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ENE 490 Applied Communication Systems Lecture 1 Backgrounds on Transmission lines and matching on Smith chart 1 Introduction How does information transfer? mixer power amp power amp mixer signal receiving signal matching network Local oscillator Transmitter BPF BPF matching network Local oscillator Receiver 2 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 ( Eand )H 3 Applications of high frequency communications Antenna gain More bandwidth Satellite and terrestrial communication links Radar communication Remote sensing, medical diagnostics, and heating methods 4 Frequency behavior of passive components 1 XC C X L L At 60 Hz 1 9 X C ( 60 Hz) 2 . 65 10 12 2 60 10 X L ( 60 Hz) 2 60 109 3.77 107 0 An equivalent circuit representation of high frequency resistor 5 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 6 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 l A / 10 (lA is the average size of the discrete component) Example of transmission lines 7 General transmission line equation (1) Kirchhoff voltage and current law representations i ( z , t ) v( z , t ) i ( z , t ) Rz Lz v( z z , t ) t v( z z , t ) v( z , t ) i ( z , t ) lim Ri ( z, t ) L z 0 z t dv( z, t ) ( R jL)i ( z , t ) dz i ( z, t ) v( z , t ) Gv( z, t ) C z t 8 General transmission line equation (2) Let VS (t ) Vs cos t v( z , t ) Re V ( z )e jt and i ( z , t ) Re I ( z )e jt V ( z ) V0 e z V0 e z I ( z) I 0 e z I 0 e z V0 z V0 z e e Z0 Z0 ( R jL)(G jC ) j V0 V0 ( R jL) Z0 R0 jX 0 () (G jC ) I0 I0 9 Lossless case Special case: Lossless line R = 0 and G = 0 j LC j L Z0 C 10 Terminated lossless transmission line Convenient representation of source and load ends for some T.L. problems V0e V0e ZL Z0 V0e x=0 -j z ZL Z0 +jz x=l j d V0e d=l -j d d=0 11 Voltage reflection coefficient Definition: jd V Reflected Voltage Wave (d) 0 e ( d ) jd Incident Voltage Wave (d) V0 e V0 j 2d ( d ) e V0 Define load reflection coefficient: V0 L (0) V0 Load reflection coefficient in terms of impedance: Z L Z0 L Z L Z0 Note: ZL is a load impedance. 12 Impedance along the transmission line V (d ) V0 e jd V0 e jd Z (d ) jd I (d ) V0 e V0 e jd Z0 Z0 Z L cos d jZ 0 sin d Z (d ) Z 0 Z 0 cos d jZ L sin d Z L jZ 0 tan d Z (d ) Z 0 Z 0 jZ L tan d Impedance anywhere on the transmission line 1 ( d ) Z (d ) 1 ( d ) 13 Ex1 Determine the input impedance Zin when a) l = /4 b) l = /2 c) ZL = Z0 14 Voltage standing wave ratio VSWR (1) V (d ) V0 (e jd Le jd ) V (d ) V (d )e jd 1 Le j 2d V (d ) max V (d ) 1 L V (d ) min V (d ) (1 L ) 15 Voltage standing wave ratio VSWR (2) I (d ) max V (d ) 1 L Z0 I (d ) min V (d ) 1 L Z0 V (d ) max V (d ) min I (d ) max I (d ) min 1 L VSWR 1 L 16 Ex2 Determine VSWR and the locations of maximum and minimum voltages for a) matched load b) ZL = 100 , Z0 = 50 17 c) short circuit d) open circuit 18 Source and loaded transmission lines L S ZS Z 0± ZL + VS IN Z in Z 0 in (d l ) 0e 2 jl Z in Z 0 OUT Z S Z0 S Z S Z0 out S e j 2l 2Z (d ) Transmission coefficient: T (d ) 1 ( d ) Z (d ) Z 0 19 Power transmission of a transmission line 1 * Average power: Pav Re VI 2 2 2 1 S 1 VS 2 For lossless line: Pin 1 in 2 8 Z0 1 S in For lossless and matched condition: we have Pavs, maximum available power provided by the source. 2 1 VS Pin Pavs 8 ZS P W Power in decibel: P dBm 10 log 1 mW 20 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. 21 Input impedance matching + ZS + VS - V Zin - Optimal power transfer requires conjugate complex matching of the T.L. to the source impedance: Zin = ZS* Similarly for output matching: Zout = ZL* 22 The Smith Chart a graphical tool to analyze circuit impedance design of matching networks computations of noise figures, gain, and stability circles. 23 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. 24 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. 25 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: a) ZL = 50 , b) b) ZL = 25+j75 , and c) c) ZL = 40 + j20 , are sequentially connected to a 50 transmission line. Find the reflection coefficients and the SWR circles. 26 Admittance transformation 1 Yin yin Y0 yin Z0 Rotations by 180 degrees convert the impedance to the admittance representation. Y-Smith chart ZY-Smith chart 27 Parallel and Series Connections (1) Parallel connection of R and L elements Yin R L ZS AC Z0 yin (L ) g j L 28 Parallel and Series Connections (2) Parallel connection of R and C elements Yin R C ZS AC yin (L ) g jZ0LC 29 Parallel and Series Connections (3) Series connection of R and L elements Z in L R ZS AC L L zin (L ) r j Z0 30 Parallel and Series Connections (4) Series connection of R and C elements Z in C R ZS AC 1 zin (L ) r j LCZ0 31 Ex6 of a T-network (operated at 2 GHz) ZL C=1.91 pF L=4.38 nH C= 2.39 pF L=3.98 nH R=31.25 Z in T-type network 32