UNIVERSITY OF MICHIGAN 2262-97-T EA16Al ENGINEERING RESEARCH INSTITUTE........ ELECTRONIC DEFENSE GROUP TECHNICAL MEMORANDUM NO. 25 Engineering Library SUBJECT: The Coupled Impedance Method of Q Measurement BY: J. F. Cline DATE: January, 1956 SUMMARY This report is a reorganization and condensation of the theory underlying the earlier work of other investigators in the measurement of the Q of a resonant circuit, cavity, or antenna by the coupled impedance method. In this method, impedance measurements at several frequencies are taken on a coil which is coupled to an unknown resonant system, with-; out knowledge of the degree of coupling, and the Q of the unknown system is computed from these measurements. INTRODUCTION It is sometimes desirable to obtain the Q of a resonant circuit, an antenna, or a cavity resonator without making any physical connection to it. This can be done by taking measurements at several frequencies (at least two) on a coil or loop which is magnetically coupled to the circuit in question. It is not necessary to know the degree of coupling, provided this is held constant during the measurements, A cavity resonator or antenna can be represented by an equivalent circuit having the same Q as that of the resonator or antenna; as far as the analysis is concerned, this has the advantage that the Q can be expressed in terms of the values of the elements in the equivalent circuit as well as in terms of the stored

i5Z ~~l,~od 0z - /td, ' V;Z:C~ A ria lc;"01I=L1 43 2-Ih\C A.,~ ~~~~//~ _qv

-2-* and dissipated energies in the unknown distributed system. CIRCUIT ANALYSIS Consider the circuit of Fig. 1, where the unknown circuit is represented by the elements L2, R2 and C2. The loop equations are E = ZlI1 ZMI2 o... (1) = - ZMI1 + Z2I2 These can be combined to show that the impedance Z = E1/I1 seen by the source is. v. 2 - Z Zl - ZM/Z2 (2) where Z1 is the impedance of the primary circuit alone and the last term is a coupled impedance Zc = -Z/Z2 (3) In all these equations M is equal to +jod4, so that -ZM = 2-2 (4) The impedance Z2 is the total series impedance of the unknown secondary circuit:. Z2= R2+ j2 5) where X = DL2 - 1/oC2 (6) The coupled impedance given by (3) can be separated into real and imaginary parts: Zc = Rc + JXc (7) By substituting (4) and (5) in (3) and rationalizing the result, we obtain Rc = R2 (8) Xc = -2Mf k R - (9) R2+X2 R2

The square of the magnitude of Zc is equal to the sum of the squares of (8) and (9): Zc2 = 2 +x2 (R2 +2 (10)) R 2 2 2 (10) (p2 + x2) Beginning with this quantity written as an identity (J4M4+(R2 2 + X 2) + X22) (11) (p2 + X22)2 + X22 we then rearrange the terms to obtain k4M14X22 (_2 _a4M4(R22 + X2) (R2 + X2 (R 2 22)2 (R22 + X22)2 We next complete the square of the last two terms by adding the term d4M4/4R2 to both sides of the equation: ( a4M4X2+ W5M4 2 cu4M4(2 + X2a) _44 _ 41,, +...... __ + (13) (R22 + X22)2 (R22 + X22)2 (R22 + X22)2 4R22 4Rb22 This is then equivalent to f1 2 x2] 2 + F2R1 -M2 [ 2 (14) LR2 2 + x2 2 2 2R2 j L 2j(22 By comparison with (8) and (9), we can see that this is the same as X ]+ R - -2M2] L — ] (15) If w were constant, (15) would represent a family of circles in the plane of Rc and Xc. Actually, if the Q of the unknown circuit is sufficiently high, most of the significant change in Rc and Xc takes place at values of o quite close to the resonant value o:Oo Consequently, if (15) is rewritten with o in place of u, we have an equation

[X] + [ 22] 2- o2M2 ] (16) which is approximately correct when Q is high and which has the analytical advantage of representing a family of true circles in the RC-Xc plane. One of the family of circles described by (16) is shown in Figo 2. From inspection of the equation it is evident that all of the circles have centers on the Rc axis and pass through the origin. For each circle there is a point D where Rc = Rc mx and X, = O; by setting (9) equal to zero, we find that at D X2 = o. (17) For each circle there are two points where Rc = IXcl, located at the intersections of the circle with the lines Xc = +Rco These two points are labelled A and B in Fig. 2. By placing RC = 1X01 (18) in (9), we find that at A and B X2 =+ 2 (19) Because the secondary is a simple resonant circuit, the usual resonant circuit relationships apply: X2 = 0 when f = fo X2= -R2 when f = fl ' fo(1 - a/i2Q) (20) 2(0) X2 = +R2 2 fo(l + 1/2Q2) fo Q2 = (21) f2 - fl Because of (9), (17),and (19), we can identify the frequency f with point D, the frequency fl with point A, and the frequency f2 with point B. We then use (21) to find Q2.

99-LI-I dr Ot7-S-v z9zz E, Z L M L2 E1 TC2 FIG. I THE CIRCUIT USED IN THE MEASUREMENT, WITH THE UNKNOWN RESONANT CIRCUIT AT THE RIGHT +c" * X t X R 0 C' FIG. 2 THE CIRCLE iN THE Rc-Xc PLANE AS REPRESENTED BY (16)

MEASUREMENT PROCEDURE Based on the foregoing analysis, the measurement procedure consists of the following steps: 1. At each of several frequencies near resonance, a. measure the impedance Z = E/I1 b. measure Z, alone, by reducing M to zero, and c. subtract to obtain Zc. 2. Plot Zc in the complex impedance plane, as in Fig. 3 or Fig. 4, a. identify the frequency fo where Xc = 0 and Rc = Rc max, and b. identify the frequencies f1 and f2 at points A and B where the curve intersects the lines Xc = _Rc. 3. Obtain Q2 from (21). SIMPLIFIED MEASUREMENT PROCEDURE It is evident from Fig. 2 that the points A and B previously defined by (18) could have been defined instead as points where Rc = 0.5 Rc max. In the case where it is known that R1 (see Fig. 1) is negligible in comparison with Re, it may be assumed that the resistive component of the measured impedance Z is equal to Rc. In this case, reactance values may be ignored entirely, and the frequencies fl and f2 may be defined simply as the frequencies where the resistive component of the measured impedance is equal to half its maximum value. The simplified measurement procedure then becomes: 1. At each of several frequencies, measure the resistive component of the impedance Z = E1/il. 2. Plot a curve of this resistance versus frequency and identify the frequencies fl and f2 where the resistance is equal to half of its maximum value. 3. Obtain Q2 from (21).

-7 -NUMERICAL EXAMPLE S Figure 3 shows two curves calculated for particular cases where Q2 = 10 (dashed curve) and Q2 = 100 (solid curve). The calculations are based on the exact equation (15) rather than the true circle equation (16), so that the circles are somewhat distorted. The values of M and R2 have been selected so that the circles have the same diameters. The same curves are redrawn in the reflection-coefficient plane in Fig. 4; the relation between the two coordinate systems is such that a circle in one coordinate set transforms into a circle in the other. The lines Xc = +Rc in Fig. 3 transform into arcs of circles in Fig. 4. In plotting the points in Fig. 3 and Fig. 4 from (8) and (9), it is convenient first to rewrite these equations in terms of /o and Q2. Since at is the value of s for which X2 = O, we use (6) to obtain oL — = l/o0C2 (22) In terms of the equivalent circuit values, Q2 may be defined as Q2 = aoL2/R2 = l/oC2R2 (23) We may then rewrite (6) as X2 = -L20Do/io - %/OOD2 = Q2R2(03/o - oo/1w) (24) If (24) is substituted for X2 in (8) and (9) Rc and Xc can then be written as _ __ (x/U1))2 c - X [Q w - %/c)] 2 (25) Xc = -%2((/oo - )o/m) Rc (26) For simplicity, the curves in Fig. 3 and Fig. 4 are plotted with,o2M2/R2 set equal to unity. In the case Q2 = 100, the following table gives the computed values of Rc and Xc used in plotting:

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-9 -D/0 0.98 0.985 0.99 0.995 1.0 1.005 1.01 1015 1.02 Rc 0.057 0.097 0.196 0.495 1.0 0.505 0.26.4.62 X +0.226 +0.291 +0.392 +0.495 0 -0.505 _0.4 -0.310 -0.245 In the case 10 the computed values are 0.8 0.85 0.9 0.95 1.0. 1.05 1.1 1.15 1.2 R 0.030.062 0.149 0.437 1.0 o.564 O.261.14 0.100 XC~~2 +0.1352 313 +0.450 0 -0.550 -0. Upon inspection of Fig.3 or Fig. 4, we note that the frequency dispersion is about 10 times as great for one curve as it is for the other, as would be expected from the difference in Q. We note further that the points corresponding to the frequencies fo(l + 1/2Q) for the Q = 100 curve fall almost exactly on the 'intersections with the XC= ~ R 0 lines,, while for the Q = 10 curve there seems to be a slight displacement. The reason for this displacement is that the frequencies f0(1 + l/2Q) given by (20) are not exact, the error becoming larger as Q becomes smaller. In connection with the half-maximum resistance points mentioned in the simplified measurement procedure,, we note that there is close agreement between these and the frequencies f0(l - 1/2Q) for the Q = 100 curve, but that there is a considerable discrepancy in the case of the Q = 10 curve. This discrepancy is caused by the substitution of a)0for co in (15) to obtain (16). However, since the freayency displacement is in the same direction and by about the same amount at both A and B,, the error in using (21) is qkite small,, so that the simplified measurement procedure is still quite accurate for values of Q as small as 10.

rNAM4E TITLE DWG. NO i-SM1TCHART FORM 82BSPR (2-4Z) KAY ELECTRIC COMPANY. PINE BROOK. N.J. O1949 PRINTED IN IDSa....DT FIG 4 THE CIRCLES OF FIG 3 IMPEDANCE OR ADMITTANCE COORDINATES PLOTTED -IN REFLECTIONCOEFFICIENT COORDINATES.2 /3 14 -". ~.," —K. s. i~.r i i.//-'7:.... —,u:- ~' '-:" I~~~~~~~~ JA~~~~~~~ t4~~~~~j i..'iO, ~ N% 7N.'~- - /~i.~t.. f Q '-.),~~~~~A 'bv...%..~..:_-j *- L5k _ TC:.-i-~-,-: R- 4..... -OARO":" — 'Ir1'ELF....::...-.:- - b'mC" ~-'.-~ ':!-'?,i'-_,., ~~~~~~~~~~~'eCYrW,?)I P I3O':~. -32i, JN 94-.AMG

-llRE]ERENCES Collins, "Microwave Magnetrons," Radiation Laboratory Series, Vol. 6, McGraw-Hill, 1st. ed., 1948, p. 178. Montgomery, "Techniques of Microwave Measurement," Radiation Laboratory Series, Vol. 11, McGraw-Hill, 1st. ei., 1947, pp. 288, 335. Terman and Pettitt, "'Electronic Measurements," McGraw-Hill, 2nd. ed.,, 1952, pp. 180-183. L. Malter and Go Brewer, 'Microwave Q Measurements in the Presence of Series Losses," J. Applied Physics, "Vol. 20, p. 918, Oct., 1949. H. J. Reich et al., V"Very High Frequency Techniques," Vol. II, McGrawHill, 1947, pp. 621-626.

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