ENE 490 Applied Communication Systems Lecture 1 Backgrounds on Transmission lines and matching on Smith chart

Introduction How does information transfer?

High frequency operation Microwave frequency range (300 MHz – 300 GHz) Microwave components are distributed components. Lumped circuit elements approximations are invalid. Maxwell’s equations are used to explain circuit behaviors ( and )

Applications of high frequency communications Antenna gain More bandwidth Satellite and terrestrial communication links Radar communication Remote sensing, medical diagnostics, and heating methods

Frequency behavior of passive components At 60 Hz An equivalent circuit representation of high frequency resistor

Transmission lines (TLs) analysis (1) At higher frequencies, voltage and current are not spatially uniform, must be treated in terms of propagating waves. The distributed-parameter model including instantaneous voltage and current

Transmission lines (TLs) analysis (2) Kirchhoff’s circuit laws fail to explain circuit behaviors at high frequency The transmission line must be viewed in terms of distributed parameters, R, L, C, and G. A transmission line theory is applied when (lA is the average size of the discrete component) Example of transmission lines

General transmission line equation (1) Kirchhoff voltage and current law representations

General transmission line equation (2)

Lossless case Special case: Lossless line R = 0  and G = 0

Terminated lossless transmission line Convenient representation of source and load ends for some T.L. problems

Voltage reflection coefficient Definition: Define load reflection coefficient: Load reflection coefficient in terms of impedance: Note: ZL is a load impedance.

Impedance along the transmission line Impedance anywhere on the transmission line

Ex1 Determine the input impedance Zin when l = /4 l = /2 ZL = Z0

Voltage standing wave ratio VSWR (1)

Voltage standing wave ratio VSWR (2)

Ex2 Determine VSWR and the locations of maximum and minimum voltages for matched load ZL = 100 , Z0 = 50 

c) short circuit d) open circuit

Source and loaded transmission lines Transmission coefficient:

Power transmission of a transmission line Average power: For lossless line: For lossless and matched condition: we have Pavs, maximum available power provided by the source. Power in decibel:

Ex3 For the circuit shown above, assume a lossless line with Z0 = 50 , ZS = 75, and ZL = 100. Determine the input power and power delivered to the load. Assume the length of the line to be /2 with a source voltage of VS = 10 V.

Input impedance matching Optimal power transfer requires conjugate complex matching of the T.L. to the source impedance: Zin = ZS* Similarly for output matching: Zout = ZL*

The Smith Chart a graphical tool to analyze circuit impedance design of matching networks computations of noise figures, gain, and stability circles.

Using of the Smith Chart Impedance transformation Step 1 – Normalize the load impedance ZL with respect to the line impedance Z0 to determine zL. Step 2 – Locate zL in the Smith Chart Step 3 – Identify the corresponding load reflection coefficient 0 in the Smith Chart both in terms of its magnitude and phase. Step 4 – Rotate 0 by the length in terms of wavelength  or twice its electrical length d to obtain in(d). Step 5 – Record the normalized input impedance zin at this spatial location d. Step 6 – Convert zin into actual impedance Zin.

Ex4 Given the load impedance ZL to be 30+j60, Determine the input impedance if the T.L. is 2 cm long and is operated at 2 GHz.

Standing wave ratio in Smith chart The numerical value of SWR can be found from the Smith chart by finding the intersection of the circle of radius with the right hand side of the real axis. Ex5 Three different load impedances: ZL = 50 , b) ZL = 25+j75 , and c) ZL = 40 + j20 , are sequentially connected to a 50  transmission line. Find the reflection coefficients and the SWR circles.

Admittance transformation Rotations by 180 degrees convert the impedance to the admittance representation. Y-Smith chart ZY-Smith chart

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

Ex6 of a T-network (operated at 2 GHz)