IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. IM-18, NO. 3, SEPTEMBER 1969

Time Domain Reflectometer Return Loss MeasurementsGEORGE D. CORMACK, MEMBER, IEEE, AND RONALD PAUL MANNING, MEMBER, IEEE

Abstract-Equations are derived for the exact identification i0,t) 121(tof a transmission line discontinuity that can be represented by an 6, Z 2'equivalent circuit consisting of one or more cascaded, series- 1 t B^inductive, parallel-capacitive, and series-transmission-line ele- Bg v,I , vI0(t) [c:D V2 i(t) V2(t) Z2ments. Equations are presented for determining frequency domain |return loss from time domain reflectometer data and the frequencylimitations for this method of discontinuity evaluation are discussed. Fig. 1. Measurement circuit.

I. INTRODUCTIONV T ARIOUS manufacturers of time domain reflee-

tometer equipment have published information[1], [2] on methods for evaluating transmission-

line discontinuities. This paper introduces more accuratemethods for signal interpretation and component identi-fication. In particular, the area of the return pulse isfound to be of major significance. Equations are derivedthat relate the area to the type and magnitude of dis-continuity and also to the return loss in the megahertzregime.

II. THEORY

The circuit shown in Fig. 1 can represent the time do-main reflectometer measuring circuit commonly used tostudy the junction between transmission lines having dis-similiar characteristic impedances, or the discontinuitydue to an electrical connector, or the discontinuity dueto a mismatched input or output impedance of a device.Each of these cases will be considered separately. Sincetime domain reflectometry is invariably used to measurehigh-frequency effects, the treatment to follow will notconsider the effects due to components that result in low-frequency effects, for example, due to coupling capaci-tors or to inductances to ground.

A. Junction Between Transmission Lines

The relationships between the voltages and currents atthe two-port network shown in Fig. 1

V10(s) = AV2i(S) + B12 i (s) (1)

I10(s) = CV2i(s) + DI2i(s) (2)reduce to A = D = 1, B = C = 0 for a network thatrepresents a junction between dissimilar transmissionlines. The junction capacitance has been neglected when

Manuscript received December 2, 1968.G. D. Cormack is with Carleton University, Ottawa, Ont.,

Canada.R. P. Manning is with the Research and Development Labora-

tories of Northern Electric Company, Ottawa, Ont., Canada.

B = C = 0, but will be considered in Section II-B. Theo and i subscripts refer to output and input quantities.The sending-end voltage-transfer function is now

V, (s) Z_Vg(s) Zi + Zg

I+ ((Z2-Zl/Z2+Z,)) exp (-2- 1, )t1 +((Zl-Z,) (Z2-Zl)/(Zl +Z,) (Z2 +Z,)) exp (-2-y1l,))

(3)or expanding in an exp (-2y1l1) power series,

Vli(s) Z f + (Z2- 2Z

Vg(s) Zi + Zg t Z2 + Zl Zg + Z,

*exp (- 2,yl ) + * -.- (4)

The method of calibrating the vertical sensitivity ofthe reflectometer determines the method of signal inter-pretation. For example, if a reflectometer having a char-acteristic impedance of 50 ohms (Zg) is connected to acable having an impedance of 75 ohms (Z1) and thevertical sensitivity vernier is adjusted so that the incidentpulse amplitude is 10 cm when the vertical sensitivitycontrol is set at 0.1 p/cm, then p, the amplitude of the re-flected signal as observed with the reflectometer that isdue to a second cable connected to the free end of thefirst, is the ratio of the magnitude of the second term in(4) to that of the first term, or

(Z2-±)( 2Z9VZ2 + Zl Zg + Z1} (5)

when the line is lossless.The alternative reflectometer calibrating procedure of

adjusting the vertical sensitivity vernier to give an inci-dent pulse amplitude of 10 cm when the vertical sensi-tivity control is set at 0.1 p/cm and when a cable ortermination of impedance Zg is connected to a reflectom-eter having a generator impedance of Z, gives differentsignal interpretation results. In particular, if the cable ortermination of impedance Z9 is removed and replaced

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185CORMACK AND MANNING: TIME DOMAIN REFLECTOMETER RETURN LOSS MEASUREMENTS

with a cable of impedance Z1 that is further connected toa cable of impedance Z2, the amplitude of the reflectedsignal due to the junction between these two cables is theratio of the magnitude of the second term in (4) to 2 or

_+4Z,2)p =Z +Z\(Zl +Zz)21 (6) Fig. 2. Typical connector circuits.

The factor of 1 accounts for the amplitude loss due tothe matching of the reflectometer during calibration.

Equation (6) can be reexpressed as

= k7wherZ + Z,)

where

(8)

When Z2 is the desired quantity, (7) takes the form

Z2= (kF P)Z (9)

B. Discontinuity Due to an Electrical Connector

An electrical connector usually possesses an equiva-lent circuit that is a cascade connection of two-port net-works of the type shown in Fig. 2. The composite two-port network that is equivalent to the cascaded net-works has an ABCD matrix that is the product of theABCD matrices of the constituent networks. For examplethe ABCD matrix of a cascaded inductance-capacitance-inductance circuit is

L + s2LIC s3L2L,C + s(L, + L2)

sC 1 +S2lL2C(10)

where L1 is the series inductance on the sending end ofthe circuit and L2 the series inductance at the load end.The circuit in Fig. 1 now has a sending end transfer

function of

Vl(s) _ Zi 1 + r, 2Z9Vg(S) Z1 + Z0 0 Zi + Z0

*exp (-2,yl) + }. (11)

when expressed as a power series in exp (-2y'l1). Thevalue of r10 is

io = Z2A + B - Zl(Z2C + D)Z2A + B + Zl(Z2C + D) (12)

The reflectometer waveform for an electrical connectoris often of a complicated shape and indicates that indeedthe connector is equivalent to a number of cascaded two-ports of the type shown in Fig. 1. In the following analy-

sis both this general case and the simpler case of single-component discontinuities are considered. If it is as-sumed that the reflectometer vertical calibration is per-formed with a matched output, that is the second methoddescribed above, the reflected signal due to the connectoris given by the ratio of the magnitude of the second termin (11) to I. Thus,

p(s) = k(s)riP(s) exp (-2'y1(s)l)V0(s) (13)

where the value of k is given in (8).This equation is the Laplace transform of the signal

that is reflected frmo the connector. The value of r10 isgiven by (12) where Z1, Z2 and ABCD are functions of s.The value of the ABCD coefficients for cascaded connec-tions of any number of the two-port networks shown inFig. 2 are

A = 1 + o(s2),m

B = s EjLi +o(s3),1=1 (14)n

C = sECi +o(.s3),i=l

D = 1 + 0(S2).

The final value theorem applied to the integral of the re-flected signal yields, from (12) to (14),

p(t) dt = lim0 s->O S

= lim {k(s)rio(s) exp (-2'y1(s)1j)}s-iO

(15)1<[ + a, + as + -...

where the input incident waveform v, (t) has been as-sumed to have a finite risetime, but an asymptotic valueof unity. This analysis therefore considers the case of anonideal step input signal that, in its most general form,is expressible as

V,(s)- + a0 + a,s + .--.S

(16)

Denoting the integral in (15) as I and considering onlycontributory terms,

z 2

k = 1 1 9.

Z, + Zg

IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, SEPTEMBER 1969

k m n

I_= k 2 Li- Z1Z2 E cjZi + Z2 i=1 i=1

+ k lim Z2-Z1 (17)s o S(Z2 + Z,)

where it is assumed that Z1, Z2, and Z, are real quantities.The last term of (17) represents the step in the timeresponse of magnitude k (Z2 - Z1) / (Z2 + ZO) that occursdue to the difference in transmission line impedance atthe connector. This step can usually be experimentallyseparated from the first term.

Equation (17) is a general equation that includes alarge number of simpler cases. For example, if Z1 = Z2,then the pulse area due solely to the connector is

m n

_ z2 ZCJtICkj 2Z= ki (18)

Further, if the connector consists of a single parallelcapacitor,

-Z1C1cI = ZiCkc 2

(19)

or a series inductor,

kLIC = 2Z (20)

Note that (18) or (17) predict that if the return pulse,due solely to the connector, has a net negative area, theconnector behaves as a parallel capacitor and that if thenet area is positive, the connector behaves as an induc-tor.

If the connector acts as a short length of lossless trans-mission line of characteristic impedance Z3, propagationconstant y-, length 1. and a one-way transit time of T,the sending end transfer function is

VIi(s) - Z + {

Vg(s) Z, + Zz, Z, + Zg

Z1 1 - r, exp (-27323)\3), Z3 X1 + F1 exp (-27t313) J

1+ Z, {1- IP exp (2,y,13)8Z3 \1 + r, exp (27)313))

when expressed as a power series in exp (-2yj1j), where

rp = Z- Z3 (22)z2 + z3

By expanding (21), it can be shown that if Z1 = Z2, thearea of the reflectometer pulse due to the connector is

I k{32 } (23)

when the reflectometer is calibrated under matchedconditions. r can be determined from the reflectometerwaveform as one-half the duration of the pulse producedby the short length of line. Alternatively, r can often beobtained from geometrical considerations if the reflec-tometer signal is not a clearly resolved square pulse. Forexample, r= V\L3C313 where L3 and C3 are, respectively,the inductance and capacitance per unit length. Equation(23) is of value for determining the characteristic im-pedance (Z3) of very short lines or of connectors.

Regardless of the waveform of the reflectometer sig-nal, the important assessment parameter for a connectoris often not the inductance or capacitance or impedanceand length it possesses but rather the return loss it intro-duces. The return loss due to a connector defined in termsof the reflection coefficient is

RL = -20 log10 F . (24)From (12), letting r Frl, s = jw, Z1 = Z2 and byconsidering only the low-frequency terms in (14),

log10t - zl C)

RL-E -20 lo -2Z2

or from (18)

RL = -20 log1

(25)

(26)

Equation (26) is valid for connectors possessing equiva-lent circuits containing any sequence of cascaded cir-cuits consisting of parallel capacitors, series inductors,and series lossless, electrically short, transmission lines.The latter consists of distributed inductance and ca-pacitance circuits and the effects thereof are thus in-cluded in (25) and (26).

C. Mismatched Input or Output of a DeviceThe circuit shown in Fig. 1 is an equivalent circuit for

this case if we consider the device to include all com-ponents on the load side of v1l (t). For example, a devicehaving an input impedance consisting of an inductancein series with a capacitor-resistor tank circuit can bemeasured with the time domain reflectometer and (17)or (18) and (26) applied directly.

FREQUENCY LIMITATIONS

Over what frequency range are the preceding equa-tions and methods valid? One upper frequency limit forapplicability of the return loss equations is the frequencyfor which the wavelength is appreciably longer than anyof the dimensions of the discontinuity. For example, (26)is valid for a discontinuity comprising a short length oftransmission line only if the wavelength of the sinu-soidal signal being considered is appreciably longer thanthe length of the line. An additional high-frequency limi-

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CORIMACK AND MANNING: TIME DOMAIN REFLECTOMETER RETURN LOSS MEASUREMENTS

tation that is usually much higher than the dimensionallimitation is that imposed by the resolving power of thereflectometer. An approximate resolving power criterionis that the upper frequency limit is less than 0.35 timesthe reciprocal of the risetime of the incident pulse. Forexample 150 ps corresponds to 2.3 GHz. Equations (25)and (26) neglect two other effects that typically occur atGHz frequencies. Specifically, high-frequency leveling ofthe return loss characteristic is not included because a

m n

jw{Z, Li + 2 Ci}

term has been neglected in the denominator of (25). Also,resonance terms are neglected in the numerator. Theseeffects can be justifiably neglected so long as the returnloss is larger than about 10 dB and the frequency consid-ered in the calculation of return loss is less than anyresonant frequency exhibited by the discontinuity. Bothof these effects are usually negligible up to about 1 GHz.The lower frequency limit for the validity of (26) is

dependent upon the acceptable error. The difficulty withthe low-frequency extrapolation of time domain reflec-tometer data is that the reflectometer pulse currents are

surface currents, whereas if the connector is used in thekilohertz region or lower, the current is distributedthroughout the volume of the conductors. The change incurrent distribution does not change capacitance valuesbut alters the resistive loss, which is neglected in thistreatment, and of most significance, it changes the in-ductance. The change in inductance due to this skin effectcan be calculated for a coaxial discontinuity by a methodbased on Skilling's analysis [3]. A modified plot of theupper graph given in his Figs. 7-9 is shown in Fig. 3.Here a is the radius of the inner conductor in meters, 8 is

the skin depth of the inner conductor in meters, LdC is thevery-low-frequency inductance due solely to flux linkageinside the inner conductor and Lac is the value of LdC athigher frequencies. From the graph it is obvious that theinternal inductance ratio La(/Ldc is related to a/8 by

Lac=

21 (27)LdC a

when 8 << a. A more exact expression calculated startingwith the assumption that the current distribution is uni-form within an annular shell of thickness 8 on the innerconductor yields

L = a (1 + 6a + ° 2-) (28)Lde 6a a2

Thus, (27) has an error of less than 5 percent when a/8 isgreater than 3. The error involved in assuming that (26)is valid for low frequencies can be evaluated by compar-

ing the return loss for a standard type of discontinuity,calculated exactly, to the return loss for this discontinu-ity as determined by using the methods outlined in this

100

.01 1 3

LAC LD C

Fig. 3. Skin effect internal inductance ratio for the center con-ductor of a coaxial cable.

paper. For example, if the discontinuity consists of a

length 13 of coaxial line over which there is an enlargedinner conductor radius of a3 whereas the normal conduc-tor radius is a,, the relevant equations become

2A In b +± 232r a3 87ra,

27rc-C3

In (b/a3)

In b/a3 lMo I 6lZ 2a7r + 4a3 In b/a3 +

In b/a1 A/ {, 6Z- 27r C l 4a, In b/a,

(29)

when a3/r > 3, and where b is the inner radius of theouter conductor. Junction capacitance [4] and skin effecton the outer conductor [3] have been neglected in theseequations.From (23) and (29)

kl3 E Iln b/a3 _ In b/a1 + a ( 1)Ic 2 In bla ln b/a3 a3 2 ln b/a,

(lIn b/a3 + Ib/a3) + o(62} (30)

The frequency-independent terms in this equation are

those that are measured by the reflectometer incidentpulse. The terms that contain 8 are frequency-dependentterms since

3 =_ IO2 (31)

where cr is the electrical conductivity of the inner con-

ductor in mhos/m, is the angular frequency in rad/s,and ,u is the permeability of the inner conductor when 8 isin meters. The terms that contain 8 in (30) are notmeasured by the reflectometer but should be included ifthe reflectometer measurements are interpreted in termsof low-frequency return loss. The error in I, caused bynot taking into account skin effect is equal to the magni-

x~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

I I

x~~~~~~~

187

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IE:EE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, SEPTEMBER 1969

Fig. 4. Oscillogram of reflection from an experimental connectorobserved with time domain reflectometer. Vertical scale: 0.02p/div.; horizontal scale: 0.4 ns/div.

tude of the frequency-dependent terms in (30) divided bythe frequency-independent terms. When a, = 0.0536 inch,b = 0.1875 inch, and a3 = 0.060 inch, this ratio is-462 S. Thus the error involved in using (26) when Icis the measured pulse area, which is uncorrected for skineffect in copper, is -30.85/f where f is the frequency inhertz. Thus the percentage error if f equals 100 kHz is9.7 percent or if f equals 30 kHz, 17.8 percent. The valueof al/3 is 3.5 or larger, when f > 30 kHz. The value forI, in this example, calculated from (30) but neglectingskin effect (letting 8 = 0), is -7.96 x 10-12 second timesreflection coefficient, where it is assumed that k = 1 and13 = 1 inch. The return loss presented by this disconti-nuity at 1 MHz is therefore, from (26), 86 dB. Thereturn loss at 30 kHz, neglecting skin effect, is 116.4 dB,or including the first-order skin effect terms given in (30),118.2 dB. It is evident that skin effect is usually of no

significance when the reflectometer method of determin-ing return loss is used.

In conclusion, the area of the time domain reflectome-ter waveforms for typical discontinuities can be reliably

interpreted in terms of component values or return lossvalues over a frequency range from approximately 30kHz to the gigahertz region. The lower frequency limit isdependent upon the permissible error.

MEASUREMENT

The time domain reflectometer waveform shown inFig. 4 was obtained with the Hewlett-Packard 1415A.The reflectometer was calibrated under 50-ohm matchedconditions. The discontinuity producing the waveformshown was of an experimental type located at the mid-point of two short lengths of 75-ohm cable. One of thefree ends of the cable was connected to the reflectometerand the other open-circuited. The area of the pulseshown is Io = -7.5 x 10-11 second times reflection co-efficient, and the return loss is given by (26). In particu-lar the return loss at 100 kHz is 86 dB, at 1 MHz, 66 dB,and at 10 MHz, 46 dB, for this particular connector usedwith 75-ohm cables.An equivalent circuit for this connector consists of at

least a parallel capacitor cascaded with a series induct-ance and another parallel capacitor. The shape of thewaveform of the return pulse could be altered substan-tially by changing the risetime of the incident pulsewith the "bandwidth" control. The area of the pulse wasobserved to be independent of the "bandwidth," in con-formity with the theory presented here and in particularproviding justification for the use of (16).

REFERENCES[1] Hewlett-Packard Appl. Note 67.[2] Tektronix Service Scope 45.[3] H. H. Skilling, Electric Transmission Lines. New York:

McGraw-Hill, 1951.[4] J. R. Whinnery, H. W. Jamieson, and T. E. Robbins, "Co-

axial-line disconthiuities," Proc. IRE, vol. 32, pp. 695-709,November 1944.

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