Colada morada

Re

|VL2|² ZL

 = 20 log 10

V_L1 V_L2

. (2.62)

Equation 2.63 follows by expressing Equation 2.62 in terms of A, B, C, D, V_S, Z_S, and Z_L. I L= −20 log 10

 ZL+ ZS

AZL+ B + ZS(C ZL+ D)

. (2.63)

Using Equation 2.46, one can also express the return loss in terms of A, B, C, D, VS, ZS, and ZL; i.e.,

RL= 20 log 10

AZL+ B + Z0(C ZL+ D) AZL+ B − Z0(C ZL+ D)

. (2.64)

2.3.9 Scattering Parameters

ABCD parameter modelling is one modelling solution out of many. The ABCD method is particularly suited when a cable transfer function made of a concatenation of several pieces of homogeneous lines is to be computed. The ABCD parameters are also very in-tuitive because they deal with voltages and currents across two-port networks. As soon as measurements are involved, it is less appropriate to deal directly with ABCD parame-ters, because it is difficult to measure them at high frequency with reliable precision. At high frequencies, engineers prefer to measure transmitted and reflected powers instead of dealing with ABCD parameters, which are subject to a considerable range of magnitude and imprecision due to sensitivity of measurement. The modelling technique that has been introduced by circuit theorists in order to deal with transmitted and reflected power across two-port networks is the scattering parameter formalism. Less intuitive than the ABCD pa-rameter modelling, the scattering papa-rameter formalism deals with travelling waves, which are linear combinations of the voltages and the current in networks. The scattering param-eters (so-called S paramparam-eters) are particularly suitable for problems of power transfer of networks designed to be terminated by resistive loads, as is the case in DSL at high fre-quencies. They involve reflection and transmission coefficients having finite range, which are measurable with high precision by modern network analyzers. This section presents the main results linked with the S parameters with a short introduction for the reader who is encountering the scattering matrix for the first time.

The two-port of Figure 2.15 is considered, which is powered with a generator of resistance RGand loaded with a resistive load RL. A normalization impedance Rnis defined, the role of which will become clear later. The incident waves at port 1 and port 2 are (respectively) arbitrarily defined asξ1andξ2such that

ξ1= V₁

R_n + i1

Rn, (2.65)

ξ2= V₂

R_n + i2

R_n. (2.66)

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52 Fundamentals of DSL Technology

V₁

The reflected waves at port 1 and port 2 are (respectively) arbitrarily defined asη1andη2

such that

The scattering matrix S is defined as the matrix providing the relationship between the reflected and incident waves; i.e.,

η1

By using elementary circuit theory, one can find the relationship between S and the ABCD parameters; i.e.,

By using the relationships in Equations 2.49 and 2.50, one can deduce from Equation 2.70 the expression of the scattering matrix in terms ofγ and Z0 for a homogeneous line of length l; i.e.,

Checking the power flow, one can now justify the definitions of incident and reflected waves. It is straightforward to show that the power flow into port 1 is

(V1i₁^∗) = [ξ1+ η1]

In a similar fashion, one can find the power flow into port 2; i.e., (V2i₂^∗) = [ξ2+ η2]

Equations 2.72 and 2.73 show that the power entering a port is the difference between the power induced by the incident wave and the power induced by the reflected wave. The two-port network of Figure 2.16 is now considered, in which the loads at both ports are such that RG = RL= Rn. The power flow across the two-port network is of interest. In the DSL

V₁

Two-port network loaded with Rn.

transmission band this resistance is the usual load impedance which is in a range between 100 and 135 . In this specific case, one obtains

ξ1 = V₁

The active power transmitted to the load of port 2 is (−V2i₂^∗) = |η2|²

4 . (2.77)

In a similar fashion, the active power reflected at port 1 is (−V1i₁^∗) = |η1|²

4 . (2.78)

Recalling that the maximum power available from the generator is ^|E|_4R²

n, and using Equa-tions 2.74, 2.76, and 2.77, one finds for the power delivered to the load

P_deliv.= |S21|²P_{max av.}. (2.79)

Similarly, power reflected at port 1 is obtained by using Equations 2.74, 2.76, and (2.78); i.e.,

P_reflect.= |S11|²P_{max av.}. (2.80)

For a passive device such as a cable, S21 and S11 are such that |S21( jω)| <= 1 and

|S22( jω)| <= 1 . The coefficient S21( jω) is defined as the transmission coefficient (from port 1 to port 2), whereas S11( jω) is the reflection coefficient (of port 1). Similar definitions hold for S12and S22by exchanging port 1 for port 2 and vice versa. Observe that the return loss already defined as the inverse ratio of reflected power to maximum power available from load is simply given as

RL= −10 log 10|S22|². (2.81)

In a similar way, if one wants to express the insertion loss using the measurement principle shown in Figure 2.14 with ZS= ZL= Rn, one finds that the power before cable insertion is

|E|²

4Rn, and the power after cable insertion is given by Equation 2.79. Therefore, one obtains the simple formula

I L= −10 log 10|S21|². (2.82)

For a homogeneous cable of length l, with characteristic impedance Z₀, propagation con-stantγ , and loaded on both sides with a resistance Rn, Equations 2.71 and 2.81 show that

RL= −10 log 10

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54 Fundamentals of DSL Technology

q′

Transfer matrix principle for a cascade of two-ports.

For the same cable, the expressions in Equations 2.71 and 2.82 show that

I L = −10 log 10

Although the scattering formalism is very practical in terms of measurements and brings simple expressions of return loss and insertion loss, it turns out to be very impractical when one has to compute the effect of a cascade of several homogeneous cables. In other words, if two two-ports are cascaded, the overall scattering matrix of this cascade is not the product of the individual matrices. Therefore, instead of working directly with the scattering matrix, engineers prefer to work with the wave transfer matrix, the manipulation of which is easier when cascading two-ports. The wave transfer matrixθ is such that

ξ1

When cascading two two-port networks, the waves at the interface port are such that the reflecting wave of one port is the incident wave of the other and vice versa. The arrangement of waves in Equation 2.85 is such that the wave transfer matrix of a cascade of two-port networks is the product of the individual transfer matrices. This is illustrated in Figure 2.17.

If one definesθ as the transfer matrix of the two-port network on the left and θ^as the transfer matrix of the two-port network on the right, one finds

ξ1

and it follows from Equation 2.86 that

ξ1

Elementary circuit theory shows that the relationship between the wave transfer matrix and the ABCD parameters is given by

θ = 1

By substituting Equations 2.49 and 2.50 into Equation 2.88, one finds that for a homoge-neous portion of cable of length l of characteristic impedance Z₀and propagation constant γ terminated in Rn, the wave transfer matrixθ is

θ = 1

By working with wave transfer matrices when cascading several pieces of homogeneous cables, one can obtain the overall transfer matrix of the overall network and compute the corresponding scattering matrix by using the following correspondence between the wave transfer matrix and scattering matrix.

S=