======================CABLE_IMPEDANCE====================== Cable impedance at RF frequencies no longer behaves like regular old wire. As approaches about 1/10 wavelength of signal cable impedance Characteristic Impedance designated Zo or"Zed nought" Zo = sqrt((R + 2*pi*f*L )/(G + j*2*pi*f*c) ) R = series resistance in ohms per length (DC resistance) G = The shunt conductance in mhos per unit length j = phase angle of +90 degres(imaginary number) pi = 3.1416 L = Cable inductance per unit lenght C = Cable capacitance per unit lenght sqrt = square root function G is small enough For materials that it can be neglected low frequencies L is so small compared with R that can be neglected Zo = sqrt ( R / (j * 2 * pi * f * L)) Polyvinyl chloride and rubber decrease somewhat in capacitance as frequency increases polyethylene polypropylene, and Teflon* do not vary significantly. When f large enough R and G may be neglected a Zo = sqrt ( (j *2*pi*f*L) / (j*2*pi*f*C) ) Which can be simplified to form: Zo = sqrt ( L / C ) -------------------------------------------------------------------------------------- coaxial cable: impedance = (138 / e^(1/2)) * log (D/d) log = logarithm of 10 d = diameter of center conductor D = inner diameter of cable shield e = dielectric constant (= 1 for air) impedance will be typically between 20 and 150 ohms. -------------------------------------------------------------------------------------- balanced pairs impedance = (276 / e^(1/2)) * log ((2D/d) * (1 + (D/2h)^2))^(1/2)) log = logarithm of 10 d = wire diameter D = distance between wires in pair e = dielectric constant (= 1 for air) h = distance between balanced pair and ground valid for unshielded balanced pair when D and h are order of magnitude larger than d. If twisted pair is far away from ground (h is nearly infinite), effect of ground is neglegtible and impedance of cable can be approximated simpler formula impedance = (276 / e^(1/2)) * log ((2D/d) twin line Zo typically between 75 and 1000 ohms old telephone pair impedance telephone poles in air has characteristic impedance of around 600 ohms. The telephone and telecommunication cables in use have typically a characteristic impedance of 100 or 120 ohms. -------------------------------------------------------------------------------------- long coaxial cable ZO = sqrt(Leff / C) freq = 1/sqrt(L*C/2*PI) _ _ _ _ _ _ _ _ _ _ _ _ / \/ \/ \ ..\ / \/ \/ \ / \/ \/ \ / \/ \/ \ ___ | () () | : / | () () | | () () | | () () | | |__| |____:___| |___| |___| |__.... |___| _|_ : _|_ _|_ _|_ ___ : ___ ___ ___ _|_ :.\ _|_ _|_ _|_ /// / /// /// /// relationship exists which makes determination of Zo rather simple with proper equipment. impedance of a length of cable is measure with far end open (Zoc), and measurement is repeated with far end shorted (Zsc), Zo = sqrt ( Zoc * Zsc ) Zoc = impedance length of cable is measure with far end open Zsc = impedance length of cable is measure with far end shorted speed_light_60%_70% Most wires 60 to 70 percent of speed of light, Normal video signal rarely exceed 10 MHz. -------------------------------------------------------------------------------------- minimize attenuation in coax if you minimize expression (1/d + 1)/ln(1/d) where d is ratio of inner conductor diameter to outer conductor ID D/d = 3.5911 is close __-----__ / d_inch \ Impedances inside coax / ___ \ | / ^ \ | | | |_ |__\| \ \___/ // \ D_inch/ \__ __/ ----- air insulated line corresponding impedance is about 76.71 ohms, but if line is insulated solid polyethylene minimum attenuation is at about 50.6 ohms. -------------------------------------------------------------------------------------- 50 ohm coaxial The most typical coaxial cable impedances used are 50 and 75 ohm coaxial used in video applications, in CATV networks, in TV antenna wiring and in telecommunication applications. 600 ohms typical impednace for open-wire balanced lines for telegraphy and telephony. A twisted pairs 22 gage wire about 120 ohms f Twin lead antenna systema are 300 ohms to match folded dipole free space impedance (However, when that folded dipole is part of a Yagi (beam) antenna, impedance is usually quite a bit lower, in 100-200 ohm range typically.). -------------------------------------------------------------------------------------- Why 50 ohm coax ? Stand coaxial line impedance for r.f. power trans in U.S. almost exclusively 50 ohms. value chosen given in paper by Bird Electronic Corp. Different impedance optimum for different parameters. 30-ohm Maximum power-carrying capability occurs at a diameter ratio of 1.65 corresponding to 30-ohms 60-ohms Optimum diameter ratio for voltage breakdown is 2.7 corresponding to 60-ohms impedance (incidentally, standard impedance in many European countries). Power carrying capacity on breakdown ignores current density which is high at low impedances such as 30 ohms. Attenuation due to conductor losses alone is almost 50 0gher at that impedance than at 77 ohms minimum attenuation impedance of 77 ohms (diameter ratio 3.6). This ratio,limited to one half maximum power of 30-ohm In early days, microwave power was hard to come by and lines could not be taxed to capacity. Therefore low attenuation was overriding factor leading to selection of 77 (or 75) ohms as a standard. resulted in hardware of certain fixed dimensions. When low-loss dielectric materials made flexible line practical, line dimensions remained unchanged to permit mating with existing equipment. The dielectric constant of polyethylene is 2.3. Impedance of a 77-ohm air line is reduced to 51 ohms when filled with polyethylene. Fifty-one ohms is still in use though standard for precision is 50 ohms. attenuation is minimum at 77 ohms; breakdown voltage is maximum at 60 ohms power-carrying capacity is maximum at 30 ohms. 50 ohm coax 50 ohm coax mechanically look good, Since almost any coax that looks* good for mechanical reasons just happens to come out at close to 50 ohms anyway, was natural tendency for standardization exactly 50 ohms. -------------------------------------------------------------------------------------- board traces Impedance of circuit board traces Z = (87 / sqrt( Er + 1.41 ))*ln( (5.98*h)/(0.8*w + t)) Er = dielectric constant(4.8 for typ fiberglass board) h = height of dielectric board thickness t = thickness of copper material in microstrip w = width of copper material in microstrip dielectric constant, Er typical 0.062 fiberglass board is 4.8. Using a trace thickness of 0.00134" gives a line width of 109 mils for a 50 ohm microstrip. W_width ____ T_thickness _____|____|______ Dielectic er h_height _________________ _________________ W = 6mil h = 4mil t = 1mil er= 4 Microstrip Z0 = (87/sqrt(er +1.4141))*ln(5.98*h/(0.8*W +t)) ...53 tpd = 85*sqrt(0.475*er +0.67) (ps/in) ...136ps/in 53.5ps/cm C0 = 0.67( er +1.414)/ln(5.98*h/(0.8*W+t) ) (pF/in) ...2.56pF/in .1pf/cm L0 = Z^2*C0 = 5071.23*ln(5.98*H/(0.8*W+t)) (pH/in) ...7185pH/in 2838pH/cm -------------------------------------------------------------------------------------- differential pairs When routing circuit board traces, should have same length trace. should also be as short as possible. Impedance matching between different impedances If two cables with different impedances are connected togerther or a cable is connected to a source which has different impedance then some kind of impdance matching is needed to avoid signal reflections in place where cables are connected together. Using transformer for impedance matching The most classical method for matching different impedances is to use a matching transformer with proper impedance tranfer ratio. -------------------------------------------------------------------------------------- impedance tranfer ratio of a transformer Za / (Na^2) = Zb / (Nb^2) Za = input impedance Na = number of turns on input coil Zb = output impedance Nb = number of turns on output coil The equation can be converted to format: Zb = Za * (Nb/Na)^2 From that equation you can see that Nb/Na is same as transformer voltage transferrign ratio between primaty and secondary. This means that when you know that ratio you can use equation without knowing exact turns ratio. Impedance matching netweork usign resistors The matching network shown below can be used to match two unequal impedances, provided that Z1 is grater than Z2. ____ ----|____|---+--------- R1 | | | Z1 | | R2 Z2 |_| | -------------+---------- The resistor for this circuit can be calxulated using following equations: R1 = Z1 - Z2*R2 / (Z2+R2) R2 = Z2 * sqrt(Z1) / (Z1-Z2) The table below will show some precalculated values for some most common interfacing situations: Z1 Z2 R1 R2 Attenuation (ohm) (ohm) (ohm) (ohm) (dB) 75 50 42,3 82,5 5,7 150 50 121 61,9 9,9 300 50 274 51,1 13,4 150 75 110 110 7,6 300 75 243 82.5 11,4 As you can see from table cost of simple resistor based impedance matching is quite large signal level attenuation in conversion process. -------------------------------------------------------------------------------------- ___\__ --> I ______|___/__|________________ ->()___________\|/______________ | |_/____| _/___ | | ________\_______|_\___|_____| | <-()_____________________|_______| <-- I |___\_| INDUCTANCE CANCELATION __ / | |__ _____ |__| | | | |*() () | | \_/\_/\_/ L1 | Lb M _ _ _ | Lb = L1 + L2 -2*M /*\/ \/ \ L2 | Lb = 0 if (L1 = L2 = M) __ | () () | | | |__| |_____| |__| ------------------------------------------------------------------------------------- ___\__ --> I _______ ______|___/__|______________| | ->()___________\|/_____________| | INDUCTANCE CANCELATION |_/____| _/___ |50 Ohms| ________\_______|_\___|_____| | <-()_____________________|_____| | <-- I |___\_| |_______| / ------------------------------------------------------------------------------------- Voltage dropping current \ -----> \ \____\ \______\ \ \ \ \ \ \ _ \ _ ->E \| \ \ _ \| \ \|/ _ /_ \ ^ _ /_ v H |_/ \ _V Voltage \ |_/ \ __/_ \ Pulse \ __/_ |_/ \ \ \ |_/ \ | _V | voltage \ \ - + \ \____\ \______\ ELECT_MAG_WAVE \ \ \ \ \ ---> \ _ \ _ | E \| \ \ _ \| \ \|/ _ /_ \ ^ _ /_ v H |_/ \ _V \ |_/ \ __/_ \ __/_ ->E |_/ \ \ |_/ \ \|/ | | v H \ \ \ \____\ \______\ \ \ \ \ current <------ Voltage increasing ------------------------------------------------------------------------------------- _____________________________ ->()____________________________) <- d_cm | D_cm Z_ohms sqrt(L/C) | _____________________________ ->()____________________________) L_cm Z_ohms = 276.0*log(2*D_cm/d_cm) C_pf/meter = 12.06/log(2*D_cm/d_cm) L_uH/meter = 0.920*log(2*D_cm/d_cm) w0 = sqrt(L*C) = sqrt(dL*dC*X^2) = X*sqrt(dL*dC) velocity = (2*pi()/sqrt(dL*dC) ------------------------------------------------------------------------------------- __-----__ / d_inch \ Impedances inside coax / ___ \ | / ^ \ | | | |_ |__\| \ \___/ // \ D_inch/ \__ __/ ----- L_R_C Coaxial(D_inch,d_inch) Z_ohms = sqrt(L/C) = 138*log(D_inch/d_inch)/sqrt(E) C_pf/ft = 7.36*E/log(D_inch/d_inch) L_uH/ft = 0.14*log(D_inch/d_inch)) Delay_ns/ft = 1.016*sqrt(E) Propagation_%_c = 100/sqrt(E) CutOffFreq_Ghz = 7.5/( sqrt(E)*(D_inch+d_inch) ) Dielectric Constant E TFE 2.1 ethylene propylene 2.24 polyethylene 2.3 cellular polyethylene 1.4-2.1 silicone rubber 2.08-3.5 polyvinylchoride 3-8 ------------------------------------------------------------------------------------- Single coaxial line <--d---> __________ / ________ \ / / \ \ / / ____ \ \ / / / \ \ \ / / / \ \ \ \ \ \ / / / \ \ \____/ / / \ \ / / \ \________/ / \__________/ <-----D-----> Z0 = (138/e^0.5) log_10(D/d) (6O/e^0.5) ln(D/d) E = dielectric constant = 1 in air ------------------------------------------------------------------------------------- Balanced shielded For D>> d, h>>d, <--h---> __________ / ________ \ / / \ \ / / \ \ / / __ __ \ \ _ / / / \ / \ \ \ d \ \ \__/ \__/ / / _ \ \ / / \ \ / / \ \________/ / \__________/ <-----D-----> Z0 =( 276/e^.5)*log( 2*vu*( (1-sigma^2)/(1+sigma^2) ) ( 120/e^.5)*ln( 2*vu*( (1-sigma^2)/(1+sigma^2) ) vu = h/d sigma = h/D ------------------------------------------------------------------------------------- Beads_dielectric e_1 _________________ _________________ | | | | __|_|____|_|____ _________________ | | | | __|_|____|_|____ __________________ ><|<-->| w s For lines A. and B., if insulating beads are used at frequent intervals call new characteristic impedance Z0_2 Zo_2= Zo/( (1+[(e1/e) � 1)*sqrt( W/S) ) ------------------------------------------------------------------------------------- Open 2-wire line in air __ __ _ / \ / \ d \__/ \__/ _ <---D---> Z0 = 120*acosh(D/d) ~276*log(2D/d) ~120*ln(2D/d) ------------------------------------------------------------------------------------- Wires in parallel near ground __ __ _ / \ / \ d \__/ \__/ _ ^ <---D---> |h _______________ V ////////////// For d< |h _______________ V ////////////// For d< where rho= D/d A= (1+0.405*rho^-4)/ (1�0.405 *rho^-4) B= (1+0.163*rho^-8)/ (1�0.163 *rho^-8) C~ (1+0.067 *rho^-12)/(l�O.067 *rho^-12) ------------------------------------------------------------------------------------- Balanced 4-wire <-d-> __ __ / +\ / o\ \__/ \__/ ^ \ / | \_/ D2 __ /| __ | / o\/ \/ +\ V \__/ \__/ <---D1-> d< __ __ | / +\ / +\ V \__/ \__/ <---D-> For d << D Z0= (173/e^0.5) log10(D/(0.933*d))