ENE 429 Antenna and Transmission Lines Theory Lecture 7 Matching network and Scattering parameters RS

Review Input impedance for finite length line Quarter wavelength line Half wavelength line Smith chart A graphical tool to solve transmission line problems Use for measuring reflection coefficient, VSWR, input impedance, load impedance, the locations of Vmax and Vmin Impedance matching

Parallel and Series Connections (1) Parallel connection of R and L elements

Parallel and Series Connections (2) Parallel connection of R and C elements

Parallel and Series Connections (3) Series connection of R and L elements

Parallel and Series Connections (4) Series connection of R and C elements

Ex1 Adding a series inductor L (zL = j0.8) to an impedance z = 0.3-j0.3.

Ex2 Adding a shunt inductor L (yL = -j2. 4) to an admittance y = 1 Ex2 Adding a shunt inductor L (yL = -j2.4) to an admittance y = 1.6+j1.6.

L matching networks

Forbidden regions Sometimes a specific matching network cannot be used to accomplish a given match.

Single- and Two-port Networks The analysis can be done easily through simple input-output relations. Input and output port parameters can be determined without the need to know inner structure of the system. At low frequencies, the z, y, h, or ABCD parameters are basic network input-output parameter relations. At high frequencies (in microwave range), scattering parameters (S parameters) are defined in terms of traveling waves and completely characterize the behavior of two-port networks.

Basic definitions Assume the port-indexed current flows into the respective port and the associated voltage is recorded as indicated.

z Parameters v1 = z11i1 + z12i2 v2 = z21i1 + z22i2 or in the matrix form

y, h, and ABCD parameters y parameters These two-port h parameters representations are very useful at low frequencies because the parameters are readily measured using short- and open- circuit tests at the terminals of the two-port network.

Two-port connected in series

Two-port connected in shunt

Two-port connected in cascade fashion

Disadvantages of using these parameters at RF or microwave frequency Difficult to directly measure V and I Difficult to achieve open circuit due to stray capacitance Active circuits become unstable when terminated in short- and open- circuits.

Introduction of scattering parameters (S parameters) Measure power and phase Use matched loads Devices are usually stable with matched loads. S- parameters are power wave descriptors that permits us to define input-output relations of a network in terms of incident and reflected power waves

Introduction of the normalized notation (1) we can write Let’s define and

Introduction of the normalized notation (2) We can also show a(x) and b(x) in terms of V(x) and I(x) as and

Normalized wave generalization For a two-port network, we can generalize the relationship between b(x) and a(x) in terms of scattering parameters. Let port 1 has the length of l1 and port 2 has the length of l2, we can show that or in a matrix form, Observe that a1(l1), a2(l2), b1(l1), and b2(l2) are the values of incident and reflected waves at the specific locations denoted as port 1 and port 2.

The measurement of S parameters (1) The S parameters are seen to represent reflection and transmission coefficients, the S parameters measured at the specific locations shown as port 1 and port 2 are defined in the following page.

The measurement of S parameters (2) (input reflection coefficient with output properly terminated) (forward transmission coefficient with output properly terminated) (output reflection coefficient with input properly terminated) (reverse transmission coefficient with input properly terminated)

The advantages of using S parameters They are measured using a matched termination. Using matched resistive terminations to measure the S parameters of a transistor results in no oscillation.

The chain scattering parameters or scattering transfer parameters (T parameters) (1) The T parameters are useful in the analysis of cascade connections of two-port networks. The relationship between S and T parameters can be developed. Namely,

The chain scattering parameters or scattering transfer parameters (T parameters) (2) and We can also write