alt="equation"/>
The following expression is obtained from equation (3.2.14):
Equation (3.2.17) gives the following reflection coefficient S11 at the port‐1, while port‐2 is terminated in the matched load Z0:
(3.2.18)
Similarly, the following expressions are obtained from equation (3.2.13), where the port‐1 is terminated in the matched load Z0, i.e.
(3.2.19)
On eliminating
(3.2.20)
If the network is reciprocal, AD − BC = 1, i.e. S12 = S21. For the symmetrical network, S11 = S22 leading to A = D. The known [S] parameters can also be converted to the [A, B, C, D] parameters. Similarly, the [Z], [Y], [ABCD] and [S] parameters are also converted among themselves [B.1, B.3, B.5].
3.2.2 De‐Embedding of True S‐Parameters
A transmission line section could be treated as a device and its performance can be evaluated by using a VNA. The device is connected to the VNA through connecting cables and connectors. The S‐parameters of a device is measured at the external circuit ports that include the effect of the cables and connectors on the S‐parameters of the device. Thus, the true S‐parameter of a device is embedded in the measured S‐parameters of the device. However, it is desired to obtain the true S‐parameters of the device under test (DUT). The line section could be the DUT. The process of extracting the S‐parameters of the device at the internal device ports (1in, 2in), from the measured S‐parameters, at the external circuit ports (1ex, 2ex) is known as the de‐embedding process. It is achieved through a calibration process in which the S‐parameters of two error boxes are quantified. The error box represents errors in the S‐parameters due to cables and connectors connecting the device to the external circuit ports [B.1]. The S or [ABCD] parameter representation of the device at internal ports (1in, 2in) along with the error boxes is shown in Fig (3.20a). The location of the measurement ports, i.e. the external ports (1ex, 2ex) and the device internal ports (1in, 2in), are further shown in Fig (3.20b).
Once the error boxes are characterized through their S‐parameters, it can be converted to the [Ae Be Ce De] parameters. Similarly, the measured S‐parameters of the device and the error box combined are available at the external ports. These can be converted to the measured [Am Bm Cm Dm] parameters. The device [Ad Bd Cd Dd] parameters are related to the other two parameters by the following equation:
Figure 3.20 Calibration process in the measurement of S‐parameters of a device.
(3.2.21)
The error box 2 is the mirror image of the error box 1 with respect to the DUT. So in the above‐given matrix sequence, the third matrix is inverse of the first matrix [B.11]. At the internal device ports, the device [Ad Bd Cd Dd] parameters are de‐embedding as follows:
(3.2.22)
The de‐embedded device [Ad Bd Cd Dd] parameters are converted to the de‐embedded S‐parameters of the device. The de‐embedded S‐parameters could be further converted to the Z and Y‐parameters. Thus, any two‐port device can be characterized through measurements using suitable parameters‐ S, Z, or Y. In the case of a transmission line section, the de‐embedded S‐parameters can be converted to the propagation parameters and the characteristic impedance of the line.
The above‐mentioned concept of de‐embedding of the device S‐parameters at the internal port of a device is equally applicable to the EM‐Simulators – both 2.5D and 3D simulators [B.10]. In EM‐simulators, the delta‐gap voltage source could be used to launch the wave on a line section or a device. It also generates the nonpropagating evanescent modes at the ports. They cause a discontinuity at the external circuit ports, i.e. at the ports of simulation. The port discontinuity affects the S‐parameters of the device that is removed by the process of de‐embedding [J.2]. The EM‐simulators could be used to extract the propagation parameters and the characteristic impedance of a line.
3.2.3 Extraction of Propagation Characteristics
The true S‐parameters of the line section, i.e. its de‐embedded S‐parameters over a range of frequencies are known either through the measurement or through the EM‐simulation. This information can be converted to the [ABCD] parameters of a line section as follows [B.1]:
(3.2.23)
Usually, S11 = S22 and S12 = S21 because a section of the transmission line is treated as the symmetrical and reciprocal network. The frequency‐dependent [ABCD] parameters of a lossy line of length ℓ are used to compute the characteristic impedance Z0 and propagation constant γ = α + j β over a range of frequencies [B.1]:
(3.2.24)
The frequency‐dependent secondary line parameters are extracted to get the dispersion characteristics of a line. Once the secondary line parameters of a line section are known, their frequency‐dependent primary constants are computed as follows:
(3.2.25)
3.3
