wave, the result is another sine wave that is 90° out of phase with the original one, as

Thus, the displacement current will be 90° out of phase with the applied electric field, while the free current will be in phase with the applied electric field. This is even easier to see with the phasor representation of fields and currents, which is reviewed in the “A recap of phasor representation section” section.

As per the final note in Idea 1, the relaxation time of the dielectric may impose another phase shift δ relative to dE/dt, such that the current becomes (90° + δ) out of phase with the applied electric field.

Frequency domain characteristics

The time domain may be useful if we’re analyzing something like the system’s step response, i.e., the output (e.g., capactior voltage) for an input (e.g., source voltage) that only changes once or a few times. However, if we have a periodic change in the input, such as an AC source voltage, it is useful to analyze the capacitor behavior in the frequency domain.

Often, you’ll see the frequency-dependent behavior of the capacitor summarized as follows:

At low frequencies (ω→0), the capacitor acts like an open circuit, blocking DC signals.
At high frequencies (ω→∞), the capacitor acts like a short circuit, passing AC signals.

In this section, we’ll explore the underlying physics of the capacitor’s frequency-dependent behavior.

A recap of phasor representation

A real-valued sinusoid with constant amplitude A, angular frequency ω, and initial phase θ has the form:

In accordance with Euler’s formula, the inclusion of an imaginary component

allows us to factor the sum of the real and imaginary components as

Thus, for a sinusoid x(t) = Acos(ωt + θ) and a corresponding phasor x̃(t) = Ae^j(ωt + θ), the real part of x̃(t) corresponds to x(t):

We can represent a sinusoid with either the analytic representation or phasor representation. Going forward, we will use the former, but either can be referred to as a phasor.

The benefit of the phasor representation is that it greatly simplifies operations on sinusoids. Firstly — and the reason why this recap is in the “Frequency domain characteristics” section — phasors inherently assume a single frequency ω, making them ideal for analyzing steady-state AC circuits and signals where all components operate at the same frequency.

Secondly, when dealing with multiple sinusoids of the same frequency, addition and subtraction reduce to simple vector addition of their corresponding phasors. Further, differentiation and integration of sinusoids translate into simple multiplication or division by jω when represented as phasors. All of these operations can be performed with just the phasors Ae^jθ, as the common factor e^jωt can be reinserted before the real part of the result.

The operations of most importance to us are phase shifts and differentiation, so these are elaborated on below (see this page for a more comprehensive overview).

Phase shift (and amplitude modulation) corresponds to multiplication by another phasor

Say we have sinusoids x(t) = Acos(ωt + θ) and y(t) = Bcos(ωt + φ) corresponding to phasors x̃ and ỹ. By taking the real part of x̃*ỹ, we get

which is the sinusoid x(t) with amplitude modified by a factor of B and a phase shift of φ.

Evidently, a phasor can convey how much a signal has shifted relative to the reference signal, i.e., the signal chosen to be the zero phase “origin.” Usually, the reference signal is the source AC voltage or current so that all the other currents and voltages have phase shifts relative to the source.

Differentiation corresponds to multiplication by jω

Say we have a sinusoid x(t) = Acos(ωt + θ) corresponding to phasor x̃. If we differentiate x̃ with respect to time, the result is as follows:

Thus, differentiation of a phasor corresponds to multiplication by jω. If we’re interested in the modification of the underlying sinusoid, we should take the real part. Using the relationship

we get:

As expected, the underlying sinusoid is differentiated, which corresponds to a modulation of the amplitude by a factor of ω and a phase shift of 90°. The same result is obtained by noting that,

As per the discussion above of how phasor multiplication corresponds to phase shift and amplitude modulation, we see that differentiating a phasor x̃ corresponds to multiplying the amplitude of x(t) by a factor of ω and adding a phase shift of 90°.

The impedance of a capacitor

The impedance Z is the effective resistance to AC current, measured in ohms. As put by molex.com,

In a direct current circuit, the impeding effect on electric current is called resistance. In the field of alternating current, in addition to the resistance, capacitance and inductance also impede the flow of current; this effect is called reactance. Reactance has the effect of resisting time-varying electric current.

Mathematically, Z is expressed as the ratio of AC voltage to current at a given frequency:

By replacing the sinusoids with their phasor representations, the impedance becomes a complex number consisting of a real part R (the resistance) and an imaginary part X (the reactance):

Thus, the impedance encodes the ratio of the voltage and current sinusoid amplitudes (Z_0), as well as the phase shift between the voltage and current sinusoid amplitudes (θ). As per the above derivation, θ indicates that the voltage V(t) always lags the current I(t) by θ/ω seconds.

Note that Z isn’t a phasor since it’s the ratio of two phasors. It has no time dependence, as the common e^jωt factor of the voltage and current phasors is eliminated by division.

Bearing in mind that for a capacitor,

the complex, frequency-dependent impedance of a capacitor is given by:

Z is usually represented by the second-to-last representation above, Z = 1/jωC, indicating that the impedance is inversely proportional to both capacitance and frequency. Equivalently, we can say the reactance of the capacitor X_C = 1/ωC.

However, the last representation tells us that 1) the ratio of the voltage amplitude to the current amplitude across a capacitor is 1/ωC and 2) that the voltage lags the current by 90°.

But what is the physical meaning of this? Essentially, the explanation is the same as that given in the section “Why can’t the voltage across a capacitor change instantaneously?” above: charging and discharging the capacitor takes time. Since we are discussing the frequency domain, we are now interested in whether the charging/discharging of the capacitor can keep up with the frequency of the source (e.g., how quickly an AC source voltage switches polarity).

In the extreme case of ω=0, i.e., when a DC voltage is applied to a capacitor, the capacitor initially draws a current to charge up. However, once fully charged, the current stops flowing and the capacitor acts like an open circuit. On the other hand, in an AC circuit, the capacitor continuously charges and discharges as the voltage polarity changes, allowing current to flow through it, with the amount of current depending on the frequency.

In particular, as the frequency of the AC signal increases, the capacitor has less time to achieve a fully charged/discharged state in which current flow stops. Since it spends less time in a current-obstructing state, the capacitor has a lower impedance at higher frequencies.

For example, consider the bridge rectifier circuit (full wave rectifier with a smoothing capacitor) below with load resistor R1 and filter capacitor C1. The AC source voltage Vin is described by frequency f = 50Hz and period T = 1/f = 0.02s.

Figure by author., captured from Multisim Live.

If we analyze the output voltage for different values of the capacitance C, we get the plots below. Each C corresponds to a certain RC time constant:

C = 500nF: 𝜏 = 0.0002s
C = 5uF: 𝜏 = 0.002s
C = 50uF: 𝜏 = 0.02s = T
C = 500uF: 𝜏 = 0.2s
C = 5mF: 𝜏 = 2s

For C << 50uF, we get 𝜏 << T. The capacitor can almost completely discharge before it starts charging again, and can almost completely charge before it starts discharging again, meaning that there is a short period when current isn’t flowing through it.

On the other hand, for C >> 50uF, we get 𝜏 >> T. The capacitor can only discharge a small amount before it starts charging again and can only charge a small amount before discharging again. This means it’s essentially always in a state of current flow.

Source voltage (blue) and output voltage (green) for different values of C. The x-axis shows 10ms/div and the y-axis shows the voltage. Figure by author.

Now we can get a better sense of X_C = 1/ωC = 1/2πfC. By increasing T relative to 𝜏 (decreasing f while holding C constant) or decreasing 𝜏 relative to T (decreasing C while holding f constant), we increase the time the capacitor has to charge/discharge before it has to start discharging/charging, and thus increase the portion of time the capacitor spends in a no current-flow state. This translates to “blocking low-frequency signals” (Z_C is high at low frequencies).

Note that this behavior is the basis for the capacitor’s “smoothing effect” in many circuits.

Non-ideal behavior

As discussed above, if we apply an AC voltage to a capacitor, the current follows 90° out of phase. The same will hold for a circuit containing purely capacitive components, as these can simply be expressed with an effective capacitance. Mathematically, because I and V are “perfectly” (90°) out of phase, the active power lost in the circuit is zero. Physically, because power is only dissipated through ohmic losses (I²R) and R=0 for a purely capacitive circuit, no power is dissipated.

If the capacitor is “leaky” in some way, such that there is a residual resistance, or the polarization of the dielectric lags behind the AC voltage such that I and V are no longer 90° out of phase, power will be lost across the capacitance. To model this behavior and quantify the non-ideality of the capacitor, we must consider two concepts: complex permittivity and the loss tangent.

Complex permittivity

Note that the derivation in this section is inspired by Föll (n.d.).

In many cases, we assume that the displacement field D and the polarization P react instantaneously to changes in E. This is apparent in the fact that each constant of proportionality (ε for D=εE, ε_0 χ_e for P = ε_0 χ_e) is real-valued: if each relationship is replaced with its phasor equivalent, then a purely real constant of proportionality indicates there is no phase shift between D and E or P and E (see “A recap of phasor representation” section). If we are to account for non-instantaneous relationships, we must include an imaginary component in our phasor representation.

The easiest way to think about this is to assume that following a sudden switch-off in E, D decays exponentially with a time constant τ equal to the relaxation time of the dielectric. (Another approach that generalizes to more complex relationships between E and D is given here.) Note that a similar derivation can be performed for the relationship between P and E by replacing D with P and ε_r with χ_e.

Imagine that E has been constant for a sufficiently long time such that the dipoles have reached a stable orientation. What happens if it’s suddenly switched off? The dipoles should begin to randomize, and accordingly, the net dipole moment (polarization P) should decay to zero. The displacement field D should follow the same decay.

The key word here is decay. Decay is not an instantaneous process — if we choose to model the decay exponentially, there will be an associated time constant (the amount of time that the quantity in question takes to decay by a factor of 1/e). In our dielectric system, we call this time constant the relaxation time τ, and both P and D can be modeled as decaying exponentially following a sudden switch-off in E.

The time-dependent behavior of D is described as follows, with D decaying starting at the time of the switch-off according to

If we are interested in the frequency dependence of D, we can apply the Fourier transform to D(t) and get

D(ω) thus is the displacement field response of the system to an electrical field given by E = E_0 · exp (jωt). This is a complex representation, but remember that the actual field corresponds to the real part.

Assuming that we’re working with linear dielectrics, the relationship between D and E is still given by D=εE. However, each of these variables is a function of frequency:

We can extract the frequency-dependent permittivity ε(ω) as,

where ε_s = is the static permittivity (the value for zero frequency), and D_0 = D(0) and E_0 = E(0) are the static displacement and electric field values, respectively.

So far, we’ve assumed that in the limit of large frequencies, P and D are zero; the dipole can’t keep up with the oscillations of the electric field and ε(ω → ∞) = 0. However, this isn’t necessarily true, and to make our equations more generalizable, we introduce a parameter ε(ω >> ω0) = ε_∞ as the high-frequency limit of the permittivity. This reasoning produces the final result for the frequency dependence of the orientation polarization, the so-called Debye relaxation model:

As with any complex number, we can decompose ε(ω) into a real and an imaginary part and write it as,

The former and latter conventions are typically used when representing the time-dependence of E as exp(-jωt) and exp(jωt), respectively, so that a lossy dielectric has a positive imaginary component of its permittivity.

Like any complex/phasor quantity, this complex permittivity carries information about phase and magnitude. In particular, the permittivity describes polarization, and the magnitude and phase describe its extent and delay, respectively. Using the ε = ε’ — jε’’ convention, we can find the real and imaginary parts of the permittivity as follows:

ε’, the real part of a complex amplitude, gives the amplitude of the response that is in phase with the driving force. It describes how well the electric dipole moments inside the material align to the electric field.
ε’’, the imaginary part, gives the amplitude of the response that is phase-shifted by 90°. It describes the EM losses in the surrounding media, attributable to the friction of movement as dipoles align to the electric field.

The frequency dependence of ε’ and ε’’ are depicted in the figure below. When the electric field oscillates at a very low frequency relative to the relaxation time (ω→ 0), ε’ ≈ ε_s and ε” ≈ 0. This condition implies that the low-frequency rotation of dipoles allows nearly total energy conservation since the dipole can easily remain in phase with E. At high frequencies (ωτ→∞), ε’ ≈ ε∞ and ε” ≈ 0. In this case, E oscillations are so rapid that dipoles cannot begin to reorient, consequently a lower limit on the dielectric constant, ε∞, is reached.

An intermediate frequency range must exist where there is maximum polarization-rotation phase lag. This condition exists when ωτ = 1; at that frequency, the dielectric loss factor is a maximum (ε” = ε”_max). Here the incident energy that was “stored” by a higher ε’ at lower frequencies is now dissipated by loss mechanisms (i.e., a larger ε”)

As briefly mentioned above, the Debye relaxation model primarily describes orientational polarization, which concerns the reorientation of permanent dipoles within a material when subjected to an electric field.

If you zoom in on the frequency range within the red dashed-line box, you get a permittivity frequency response similar to that depicted in the figure above. Figure adapted from source.

If we are concerned with other types of dielectric polarization (e.g., ionic polarization, responsible for the high dielectric constants in ceramics popularly used in capacitors), we must consider the resonance-dependent permittivity. A good overview of the relevant math is given by Morton (n.d., pp 10–12), and a good overview of the underlying physical phenomena is given by Mertens et al. (2023), in particular Figure 4 in their article. For now, it suffices to understand that the lagging of the dielectric’s polarization behind the AC voltage contributes to an imaginary component of the complex permittivity.

Loss tangent

Recall that our goal is to quantify the non-ideality of a “leaky” capacitor, i.e., one where there is a residual resistance or the polarization of the dielectric lags behind the AC voltage such that I and V are no longer perfectly out of phase, resulting in power loss across the capacitance. Now that we’ve discussed the complex permittivity, we can understand why the loss tangent is a good candidate parameter. There are several ways to think about it.

Approach 1: loss tangent represents the ratio of the resistive power loss to the reactive power in a leaky capacitor

For this approach, we begin by modeling the leaky dielectric as a perfect capacitor with a resistor in series:

The loss tangent can be expressed as the ratio of the resistive power loss in the resistor to the reactive power oscillating in the capacitor, such that it is zero for an ideal capacitor (R=0).

Since the same AC current flows through both resistor and capacitor, the ratio of resistive to capacitive power is simply the ratio of |R| to |Z_C|:

Thus, a perfect capacitor (R=0) has a loss tangent of zero.

Approach 2: loss tangent represents the ratio of ε’’ to ε’

Another way of thinking about this is allowing a complex relative permittivity that incorporates this loss tangent. The imaginary part of the permittivity ε’’ is then directly responsible for the effective resistance.

Consider the complex power S for a parallel plate capacitor with a complex permittivity ε = ε’ — jε’’:

Thus, the loss tangent tan δ = P/Q represents the ratio of the active power P (results in net transfer of energy, i.e., loss or gain) to the reactive power Q (doesn’t result in net transfer of energy so is considered to be “stored,” i.e., it oscillates between the source and load).

In line with Approach 2, if the polarization change is in phase with the applied electric field (ε’’=0), the material appears purely capacitive, and the loss tangent is zero. If there is a lag (e.g., for some of the reasons mentioned in the “Complex permittivity” section above), then the relative permittivity acquires an imaginary component, the material acquires resistive character, and the loss tangent becomes non-zero.

Approach 3 (more general): loss tangent represents the ratio of the imaginary part of the total displacement current to its real part

Above, we assumed that the conductivity of the dielectric σ=0. The more general definition of loss tangent in a medium with complex permittivity ε = ε’ — jε’’ and non-zero conductivity σ is the ratio of the imaginary part of the total displacement current to its real part.

By factoring the phasor form of Ampere’s law (with Maxwell’s modification) as follows and defining an “effective displacement field” D_eff,

we can define the loss tangent as:

Thus, the loss tangent represents the ratio of the energy lost as heat (conduction current) to the energy stored in a dielectric material due to polarization (displacement current) when exposed to a time-varying electric field, essentially indicating how much of the electric field contributes to creating a current that results in energy dissipation rather than being stored within the material.

In simpler terms, it is the ratio (or angle in a complex plane) of the lossy reaction to the electric field E in the curl equation to the lossless reaction. A higher loss tangent means a larger portion of the electric field is used to generate conductive current, leading to greater energy loss.

Upon comparison with Approach 2’s representation of the loss tangent, we see that the final term in the equality above is quite similar; the only difference is that the numerator has a “+ σ” term. Essentially, this means that the energy loss due to conduction current (the σ term) is indistinguishable from the energy loss due to the dielectric relaxation (the ε’’ term), which results from a delay in the dipole orientation with the applied field.

From a more mathematical point of view, we can see why the loss tangent is called a “tangent.” Again consider the phasor form of Ampere’s law (with Maxwell’s modification):

By matching up the cos δ and sin δ terms as above, we see that δ is the angle between the current and the time-varying electric field, or alternatively, 90° + the angle between the current and the electric field.

Summary

Capacitor “main ideas” (black boxes) and their relation to concepts from physics and mathematics. Solid-line boxes represent fundamental relations between physical quantities and dashed-line boxes represent “connecting ideas.” Red and blue boxes correspond to ideal and non-ideal capacitor behavior, respectively, with the latter likely beyond the scope of most intro-level engineering classes. Figure by author.

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👋 I am a senior at Yale University majoring in electrical engineering. I’m interested in neuroscience, computer science, math, and everything in between!

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Gmail: danielle.l.gruber@gmail.com