Equalizers in High-Speed Application

In the multimedia era, size of source data is increasing dramatically to get high definition video or better audio quality. At the same time, high-speed trans-mission for related application becomes more and more important to provide sufficient hardware ability. However, the physics characteristics of channels such as wire between TV and DVD player, wire of earphone, introduce several negative impacts on the correctness of data. Therefore, we will give a brief introduction about the factors that cause these negative impacts from physics point of view.

2.1.1 Physical Limitation of High-Speed Transmission

There are two main phenomena that will alter the characteristic of current or voltage on conductors when we take frequency of current or voltage into con-sideration. One of the phenomena is skin effect that is mentioned in chapter 1, the other phenomenon is dielectric loss. We will give a short introduction and explanation about these two roles and will get a whole picture of the relationship between signal frequency and characteristics of channel.

The skin effect is the tendency of alternating electric current (AC current) to distribute itself within a conductor so that the current density near the surface of the conductor is greater than that at its core. That is, the electric current tends to flow at the skin of the conductor. The way to measure the depth current flows at the skin of the conductor is skin depth. By definition, the skin depth is the

measure of the distance over which the current density falls to 1/e of its original value.

Skin depth is a property of the material that varies with the frequency.

Skin depth is affected by the material relative permittivity, conductivity of the material and frequency of the wave. First, we let the complex permittivity ε_c of an material is

ε

= ε(1 − j σ

ωε ) (2.1)

where:

ε: permittivity of the material of propagation

σ: electrical conductivity of the material of propagation ω: angular frequency of the wave

j: the imaginary unit

Thus, the propagation constant kcof the signal propagating on the conductor will also be a complex number

k

= ω √

Before we get the final equation of skin depth, there is one more simplifica-tion. For a good conductor, we can say that 1¿ _εω^σ . Therefore, the propagation constant in equation (2.2) can be simplified as

k

= p

jµωσ = 1 + j

√ 2

p 2πf µσ = (1 + j) p

πf µσ (2.3)

then the above equation can be separated into real part, α, and imaginary part, β,

k

= α + jβ = p

πf µσ + j p

πf µσ (2.4)

Now, assume a uniform wave, that can be voltage or current, propagating in the +z-direction,

E

= E

e

= E

e

e

(2.5)

then β gives an exponential decay as z increases. For this reason β is also called attenuation constant of a propagating wave. By the definition of skin depth, it is the distance over which the current density decays to 1/e of its original value, that is βz = 1. Therefore, the skin depth δ is

δ = 1

β = 1

√ πf µσ (2.6)

From equation (2.6) we can find that the skin depth decreases while the signal frequency increases. That means the high frequency signal has less effective cross area than the low frequency signal. Moreover, the effective resistance is inverse proportion to effective area. So that high frequency signal will suffer more effective resistance and will get more loss than low frequency signal will. Table 2.1 lists the skin depths of several types of materials at various frequencies [11]

Table 2.1: Skin depth of several material at various frequencies

Material f =60(Hz) 1(MHz) 1(GHz) Silver 8.27(mm) 0.064(mm) 0.0020(mm)

Copper 8.53 0.066 0.0021

Gold 10.14 0.079 0.0025

Aluminum 10.92 0.084 0.0027

The dielectric loss is the signal power will loss due the dielectric materials in

the wire. When an time-varying electric field passes to a material, the particles in the material will be polarized. The polarization vector will change with the varying of electric field. With the input field frequency increases, the internal polarized charge cannot follow the varying of field in time and the polarized charge becomes out of phase. This phenomenon leads to a frictional damping mechanism that causes power loss and generates heat. We can model the phenomenon in the imaginary part in a complex permittivity ε_c

ε

= ε

− jε

(F/m) (2.7)

where both ε⁰ and ε⁰⁰ can be functions of frequency. Here, we also define an equivalent conductivity σ representing all loss

σ = ωε

(S/m) (2.8)

The ratio ε⁰⁰/ε⁰ is called a loss tangent because it is a measure of the power loss in the medium:

tan δ

= ε

ε

∼ = σ

ωε (2.9)

The quantity δc in equation (2.9) is called the loss angle. We know that a medium is say to be a good conductor if σÀ ωε, and a good insulator if ωε À σ. For an fix medium, ω and σ are almost constants. When the frequency of signal increases, the medium tends to be like an insulator. That means the signal suffers more resistant force to pass the medium.

2.1.2 Equalizer and Compensation

An ideal channel should have uniform gain for any band of frequency. How-ever, from the previous introduction of physical characteristics of a channel we can find that high frequency signal owns a unfair treatment at amount of gain.

The behavior of having much loss at high frequency than at low frequency is just like a low pass filter. Therefore, we often model the channel frequency response as a low pass filter.

Equalization is a process to compensate or to make equal the frequency response of the channel. This technique was first used by the Bell Labs for correcting audio transmission losses on the telephone system [12]. The block to complete the equalization process is called equalizer.

An equalizer can be understood as an high pass filter. This filter has a trend of increasing gain in high frequency to compensate gain decreasing of the channel.

After adding equalizer between the channel and receiver, we hope the high pass response can just compensate the losses of channel at high frequency to flat or to equal the total effective response. If we can not flat the response, at least we can make the band nearly flatten. Fig. 2.1 roughly illustrates the goal of equalizer.

Figure 2.1: Channel response and equalizer response

Fig. 2.2 shows the location of equalizers in a system. As mentioned in chap-ter 1, we can put the compensation block in the transmitchap-ter side also called pre-emphasis or do the mechanism in the receiver side. No matter in transmitter side or in receiver side, the goal is that the channel response plus the responses of the two compensation blocks can be almost flat at the target frequency.

Figure 2.2: Typical system view with equalizer