Introduction to Dielectric Loss in Transmission Strains

Two of crucial loss mechanisms in transmission strains are conductor loss and dielectric loss. The conductor loss is influenced by the pores and skin impact phenomenon that we launched beforehand. The dielectric loss is said to the properties of the insulating materials positioned between the conductors. On this article, we’ll learn the way the dielectric’s relative permittivity and loss tangent have an effect on the efficiency of transmission strains.

Equal Circuit of a Transmission Line

The equal circuit of a two-conductor interconnect at excessive frequencies is proven in Determine 1.

Determine 1. Equal circuit of a two-conductor transmission line.

As mentioned in a earlier article, the resistance R fashions the loss from the sequence resistance of the conductors, whereas the conductance G accounts for the dielectric loss. In different phrases, G characterizes the electromagnetic energy that the dielectric absorbs and turns into warmth. This impact turns into extra pronounced at increased frequencies.

Earlier than diving in too far, let’s assessment the traits of a lossless capacitor.

Ideally suited, Lossless Capacitor

Think about the acquainted capacitance equation beneath:

$$C = epsilon_0 epsilon_r fracAd$$

Equation 1

the place:

• C is the capacitance
• ε₀ is the permittivity of free house
• ε_r is the relative permittivity of the dielectric materials
• A is the parallel plate space
• d is the space between the 2 conductive plates

The relative permittivity, ε_r, often known as the dielectric fixed denoted by Dk, characterizes how properly the dielectric materials shops an electrical discipline. The relative permittivity may be outlined because the ratio of the capacitance {that a} capacitor produces with a given dielectric to that obtained when the identical capacitor has vacuum as its dielectric.

The dielectric fixed for many PCB supplies is within the 2.5 to 4.5 vary. Some ceramics, equivalent to barium titanate, exhibit dielectric constants as excessive as 7,000. The dielectric fixed usually decreases as frequency will increase.

The admittance, Y, of an excellent capacitor is given by:

$$Y_ideal=jC omega = j epsilon_0 epsilon_r fracAd omega$$

Equation 2

the place:

• ω is the frequency in radians = 2πf

The relative permittivity, ε_r, is an actual worth; therefore, a capacitor’s admittance is a purely imaginary worth. We are able to additionally see from this equation that the capacitor admittance is a operate of frequency. Equation 2 exhibits that with an actual ε_r, the present going by the capacitor leads the voltage waveform by exactly 90°, as illustrated in Determine 2.

Determine 2. The part relationship between voltage and present waveforms of an excellent capacitor.

It’s price mentioning that the impedance and wave velocity of a transmission line are affected by the dielectric fixed of the insulating materials. Additionally, for an excellent capacitor, the typical energy dissipated is zero.

How Does a Dielectric Improve Capacitance?

Think about a dielectric made up of polar molecules. With polar molecules, there’s a everlasting separation between the middle of constructive and unfavourable electrical fees within the molecules. Within the absence of an exterior electrical discipline, the electrical dipoles of the molecules are randomly oriented, as proven in Determine 3(a).

Determine 3. (a) Dielectric dipoles with out an exterior electrical discipline and (b) within the presence of an exterior discipline.

Nevertheless, within the presence of an electrical discipline, as in Determine 3(b), the dipoles expertise a torque and partially align with the sphere. As you’ll be able to see, the alignment of the electrical dipoles results in some induced floor cost of reverse polarity on the alternative faces of the dielectric. This conduct is the essential property that allows a dielectric to extend the capacitance of a capacitor.

To know this, think about a charged capacitor with voltage V₀ and cost Q₀ on every plate, as Determine 4(a) depicts. The electrical discipline between the plates is E₀. Assume that the capacitor is disconnected from its charging voltage supply in order that the costs are caught on the plates and don’t have a path to discharge.

Determine 4. The impression of a dielectric on capacitor electrical discipline

If we place a dielectric between the plates, as proven in Determine 4(b), the electrical dipoles of the dielectric align with the exterior discipline and produce some induced floor cost on the alternative faces of the dielectric (Q_ind on the appropriate face and -Q_ind on the left face). Notice that the dielectric’s unfavourable floor cost faces the capacitor’s constructive plate and vice versa.

The induced floor fees produce an inner electrical discipline that opposes the preliminary electrical discipline between the capacitor plates. Because of this, the general electrical discipline (which is the same as E₀ − E_ind) and the voltage between the capacitor plates are lowered (V₁ < V₀).

Notice that the cost on the capacitor plates has not modified, however their contribution to the capacitor voltage is partially canceled by the floor fees of the dielectric. How are these adjustments going to have an effect on the capacitance? This may be answered by contemplating the capacitance definition in Equation 3:

$$C= fracQV$$

Equation 3

We observe that the cost saved on the capacitor plates is similar whereas the voltage throughout the capacitor is lowered. This corresponds to a rise within the capacitance.

Within the above dialogue, we’ve assumed that the dielectric molecules are polar; nonetheless, a dielectric with nonpolar molecules additionally will increase the capacitance. With a nonpolar dielectric, the exterior electrical discipline produces some cost separation and an induced dipole second. These induced dipoles then align within the course of the exterior discipline, much like the case of everlasting dipoles.

Lossy Dielectrics

The interplay between the dielectric molecules and the exterior electrical discipline may be far more sophisticated than the above dialogue. For instance, when a molecule is displaced from equilibrium by an electrical discipline, the molecule can oscillate for a short while after eradicating the exterior discipline. Alternatively, the molecules might not be capable to align at sufficiently excessive switching frequencies for the exterior electrical discipline.

Two several types of movement can describe the general movement of the dipoles. First, if the movement of the dipoles contributes to a present part that leads the capacitor voltage by exactly 90°, the dielectric will increase the capacitance. Second, dipole displacements that result in a present in part with the capacitor voltage may be modeled by a resistive part. This latter kind of dipole displacement really contributes to the losses. This loss time period is modeled as a parallel resistor (the conductance G within the transmission line mannequin of Determine 1).

The Complicated Dielectric Fixed

As famous above, the dielectric fixed of a lossless dielectric materials is an actual worth. Nevertheless, to fully describe {the electrical} properties of a lossy dielectric, we outline ϵ_r as a fancy worth:

$$epsilon_r = epsilon_r ^prime -jepsilon_r ^prime prime$$

Equation 4

Combining Equation 4 with Equation 1, the admittance of a capacitor with a lossy dielectric is:

$$Y_lossy= j epsilon_0 epsilon_r^prime fracAd omega + epsilon_0 epsilon_r^prime prime fracAd omega$$

Equation 5

By assuming a fancy dielectric fixed, two phrases seem on this admittance equation. The imaginary a part of the admittance corresponds to an excellent lossless capacitor, whereas the true a part of the admittance accounts for the dielectric loss.

Losses of the dielectric materials are often expressed by the dielectric loss tangent or generally merely the loss tangent, which is outlined because the ratio of the true and imaginary elements of the permittivity:

$$tan(delta) = fracepsilon_r ^prime primeepsilon_r ^prime$$

Equation 6

The loss tangent at varied frequencies may be present in varied handbooks. PCB laminate information sheets additionally present a dielectric loss tangent worth and a Dk worth. By multiplying these two values collectively, we are able to discover the imaginary a part of the dielectric fixed utilizing Equation 6.

The traits of some frequent PCB laminates are listed beneath in Desk 1. For a dielectric to be categorised as low loss, the loss tangent must be beneath 0.004.

Desk 1. Traits of frequent PCB laminates. Knowledge used courtesy of Altium

Frequency Dependence of Dielectric Properties

The dielectric fixed’s actual and imaginary elements can range to some extent with frequency. That is illustrated in Determine 5 beneath. The determine additionally highlights that there are a number of totally different dielectric loss mechanisms that dominate at totally different frequencies.

Determine 5. Dielectric fixed actual and imaginary elements as a operate of frequency. Picture used courtesy of Dr. Ken Mauritz

Discovering the Conductance of a Transmission Line

Now, we are able to use Equation 5 to search out an expression for the conductance G in our transmission line mannequin:

$$G = epsilon_0 epsilon_r^prime prime fracAd omega$$

Equation 7

The above expression may be simplified by noting that the road’s capacitance is given by:

$$C = epsilon_0 epsilon_r^prime fracAd$$

Equation 8

Combining these two equations with Equation 4 produces:

$$G = C tan(delta ) omega$$

Equation 9

From Equation 9, we are able to see that the conductance is straight proportional to the capacitance, loss tangent, and frequency.

The Impression of Dielectric Loss on Sign Transmission

Hopefully, this has supplied you with a greater understanding of dielectrics, together with the significance of relative permittivity and loss tangent. Within the subsequent article on this sequence, we’ll broaden on this dialogue and see how the above outcomes can be utilized to characterize the impact of dielectric loss on the sign attenuation because it travels down the road.

Except in any other case famous, all pictures are the writer’s.