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The Twisted-Pair Telephone Transmission Line

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The Twisted-Pair Telephone Transmission Line High Frequency Design From November 2002 High Frequency Electronics Copyright © 2002, Summit Technical Media, LLC TRANSMISSION LINES The Twisted-Pair Telephone Transmission Line By Richard LAO Sumida America Technologies elephone line is a This article reviews the prin- balanced twisted- ciples of operation and Tpair transmission measurement methods for line, and like any electro- twisted pair (balanced) magnetic transmission transmission lines common- line, its characteristic ly used for xDSL and ether- impedance Z0 can be cal- net computer networking culated from manufactur- ers’ data and measured on an instrument such as the Agilent 4395A (formerly Hewlett-Packard HP4395A) net- Figure 1. Lumped element model of a trans- work analyzer. For lowest bit-error-rate mission line. (BER), central office and customer premise equipment should have analog front-end cir- cuitry that matches the telephone line • Category 3: BWMAX <16 MHz. Intended for impedance. This article contains a brief math- older networks and telephone systems in ematical derivation and and a computer pro- which performance over frequency is not gram to generate a graph of characteristic especially important. Used for voice, digital impedance as a function of frequency. voice, older ethernet 10Base-T and commer- Twisted-pair line for telephone and LAN cial customer premise wiring. The market applications is typically fashioned from #24 currently favors CAT5 installations instead. AWG or #26 AWG stranded copper wire and • Category 4: BWMAX <20 MHz. Not much will be in one of several “categories.” The used. Similar to CAT5 with only one-fifth Electronic Industries Association (EIA) and the bandwidth. the Telecommunications Industry Association • Category 5: BWMAX <100 MHz. Have tightly (TIA) define some North American standards twisted pairs for low crosstalk. Used for fast for unshielded twisted-pair cables and classify ethernet 100Base-T, 10Base-T. #22AWG or them into categories based upon deployment #24 AWG wire pairs. requirements. EIA/TIA-568A, for example, is a • Category 6: BWMAX <350 MHz. Under devel- commercial building telecommunications opment. wiring standard. • Category 7: BWMAX <600 MHz. Under development. • Category 1: BWMAX <1 MHz. No perfor- mance criteria. Good only for analog voice A Bit of Derivation communication, RS232, RS422 and ISDN, Recall a transmission line model composed but not for data. Used in POTS (Plain Old of discrete resistors, capacitors and inductors. Telephone System). A length x of transmission line can conceptu- • Category 2: BWMAX <1 MHz. Used in tele- ally be divided into a large (infinite) number of phone wiring. increments of length ∆x (dx) such that the 20 High Frequency Electronics High Frequency Design TRANSMISSION LINES series and shunt Rs, Ls and Cs are given as shown in Figure 1. G (=1/Rshunt) could have been given a resistor “Rshunt” label, but analytically it is often simpler to deal with reciprocal resistance values for shunt resistors. There are many old and new textbooks on electricity and magnetism in which transmission line derivations can be found [for Figure 2 · A balanced line model in three increments. example, References 6, 7, 8 and 9]. A real twisted pair telephone line has resistance, capacitance and inductance distributed equally between both wire members of the pair as shown above (Figure 2) in a segment of line approximated by three segments of length ∆x. For math- ematical modeling, however, the single-sided model of Figure 1 is easier to comprehend and analyze. An ideal transmission line would be lossless; that is, R = 0 and G = 0 (i.e., Rshunt = ∞), as represented in Figure 3. For such a line, the characteristic impedance is given by the familiar formula Figure 3 · Lumped element model of an ideal lossless line. L′ L Z = = (1) 0 C′ C value due to frequency dependent dielectric loss. Consequently, |Z0| measured at one frequency will not However, for a real transmission line, the characteris- necessarily be the same as |Z0| measured at another fre- tic impedance is quency. This deviation is especially notable at the audio frequencies (up to 100 kHz). As a rule of thumb, it is advisable to conduct characteristic impedance measure- RiL′ + ω ′ Z = (2) ments at frequencies greater than or equal to 100 kHz, 0 GiC′ + ω ′ where |Z0| begins to approach an asymptotic value. From Equation 2, we can calculate the complex valued Z0. where Introduce u as a temporary additional variable to avoid clutter and confusion in the derivation. Then, ′ = dR ′ = dL ′ = dC ′ = dG R ,,,L C G ′ + ω ′ ()′ + ωω′ ()′ + ′ dx dx dx dx ==2 RiL= RiLGiC uZ0 GiC′ + ω ′ GC′222+ω ′ are the longitudinal derivatives of resistance, inductance, (3) capacitance, and admittance along the line. Keep in mind ()RG′′+ω 2 L′CiLGRC′′ + ω ()′′− ′′ that Z is a complex valued function. The values of these = 0 GC′222+ω ′ parameters and their tolerances can be found tabulated in the catalogs of cable manufacturers, usually under the 2 designation “RLCG parameters.” Some engineers make ()RG′′+ωω2 LC′′ + 2 () LG′′− RC′′2 uu* == u 2 impedance magnitude measurements at 1 kHz and are 2 (4) puzzled why their measured values do not agree with ()GC′222+ω ′ manufacturers’ data. Each of the parameters R', L', C', and G' is frequency dependent. For example, R' will change in value due to skin effect, and G' will change in 22 High Frequency Electronics High Frequency Design TRANSMISSION LINES 2 ()RG′′+ωω2 LC′′ + 2 () LG′′− RC′′2 ∴=u (5) GC′222+ω ′ 2 4 ()RG′′+ωω2 LC′′ + 2 () LG′′− RC′′2 == Zu0 (6) GC′222+ω ′ We now can use Equation 6 to calculate the magnitude of Z0. Remember that Z0 is a complex quantity, Z0 = R0 + iX0 (7) Figure 4 · A plot of characteristic impedance versus fre- where R0 is the real part (resistance) of Z0 and X0 is the quency. imaginary part (reactance) of Z0. If X0 > 0, then the reac- tance is inductive, and if X0 < 0, then the reactance is capacitive. Computer Calculations A computer program (listing in Table 1), written in Frequency Dependence of Transmission Line TrueBASIC, calculates and tabulates values of |Z0| and Parameters plots them as a function of frequency. As frequency gets There are a number of continuous equations that can larger and larger, the value of |Z0| approaches an asymp- be used to express R', L', C', and G' as functions of fre- totic value. The source code is also useful as pseudo-code quency. Here, we use some equations and parameter val- for writing a Z0 program in any programming language. ues found in Reference [4] and consider #24 AWG twisted This impedance is that of a real transmission line; that is, pair copper wire. the transmission line has electrical leakage characterized by the conductance per unit length, G'. The program per- forms these functions: 1 Rf′()= (8) 11 + • Use Equations 8 through 11 to compute R', L', C' and G' 4 ′42+ 4 ′42+ RafRaf0cc0 ss at specific frequencies. • Substitute these values into Equation 6 to obtain the b magnitude of the characteristic impedance for various f LL′ + ∞′ frequencies. 0 fm Lf′()= (9) • Plot the magnitude of cable characteristic impedance as b f a function of frequency (as in Figure 4). 1 + fm Measuring Characteristic Impedance Z0 Use an impedance measuring instrument such as a − ′ = ′ + ′ ce Cf() C∞ Cf0 (10) Hewlett-Packard (Agilent) HP4395A network analyzer and HP43961A impedance test adapter. Manufacturer’s specification guarantees impedance measurement accu- ′ = ′ ge Gf() Gf0 (11) racy down to 100 kilohertz; however, lower frequencies can be included with reduced accuracy, especially to com- pare data with that in Figure 4 or to focus attention on Similar equations can be found in Reference [3], [5] the lower frequency range occupied by ADSL signals. For and in a number of other sources. example, we set the lower frequency to 10 kilohertz to 30 In the above equations we use primes to express kilohertz in our laboratory. For the purposes of this paper, derivatives. More often than not, in texts on electromag- however, set up the instrument to measure magnitude of netic theory, transmission lines and in other literature impedance from 100 kilohertz to 1 megahertz. Measure (and in References [3] and [4]), the primes are omitted open-circuit impedance, Z (with the other end of the line and the derivative sense is understood. open; that is, not connected to anything). Then measure 24 High Frequency Electronics High Frequency Design TRANSMISSION LINES the short-circuit impedance Z' (with ! The name of this TrueBASIC program is Z0.TRU. ! The purpose of this program is to compute the characteristic impedance, Z0, the other end of the line short-circuit- ! of a twisted pair wire line as a function of frequency. The resistance per ed). The characteristic impedance of ! unit length, capacitance per unit length, and inductance per unit length the line, Z0,is ! as a function of frequency are first computed. = ′ ! The values provided below in the initialization of this program Z0 ZZ (12) ! are found in Reference 4. ! r = dR/dx, l = dL/dx, c = dC/dx, g = dG/dx, etc. R', L', C', and G' all vary with fre- ! #24 AWG twisted pair copper wire for telecommunications. quency. Hence, Z0 also varies with frequency. However, Z (f) monotoni- LIBRARY "D:\TB Gold\tblibs\sglib.trc" 0 cally decreases and approaches an asymptote, Z . If only one or two fre- DIM x(2,1000), y(2,1000), legends$(2) 0 MAT READ legends$ quencies are available for measure- DATA A,B ment, choose frequencies greater option nolet than or equal to 100 kHz. Consider the following twisted- r0c = 174.55888 ! Ohms, resistance/km. pair cable: r0s = 1E20 ! infinity Berk-Tek telephone cable, part ac = 0.053073481 number 530462-TP, Category 3, PVC, as = 0 non-plenum, # 24AWG, 4 unshielded 5 twisted-pairs (UTP), Nominal charac- l0 = 617.29539E-6 ! Henries, inductance/km.
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