I 7956-6-T Air Force Avionics Laboratory Research and Technology Dvision Air Force Systems Command Wright-Patterson Air Force Base, Ohio TRANSMITTER IMPEDANCE CHARACTERISTICS FOR AIRBORNE SPECTRUM SIGNATURE Interim Technical Report No. 6 1 July - 30 September 1967 W. B. Henry, W. R. DeHart and J. E. Ferris 15 October 1967 7956-6-T = RL-2171 Contract AF-33(615)-3454 Contract Monitor: K. W. Tomlinson, AVWE THE UNIVERSITY OF MICHIGAN COLLEGE OF ENGINEERING DEPARTMENT OF ELECTRICAL ENGINEERING Radiation Laboratory Administered through: OFFICE OF RESEARCH ADMINISTRATION * ANN ARBOR

THE UNIVERSITY OF MICHIGAN 7956-6-T TABLE OF CONTENTS Page ABSTRACT I. INTRODUCTION II. POWER TRANSFER THROUGH A LOSSY TRANSMISSION LINE 3 2.1 Power Transfor for Line Lengths of Integer Multiples of Half Wavelengths 5 2.2 Power Transfer for any Length Lossy Line 17 III. COMPUTER ANALYSIS OF A CLASS C AMPLIFIER SPECTRUM 21 3.1 Spectrum I 21 IV. EQUIVALENT CIRCUIT OF A CLASS C TRIODE 29 V. CONCLUSIONS 35 APPENDIX I 36 REFERENCES 37 i

THI UNIVERSITY OF MICHIGAN 7956-6-T ABSTRACT Simplified expressions for the power transfer through a real (lossy) transmission line have been developed. These expressions are presented in two forms. The first is designed for an exact solution to the power transferred and requires accurate knowledge of the transmitter and source impedances, transmission line length, and transmission line loss. The second form is designed to yield the maximum and minimum power transfer and the probability that the power will exceed some intermediate point. This form requires only the transmitter and source VSWR and approximate transmission line length. A computer program which solves for the spectrum output of a class C amplifier is presented. This is followed by an equivalent model of a high frequency triode which will allow the program to be extended to this type of device. ii

THE UNIVERSITY OF MICHIGAN 7956-6-T I. INTRODUCTION The statement of problem as set forth in the contract which provides for the present investigation is as follows: 1) A determination of the power delivered to the antenna for "spectrum signature" purposes will require a measurement of the antenna impedance, transmission line characteristics, transmitter maximum power output, and transmitter output impedance at the fundamental, spurious and harmonic frequencies. The transmitter output impedance at the spurious and harmonic frequencies is not well understood and therefore, requires further study. The prime payoff in this study will be better "spectrum signatures" for more accurate predictions of interference between systems. 2) There is a requirement to verify the results of the earlier successful program, Contract AF 33(615)-2606 "simplified Modeling Techniques for Avionic Antenna Pattern Signatures", with a mock-up of an aircraft transmitter system. 3) The stated objective of the contract is: To conclude the development of "1simplified"techniques for determining the RF spectrum signatures of flight vehicle electronics systems. To establish the validity of the techniques by comparing the results of data obtained by the "simplified" techniques with data obtained from tests employing a typical transmitter system in a mock-up. 4) The present phase of the contract is concerned with developing a technique for the accurate prediction of the power output of a typical transmitter. The realization of such a technique requires a thorough knowledge of a transmitter's output as a 1

THE UNIVERSITY OF MICHIGAN 7956-6-T function of the parameters most likely to vary in a practical situation at not only the fundamental frequency but the harmonic and spurious frequencies as well. 2

THE UNIVER SI T Y OF MICHIGAN 7956-6-T II. POWER TRANSFER THROUGH A LOSSY TRANSMISSION LINE It has been previously shown that the power transferred from a transmitter to a load is a function of the transmission line length (DeHart, 1966). Expressions have been developed and a graph presented which solve the power transferred in terms of power available from the source, the source VSWR, load VSWR and transmission line electrical length. The solution was however for a lossless transmission line. In many practical situations, the transmission line may introduce substantial losses. The following portion of the text devotes itself to providing a solution to the power transfer problem for those instances when transmission line losses cannot be neglected. The lossy line problem is not a new one and formulas are presently available in handbooks to handle this situation, however, they generally appear in an unusually cumbersome form. It is possible to simplify the problem considerably without sacrificing accuracy or generality by dividing the problem into two parts. In part 1 we consider the power transfer for line lengths of integer multiples of half wavelengths. In part 2, the line is allowed to assume any length by approximating thbe entire transmission line with the appropriate number of integral half wavelengths of lossy line connected to a short section < A ) of lossless transmission line. 2 3

THE UNIVERSITY OF MICHIGAN 7956-6-T The characteristic impedance of a transmission line with losses is given by, -' R + j L z o _- * (2.1) o j G+ j w C In equation (2. 1), R = series resistance of the line per unit length L = series inductance of the line per unit length G = shunt conductance of the line per unit length C = shunt capacitance of the line per unit length. In the consideration of the loss in transmission lines, it is convenient to assume that the characteristic impedance of the line is real. The exact condition is, R G - G -(2.2) L C The characteristic impedance, Z, is then Z =R +jO (2.3) o o and 0R L/C. (2.4) 4

THE UNIVERSITY OF MICHIGAN 7956-6-T For most radio frequency lines the reactive component of the characteristic impedance will be very small so that equation (2. 2) is sufficiently satisfied for acceptable accuracy in the computation of power transfer. 2.1 Power Transfer for Line Lengths of Integer Multiples of Half Wavelengths An equivalent circuit for the general power transfer problem is shown in Fig. 2-1. E g R a FIG. 2-1: Equivalent Circuit of the Power Transfer Problem Let R = resistance of the termination of the atransmission line R - characteristic impedance of the transmission line R 3 internal resistance of the equivalent generator 5

THE UNIVERSITY OF MICHIGAN 7956-6-T E = internal voltage of the equivalent generator I = current into the load resistance a I, = current into the transmission line El = voltage applied to the transmission line. Note that the generator impedance and the load impedance have been chosen as real. No loss of generality has been sacrificed because an antenna (or generator) can be connected to an appropriate short length of lossless transmission line to transform any complex impedance to a pure resistance. In this discussion the transforming line will be assumed to be of such a length that R > R. R > R In this case, let R a =r (2. 5) 0 R X r (2.6) R g 0 in which r and r are the standing wave ratios for the antenna and generator impedances. Consider first the power transfer characteristic of the transmission line alone. 6

THE UNIVERSITY OF MICHIGAN 7956-6-T I E1 R =R a 0 0 x -- FIG. 2-2: Lossy Transmission Line and Load. From the transmission line equations r z 1 I(x)=Ia cosh(yx)+ a sinh (yx) (2.7) a Z Z cosh (yx) + Z sinh ('yx) a j Z (x)= Z (2 8) Zo cosh (YX)+ Z sinh (ax) ' In equations (2. 7) and (2.8) = propagation constant for the transmission line. These equations represent our circuit in Fig. 2-2 for any line length x. If we restrict the values of x that we use to exact multiples of a half wavelength, then x = ax + jfx = acx + jpnr (2.9) and the hyperbolic functions are: cosh (ax + j3n7r) - cosh ax (2.10) sinh (ax + jfn7r)= sinh x. (2.11) 7

THE UNIVERSITY OF MICHIGAN 7956-6-T Under this condition equation (2. 7) and (2. 8) contain no complex quantities. Note that we can come within about 20 inches of the actual line length at 300 MHz without violating that condition. As indicated previously the additional line length necessary can be assumed without loss, though not without a transformer effect. This transformer effect was discussed in a previous paper (DeHart, 1966). The power into the line at x = is 2 P I R1 (2.12) and the power delivered to the load resistance is 2 P =I R. (2.13) a a a The power transfer characteristic of the transmission line is the ratio of the power delivered to the load to the power input to the line, P I R a a a -- (2. 14) 1 I1 R1 1 1 For this case Z = R, Z= R, and I -- as previously discussed so that o 0 a a 2 (2.7) and (2.8) become r A + B Z()=R R r B (2 15) L a and I()=I =Ia (A+ r B) (2.16) l ~a a 8

THE UNIVERSITY OF MICHIGAN 7956-)'-T where A = cosh ax, B = sinh ax for convenience. Substituting equations (2.15) and (2.16) into (2. 14) P r P r aAr (2.17) P (r A + B 'A + r B * 1 k a. This result may be further simplified by rewriting the hyperbolic function in terms of exponentials ax+ -x cosh (ax) = e (2.18) 2 ax -ax sinh (ax) e (2.19) 2 so that P 4r A /(r + 1'2 a o a (2.20) 1 1- P A2 where A e2ax (2. 21) 0 r -1 a and p = the magnitude of the load voltage reflection coefficienti r + - a / Note that (2. 20) includes the losses due both to line attenuation and load mismatch. When R 3= R, p becomes 0, r becomes 1 and (2. 20) reduces to a o a a 9

TH-E UNIVERSITY OF MICHIGAN 7956-6-T P a A for R =R. (2.22) P o a o 1 Equation (2. 20), while useful, requires that P1, the power input to the line, be known before P can be found. A more practical expression would be one a giving P in terms of the generator power delivered to a matched termination, (i. e. a matched to the characteristic impedance of the transmission line, R ). Referring to Fig. 2-1, the power input to the transmission line is E 2R E r +R R + rR (r +r If R1 is replaced by a matched termination, R~ E 2 R E2 P= - (2.24) o /2 \2 +R + R R r R +1/ Rg 0/ o', g I and P P P a a 1 a _ a 1 (2.25) o 1 0 p1 P P From (2. 25), in order to solve for a it is necessary only to find P p o o and combine this expression with (2. 20) 10

THE UNIVERSITY OF MICHIGAN 7956-6-T Substituting (2. Rewriting in terms of P r + +1 -1= 1,, P,2 o r +r g 1i 15) into (2. 26) and simplifying one has 2 r A+B (A r B) +r + 1 2 1 a a g e a b J exponentials as before, (2. 27) becomes (2.26) (2.27) P 1+A p 1 -A p) 1 * o a)1 o a) p r 1 o 0 - paPgAoj a i/ 0i (2.28) where again -2ax A = e, the nominal line attenuation o Pa the magnitude of the load voltage reflection coefficient, r -1 a r +1 a p = the magnitude of the generator reflection coefficient, r - 1 r +1 g 11

THE: UNIVIERSITY OF MICHIGAN 7956-6-T Setting R = R, p becomes 0 and a o a P - 1 for R=R. (2.29) P a o o To find the power transferred to the load in terms of load VSWR, generator VSWR transmission line length and generator power into a matched load, it is necessary to combine results (2. 28) and (2. 21) as indicated by (2. 25) such that P 4 r A a --- (2. 30) P,. 2 fi A 2 o r + " a /g 0 or 4 P r A P a o (2.31) a 2 2 ir + 1 - p p A 2 a g 0a Tt is interesting to consider the exponential factors in equation (2. 31). The factor -2vx A e O 1-p p A 1-n n e ag o a g e 2^x r 2^x e - n n 2 a g 2vx varies linearly for changes in n n That is, the curve of e -n n for a g a 12

THE UNIVERSITY OF MICHIGAN 7956-6-T n n fC1 is exactly the same curve as for n n C2 only with the origin shifted. Thus, the power transfer through a lossy line varies in the same fashion for changes in line length irrespective of source or generator VSWR. A second important observation is that thus far the results derived have been valid only for line lengths of integer multiples of half wavelengths. However, R (equation (2. 15) is real for line lengths which are integer multiples of quarter wavelengths. Choosing an odd multiple of a quarter wavelength, is equivalent to solving the same problem with R < R or R <. This causes the sign in the l1-n n A a o g o a g O factor of '2. 31) to change resulting in a minimum power transfer. The result is unchanged if both R <P and P <. a o g o Figures 2-3 through 2-5 have been prepared to aid in solving (2. 301 and '2. 31'. T etting ~-2 F 10 log -n n A 1 *db) F 1 L a g oj F 10 log il+n n Ao 2 tdb1 I L a g o A = nominal transmission line loss (dbl (loss per foot x no. of feet) x 10 log A 0 0 A 10 log A n (dbl 1 a g A a 10 log 4r/(r + 1) (db), 2 '2. 30' becomes, P a ( db A A FH" fdb for I --.2n (2. 321 0 // 13

-- 1 2 3 4 5 6 7 8 9 10 20 30 40 50 r a FIG. 2-3: A as a function of r. 1 a 1,00

THE UNIVERSITY OF MICHIGAN 7956-6-T p p o g -r-.92 5S 9 q0..85 1).8.75 1s..7 10.6 9 r g r g.5 6 &.4 I.3 3 r -1 r - 1 a g where Pa g r +1 r +1 a g 2 1 VI FIG. 2-4: p p as a function of a g r and r. a g 15

7956 -43-T p ap i I I 1 i i - -10 i-20 — 30 -— 40.-50 F H 20 O 10 6. 6 -3. 0 -- 1.75 -0.80 I.41.21. 085-. 043-. 022-. 0087.0043.0022.00087.00043.00022.0000 9 F I -3. 010 PI -2. 22 - -. 97 -. 46 —. 22 -. 088 —. 044 —. 004 —. 004 -. 0009 -. 0004 —. 00009 -1. 5 ~. 1.05 01 ~-.005.00001 -—.00001 FIG. 2-5: F ' F' as a f'nof p pa and A 16

THE UNIVERSITY OF MICHIGAN 7956-6-T /D (2 n - 1 a At + A + F (db for l= 4.2. 33) o o 2 4 O P a nne may now solve for as follows O o 1) Pnter r in Fig. 2-3 to find A1. a 2) Rnter r and r in Fig. 2-4 to find n n. a g a g 3) Enter p p and A' in Fig. 2-5 to find F and F. ag o H 0 A' is given in standard tables of transmission line data. We have now all the necessary O information to solve (2. 32) and (2. 33). 2.2 Dower Transfer for any TLength T.ossy T.ine The previous section derived an expression for the power transferred through a lossy line when the line length was an integer multiple of quarter wavelengths and the terminations were real. This section will generalize those results to include all of other physically realizable systems, the only constraint being that the total line length be of the order of magnitude of one wavelength or greater. An equivalent circuit of the generalized problem appears in Fig. 2-6. Referring to Fig. 2-6 the total length of transmission line l1 is divided into four lengths, l1, 12, A3, and 14. Length 1 transforms Z to R; length 4 transforms Z to Ra; L is the remaining integer 1 g g' 4 a a 2 number of quarter wavelengths of line, and 13 is what is left over. Since the 38 these lengths will be assumed lossless. maximum length of (A1 + I + A)m -I these lengths will be assumed lossless. 17

Loss-. Lossy Los- LossLess Line Less Less Line - nX Line Line 2 4 g Z=R 6 Z =R - Z =R Z =R a Z R 0 0 0 0 0 0 0 0 I t —' 1 2 3 4 FIG. 2-6: Equivalent Circuit of the General Power Transfer Problem. z a -} -21 -. I - r i>

THE UNIVERSITY OF MICHIGAN 7956-6-T Assuming 13 = 0 for the moment, P al A +A + F for (2. 34) P o 1 H 0 o a) R >R R >R, l 2 2n a o' g o 2 4 b) R <Ro R <Ro, 2-n a o g o 2 4 c) R >Ro, R < R, (2n -1) a o g 0 2 4 d) R <R R >R, 2x2 n-1) a 0 g 1 2 4 and a2 p A +A + F for (2.35) 0 1.o O a) R <Ro, R >Ro, 2 4 a o g o 2 4 b) R >Ro R <Ro, 2I n a o g o 2 4 c) R >R R >R, (2n-1)X a o' g o 2 4 d) R <R R <R 2 (2 n 1) a o g 0 2 4 The effect of length 13 will be to cause the power delivered to the local to vary between (A' +A1 +FH) and (Ao+A1 + F) o 1 H 1 19

THE UNIVERSITY OF MICHIGAN 7956-6-T The ratio of the maximum to the minimum power available 5 al pa (FH - F) (2.36) a2 H 2 This ratio becomes a in DeHart's expression (DeHart, 1966) so that P a 1 (2.37) al sin 2S3 + cos2 3( for those conditions given under (2. 34), and P a 1 (2.38) Pa2 C cos2 3 +sin2 3 for those conditions given under (2.35), where a2 log FH - FI /10 Where an exact power level is not required, it may be sufficient to solve (2. 32) and (2. 33) to find the maxima and minima power levels available and use (2. 37) or (2. 38) to determine the probability that the power level will exceed some point between those levels given by (2. 32) and (2. 33). 20

THE UNIVERSITY OF MICHIGAN 7956-6-T III. COMPUTER ANALYSIS OF A CLASS C AMPLIFIER SPECTRUM A program has been written which computes the output voltage of a class C amplifier as a function of both frequency and time. The time domain waveform is plotted by the computer and the frequency domain solution appears as the magnitudes of the coefficients a and b of a Fourier series where n n I I E (t)= a + X a cos nwt+ ) b sinnwt (3.1) n 1 n1 The necessary inputs include a piecewise linear device model, device input waveform, device static operating conditions (e. g., plate supply voltage, grid bias, etc.), and the complex load impedance as a function of frequency. 3. 1 Spectrum I A flow diagram of the spectrum analysis program appears in Fig. 3-1. The development of the iteration technique utilized in this program was presented in a previous report (J. E. Ferris, et al, 1967). Thi s report will summarize that discussion and include a detailed description of Spectrum I. Briefly, the output spectrum of the amplifier is found by calculating the frequency components of the plate current waveform and multiplying these components by the tube load impedance at the corresponding frequency. The device is modeled 21

F ELC=RES5HN ZHN 5HN HN ELS=RES5HN+I Z 5HNtI Z.H ELT-ELC x COS(CA ELS x SIN(CAA x T~RT=ERTR-E T.T SUMEB = SUMEB + EBT SUMEL=SUMEL + EiLT FIG. 3-1: Spectrum I

THE UNIVERSITY OF MICHIGAN 7956-6-T by a controlled current source as shown in Fig. 3-2. eb t) '7,(, 1 ----;7z (1,.<\ e, (t) e (t) g X 0 ib(t) Ebb - -- [ FIG. 3-2: Pentode Equivalent Circuit One may now write ib (t) =A e (t)+ B(r eb (t)+C br g B(r, s) br (3.2) where e (t) instantaneous grid voltage g eb (t) = instantaneous plate voltage ib (t) instantaneous plate current eL (t) = instantaneous load voltage Zt ) complex load impedance Ebb plate supply voltage E bb,2 plate supply voltage 23

THE UNIVERSITY OF MICHIGAN 7956-6-T and Ar, Cr, and B( ) are constants, appropriately chosen over the region e(r 1 < e < e < e(r) and e (s < eb < eb (s) to provide the best c(r-1) c - c(r) b(s-1) b - b(s) linear approximation to the tube static characteristics in that region. Assuming that e (t) is known, it is necessary only to find eb (t) in order g b to solve (3.2). From the model in Fig. 3-2, eb (t)= Ebb -e (t) ib (t) Z(t). (3 3) Z (t), however, is not defined so that it is impossible to solve (3. 3) directly. The iteration technique used in Spectrum I assumes temporarily that Z (t) is defined and is equal to a constant Z'. Substituting this result into equation (3.2) and (3. 3), one has A e (t)+B B Eb +C ib(t) r 1 (r,s) bb r (3.4) + B(r,s) Z If e (t) is known, (3.4) may be solved for ib (t). This provides a first approximation to the plate current waveform. This approximation may be improved by transforming ib from the time domain to the frequency domain and solving eb (u) Ebb- et () Ebb - i () Z (). (3.5) Transforming eb (w) back to eb (t) allows an exact solution of (3.2) and a successively better approximationto ib (t). This process is continued until successive 24

THE UNIVERSITY OF MICHIGAN 7956-6-T calculations of ib (t) produce some arbitrarily small change in eA (w). * Referring to Fig. 3-1 Spectrum I accomplishes the above in the following manner: A variable, x, which counts the number of times the time variable, CA, has been incremented is set to 1. Next, CA is set initially to zero and CA is tested to determine if it is greater than the maximum time of interest, MF. If CA < MF, XlEC is set to the current value of grid voltage, EC (CA), and X1EC is compared to ECOR for 1 < R = P. The ECOR represent ranges of grid voltage for which the factors A and CI in (3.2) assume a particular value. If there is no ECO corresponding to the value of EC (CA), the loop will continue testing R until R exceeds P, the number of ECO's available, and will cause an error flag to be printed out. Once the proper value of R has been established, the corresponding factors A and C are selected from the data and the approximation to the plate current waveform given in (3.4) is calculated for that value of CA, and stored in memory. CA is then incremented by S and the process continued until (CA + S) > MF at which time N, the number of times that the plate current waveform has been calculated, is set to 1. N is then tested to determine if it is greater than I, the number of times it is to be calculated, and if it is not, tested again to see if it is I. See Table 3-1 for a complete input-output list. 25

THE UNIVERSITY OF MICHIGAN 7956-6-T When N 1, INPUT, the time domain input to the Fourier series sub-routine, HAS 1, is set to IBO and HAS 1 is executed to find the coefficients a and b of the Fourier n n series given by (3. 1). At this point, the time variable is renamed CAA and set to zero. EC (CAA) is calculated for the current value of CAA and renamed X2EC. The Fourier coefficients a (called RES ) and b (called RES(5 + are n (5hn) n (5hn + 1) multiplied by cos (HN) (CAA) and sin (HN) (CAA) respectively, Z (w), and summed. This is repeated for all values of HN (harmonic number) up to the highest harmonic desired (H). The sum is then subtracted from the plate supply voltage, EBB, This result, EBT, is the plate voltage waveform at time CAA. Equation (3. 2) is now solved by an internal function IBB for ib at time CAA. This process is repeated for all CAA, 0 CAA = MF, resulting in ib (t). ib(t) is then reentered into HAS 1, and cranked through the N loop, improving the approximation of ib (t) until, at least, N = 1. The last time through the loop, the Fourier coefficients are stored and printed out, and the final load voltage waveform as a function of time is plotted by the computer. It mus t be noted that this program was only recently completed and no results are currently available for publication. While the example presented he computes the e r co spectrum of a tetrode vacuum tube operating in the class C mode, the technique is very general and could be used with a variety of devices such as triodes, pentodes, and transistors, so long as the results are not effected by physical phenomena such as electron transit time, etc. 26

THE UNIVERSITY OF MICHIGAN 7956-6-T The next section devotes itself to the development of a corresponding model of a triode, taking into account the effect of inter-electrode capacitances. 27

THE UNIVERSITY OF MICHIGAN 7956-6-T TABLE 3-1 INPUT-OUTPUT LIST FOR SPECTRUM I Inputs: EC magnitude of the excitation voltage EBB plate supply voltage Z (o) plate load impedance as a function of frequency ECO (R) range of grid voltage for the R'th piece of the piecewise linear static characteristic model EBO (R, S) range of plate voltage for the (R, S) piece of the piecewise linear static characteristic model A (R), C (R), B (R, S) piecewise linear model coefficients P, Q the range of R and S respectively M F range of wt (i.e., CA, CAA) Delta increment of CA, CAA I number of times iteration process is to be run H maximum number of harmonics desired Outputs: E L (T) a plot of the load voltage waveform as a function of time ELCC, ELSS magnitudes of the coefficients of the Fourier series of E L (T) 28

THE UNIVERSITY OF MICHIGAN 7956-6-T IV EQUIVALENT CIRCUIT OF A CLASS C TRIODE The spectrum analysis program presented in the previous section was developed around a piecewise linear model of a tetrode. All inter-electrode capacitances were neglected. These capacitances cannot be neglected in a high frequency triode. Fig 4-1 is an equivalent circuit of a triode including inter-electrode capacitances and driving source resistance. R C (1) (2) /. e g e. (4.1). E c Cc (r b ib p FIG. 4-1: Equivalent Circuit of a Triode The instantaneous plate current is found by a piecewise linear approximation to the static characteristics as before. ib =A e +B eb +C. (4.1) b r c (r, s) b r Again, an exact solution is not possible because the plate voltage is related to the plate current by impedances defined only as a function of frequency. Writing node 29

THE UNIVERSITY OF MICHIGAN 7956-6-T voltage equations at nodes (1) and (2) in Fig. 4-1, At node 1, E - e (t) 0 - e (t) WC gk+ (eb(t) - e (t)) jwC. (4.2) s At node 2, 1 ib(t) e ((t) - eb(t)) jCgp - eb(t) jWCpk + (Ebb - eb(t)) Z) (4.3) The first step is to obtain an approximation to ib. Perhaps the simplest approach is to neglect the inter-electrode capacitances and assume ZL is constant with frequency. Under these conditions, b(t) bb - ib(t) Z' (4. 4a) and ec (t) E (t) (4.4b) c s Thus (4.1) becomes A E (t)+BEb+ C s bb (4.5) ib(t) - 1 + B Once the first approximation to ib(t) has been found, ib must be transformed from the time domain to the frequency domain, See Table 4-1 for a list of symbols used in section 4. 30

THE UNIVERSITY OF MICHIGAN 7956-6-T H ib(t) Ib + n I sin (mnt + p ) b Z 1 n n n=l Let (4.6) Rewriting (4. 3) in the frequency domain and equating terms of the same frequency, (4. 3) becomes Ip Ecl pi cl jC - E juC 1 z (o) + (Ebb K' /Ip2 c2 p2 kc2 \ C - E j2]'C - pl gP E 1 j2wCpk + Ebb //* -pE 1 z(w2) 0 0 0 pH cH - EH) jHwCgp PEI 9 P - EH jHCpk pH - pk + bb, —" E = pn p Z(wH) E jnwC + E cn gp bb (4.7) or 1 Z -I o pn Z) + jnu(C + p) n gp pk) (4.8) 31

THE UNIVERSITY OF MICHIGAN 7956-6-T The term E jnwC is the component of plate current fed back to the grid. This cn gp current is very much smaller than the total current I in (4.8) and may be neglected from this expression without appreciably affecting the result. Thus, E - = bb Z pn E - -....... * (4.9) pn Z( ) + jn (C +C ) Z(un) gp pk Expressing (4.2) in the frequency domain in a manner similar to that described above, E ~ / — n - - E jn C + E jn C (4.10) R cn n gk + Epn p on solving for E c E -S+E j nwC E pn gp E - (4.11) cn 1 R + j nw(Ck + Cpk) Combining (4. 9) and (4. 11), bb -I. X pn E 0 R '1 s +j nw(C +C ) cn a (4.12) c +j nw (Cgk+ C ) Equations (4. 9) and (4. 12) are expressions for the plate and grid voltage in the frequency domain as a function of the plate current. Using a frequency domain transform of the time domain approximation given for the plate current as a function of time 32

THE UNIVERSITY OF MICHIGAN 7956-6-T equation (4. 5), (4. 9) and (4.12) can be solved. Transforming these quantities back to the time domain (the inverse of (4. 6)) (4. 1) can be solved for the next successive approximation to ib(t). This process is repeated until ib(t) has been found to the required degree of accuracy. 33

THE UNIVERSITY OF MICHIGAN 7956-6-T TABLE 4-1 SYMBOLS USED IN SECTION IV b total instantaneous plate current average (DC) plate current I n magnitude of the n'th component of plate current eb total instantaneous plate voltage E average (DC) plate voltage E plate supply voltage nE magnitude of the n'th component of plate voltage e total instantaneous grid voltage c E average (DC) grid voltage c E grid supply voltage cc E n magnitude of the n'th component of grid voltage i total instantaneous grid current c I average (DC) grid current cI n magnitude of the nth component of grid voltage 1Ig n magnitude of the n'th component of grid voltage 34

THE UNIVERSITY OF MICHIGAN 7956-6-T V. CONCLUSIONS The text of this report is comprised of two phases of spectrum signature prediction. Section I predicts the output of the antenna once the transmitter behavior has been determined and Section III and IV discuss methods for predicting the behavior of a particular class of transmitter. If one is willing to assume that a given transmitter behaves in a linear fashion with respect to changes in the load impedance, the results in Section II have an immediate application to spectrum predictions. No experimental evidence has been given to support any of the results. It is felt, however, that such evidence is necessary and experimental data should be gathered as soon as possible. 35

THE UNIVERSITY OF MICHIGAN 7956-6-T APPENDIX I Errata Sheet for 7956-5-T The following corrections should be made to the equations corresponding to the numbers given. 27r 1 rf a - - n T J 0 ib(t) cos (nut) dut (3.5) 27r b = n ir J 0 ib(t) sin (nut) dut ib A e +B eb +C b r c s D r b-0 EL = I Ln p fe (r-l >e >e c (r-1) >ec - c (r) ~e(s-l) > eb > eb (s) e <e c- co (3.7) z n n (3.8) 00 ib -B eb =I (1 -B R )+ s b s o n= l I sin (nwt + O ) (1 - B Z) = A e +C p n (3.n rc r (3.9) The screen grid and control grid connections are shown interchanged in Fig. 3-11, page 29. 36

THE UNIVERSITY OF MICHIGAN 7956-6-T REFERENCES Ferris, J.E., W.R. DeHart, and W.B. Henry, (January, 1964) "Transmitter Impedance Characteristics for Airborne Spectrum Signature," Interim Technical Report No. 3, The University of Michigan Radiation Laboratory Report 7956-3-T, 14 pages. Dehart, W.R. (1966), "Transmitter-to-Antenna Power Transfer Under Unmatched Conditions," IEEE Trans. EMC-8. No. 2, pp. 74-80. 37