For most of us, discoveries in the art are

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1 Filtes and an Oscillato Using a New Solenoid Model A tansmission-line model enables a wide ange of filte and oscillato designs By Randy Rhea Eaglewae Fo most of us, discoveies in the at ae ae. My books eviewed the contibutions of many enginees and contained a limited numbe of oiginal ideas. I ecall many excited moments duing my caee that faded upon the ealization that a discovey was actually a measuement eo, misundestanding o ediscovey. In 1997, an enginee equested that I examine his measued data fo an inducto. The data was unexpected, so I decided to measue a simple solenoid with a netwok analyze. The analyze display stunned me. This moment of excitement would not fade. The industy s accepted and centuy-old inducto model was wong. Why had no enginee befoe me pefomed this simple expeiment and witten about it? The significance of the expeiment began to unfold at my desk as I examined the poblem mathematically. As is often the case, new knowledge is both satisfying and useful. The pupose of this pape is to descibe new filte and oscillato stuctues suggested by the new model. Fist, I will eview the histoic inducto model, then the new one. The histoic model One need only ponde a solenoid inducto to ealize that the close spaced tuns ae capacitively coupled. But how do you measue capacitance that is shoted by a tun of wie? Webste [12] solved this poblem by building a paallelesonant mode oscillato using an inducto and its self-capacitance. Knowing the inductance (measued at low fequency) and the oscillating fequency, you may deduce the capacitance. Fom this capacitance, and seies esistance fom the wie loss, the model of the inducto Figue 1. lassic inducto model with capacitance fom Medhust (a) and the tansmission amplitude and phase esponses (b). given in Figue 1(a) was bon. The tansmission amplitude and phase vesus fequency of this model that is simulated fom the schematic and computed by the GENESYS [2] softwae pogam is given in Figue 1(b). Numeous attempts at mathematically solving the self-capacitance failed. Medhust [6] 3 APPLIED MIROWAVE & WIRELESS

2 abandoned mathematical attempts in favo of an empiical appoach with one end of the solenoid gounded. The accuacy of his capacitance is veifiable. While Medhust s capacitance is coect, the model he inheited is wong. You may easily obseve that the wold is flat. It is equally obvious, and incoect, that inte-winding capacitance is impotant. The phase shift fom one tun to the next is small. With tuns at nealy equal voltage potential phase, the effect of coupling capacitance is negligible. The capacitance between tuns is effectively shoted. Since expeiments showed that close spacing did not incease the capacitance, it should have been suspected yeas ago that the capacitance is not tun-to-tun. Some athe esoteic explanations fo this anomaly wee pomulgated. The capacitance that Medhust quantified is capacitance of the solenoid to gound, not capacitance tun-to-tun. To undestand how this eo occued, conside the analyze data and a poposed new model. A new solenoid model Given in Figue 2 is a 12.9 tun solenoid mounted ove a gound plane [1]. It is wound with 18 gauge coppe wie. The mean adius is.268 inches, the mean length is.89 inches, and the outside of the wie is.135 inches above the aluminum gound plane. The esulting tansmission amplitude and phase ae given in Figue 3. One stiking featue is the peiodic natue of the esponse, which clealy suggests a tansmission-line model fo the solenoid. This is not pedicted by the classic model. Equally evealing is the tansmission phase shift at the fist anti-esonant mode of the classic model tansitions fom 9 to +9 degees, while the measued data is continuous at 9 degees. The classic model completely fails to pedict the high-fequency behavio of the solenoid. Shown in Figue 4 ae the tansmission amplitude and phase of a tansmission line with Z 796 ohms and an electical length of 9 degees at 195 MHz. Notice impoved coelation to the measued solenoid esponses. A tansmission line may be modeled by distibuted seies inductance and shunt capacitance to gound. The chaacteistic impedance and electical line length ae elated to the pe unit seies inductance and shunt capacitance by o Z L ohms θ πf L adians 2 θ 36F L (1) (2) (3) This model pedicts highe-ode modes by static Figue 2. The 12.9 tun coppe-wie solenoid inducto mounted ove a gound plane and configued fo tansmission amplitude and phase measuement. Figue 3. Tansmission amplitude of the solenoid ove gound (top) fom.5 to 13 MHz with a vetical scale of 4 to +1 db and the tansmission phase (bottom) with a scale of 27 to +63 degees. seies inductance and shunt capacitance measued at low fequency. The model s elegance is futhe confimed by the fact that these static paametes ae easily calculated using existing mathematical techniques and that these techniques ae equally applicable to divese configuations, such as a solenoid above a gound plane o coax with a helical inne conducto. Mathematical calculation of the capacitance is now staightfowad. If the classic model assumed by Medhust is wong, how did he obtain the coect value of capacitance? 32 APPLIED MIROWAVE & WIRELESS

3 Figue 4. Tansmission amplitude of the solenoid ove gound (top) fom.5 to 13 MHz with a vetical scale of 4 to +1 db and the tansmission phase (bottom) with a scale of 27 to +63 degees. Medhust s paallel-mode esonato had one end of the inducto gounded. By chance, the inte-winding capacitance was effectively capacitance to gound. Models fo common configuations The static inductance may be estimated using the popula fomula fom Wheele [13], na 2 2 L (4) 9a + 1c micohenies whee n is the numbe of tuns, a is the solenoid adius and c is the solenoid length. Wheele s fomula is accuate to ±1.5 pecent fo small wie diamete. Fo smalle gauge wie it oveestimates the inductance. To compensate, I pefe to use the inside adius of the winding. Table 1 gives the computed capacitance fo common solenoid inducto configuations. The model tansmission line impedance and electical length ae then computed fom Equations (1) and (3) using these static inductance and capacitance. Altenatively, the static inductance and capacitance may be measued with low fequency instumentation. The capacitance is estimated assuming the solenoid is a solid cylinde. Fo example, to find the capacitance of the solenoid in Figue 2, the capacitance of a cylinde ove gound is used. In Table 1, the chaacteistic impedance fomula fo vaious configuations [3] is used to compute the capacitance by the fomula c ε faads (5) VZ Figue 5. S/FILTER Specification tab (top) and schematic (bottom) fo the 3d ode (6th degee) seiesesonato bandpass filte. whee V is the velocity of light in a vacuum and Z is the chaacteistic impedance of the configuation. Effect of a shield on inductance Wheele s inductance fomula assumes an unshielded solenoid. Bogle [1] gives an inductance eduction facto based on a conducting, non-magnetic, cicula shield. a as LF 1 as a ( ) c 2 The shielded inductance is found by multiplying the unshielded inductance by Bogle s facto. A shield adius of twice the solenoid adius and a solenoid length to adius atio of 4 yields an inductance eduction of 18 pecent. Fo squae shields, a adius equal to.6 times the side dimension may be used. Shielding is ignoed fo a solenoid ovegound as data egading the inductance eduction of a flat, adjacent, gound plane is unknown to this autho. (6) 34 APPLIED MIROWAVE & WIRELESS

4 The model below the 1st-esonant mode Fo fequencies well below the 1st-anti-esonant mode, the capacitance is immateial and the solenoid inducto is accuately modeled as a simple inducto. The eactance of a shoted tansmission line is nealy linea (modeled by an inducto) fo electical line lengths up to λ/16 o 22.5 degees. The fequency limit associated with this line length is Flimit 1 (7) π L hetz Above this fequency limit the new model is suggested. Next, we will exploit the accuacy of this new model to ceate new classes of bandpass filtes that use the seies-esonant 2nd mode. apacito-coupled 2nd-mode bandpass filte Figue 5 gives the schematic of a thid-ode (sixthdegee) shunt capacito-coupled, seies-esonato lumped element hebyshev.1 db ipple, 75 MHz to 8 MHz bandpass filte. This filte is often designed using an appoximate method descibed in Matthaei [5]. The accuacy of this outine degades with inceasing bandwidth. In this case, exact synthesis [11] was used to find element values. This filte has five tansmission zeos at D and one tansmission zeo at infinite fequency. An S/FILTER pogam sceen, with the Specification tab active, is given at the top of Figue 5. The L- seies esonatos in Figue 5 may be eplaced with equivalent tansmission lines [8] that ae 18 degees long at the esonant fequency of that banch and with a chaacteistic impedance of Z 16 2ωL 4FL ohms π whee L is the inductance in each espective banch. The inducto and capacito in each seies esonato (8) ylindical shield Solenoid ove plane Squae shield Tough shield Slabline shield pf b V ln a 2H 2H V ln + a a 1 787H V ln. a 2 1 pf 4b πh V ln tanh πa b pf 4 X + Y R V ln + 1. R X Y 3 π a R 2H 2 2 X 1+ 2sinh R, Y 1 2sin R Figue 6. Shunt-capacito coupled bandpass with seies L- esonatos eplaced with 2nd-mode solenoids modeled by tansmission lines. Physical solenoids ae designed using Equations (1) and (3) with inductance fom Equation (4) and capacitance fom Table 1. Table 1. Static capacitance computed fom the chaacteistic impedance of solid-cylinde models of solenoid configuations. V fo dimensions in inches and V fo dimensions in millimetes. 36 APPLIED MIROWAVE & WIRELESS

5 above the passband. The filte in Figue 5(b) has no tansmission zeo at D, which esults in educed lowfequency ejection. Figue 7. Shunt-inducto coupled seies-esonato bandpass filte (top) designed by S/FILTER and with esonatos eplaced with 2nd-mode solenoids (bottom). Seies inductos model the wie connecting the solenoids. The helix paametes ae fo solenoids in a squae shield. Figue 8. Simulated and measued amplitude esponse of the shunt-inducto coupled 2nd-mode solenoid bandpass. should not be confused with the static inductance and capacitance of the solenoids used to fom the esonatos. The schematic of the final shunt-capacito coupled 2ndmode bandpass is given in Figue 6. The solenoids ae opeated at the seies-esonant 2nd mode and ae depicted in Figue 6 as tansmission lines. The substitution of the seies L- esonatos with solenoid tansmission lines convets tansmission zeos at infinite fequency to eentant modes at fequencies 2nd-mode bandpass filte using only solenoids Figue 7 gives the schematic of a thee-section seiesesonato bandpass using coupling inductos athe than coupling capacitos. This stuctue has five tansmission zeos at D and one at infinite fequency. On the bottom in Figue 7 the seies L- esonatos have been eplaced with 2nd-mode solenoids. Figue 8 gives the simulated esponse of the inductive-coupled 2nd-mode bandpass. A photogaph of a pototype filte with solenoids wound using 18-gauge wie is given in Figue 9. Measued data is supeimposed as cicula points on the simulated esponse in Figue 8. The fequencies of the 2nd-mode esonatos in the pototype filte wee tuned using bendable metal tabs soldeed to the side of the squae shield. This intoduces a small amount of capacitance, thus loweing and adjusting the fequency of each esonato. Notice that this filte equies no capacitos: esonatos ae fomed by the seies-esonant 2nd-mode of the new solenoid model. Solenoid unloaded Q omponent Q (unloaded Q) is defined as the atio of stoed to dissipated enegy. Moe enegy is stoed in a lage magnetic field, and inducto Q can be inceased by inceasing the adius of the solenoid. Unfotunately, inceasing adius also esults in inceased solenoid capacity. lassic inducto theoy dictates that this capacitance must be much less than the inducto eactance. This limits the solenoid size and thus the available inducto Q. Resonatos constucted using the theoy of the new model do not suffe this limitation. This capacity is a natual and desied element of the esonato. Theefoe the esonato can be physically lage and can achieve highe unloaded Q. My wok with the new model has not yet specifically addessed unloaded Q. Nevetheless, the insetion loss of esonatos and filtes that have been constucted suggest that the unloaded Q pedicted by Medhust [6] is valid fo the new tansmission line model. As solenoid size inceases, adiation loss becomes moe significant. This is typically not significant fo inductos designed using the classic theoy because inducto capacitance limits the maximum size. The new model pemits lage solenoid size, and adiation is moe likely to become significant. While solenoids enclosed by a squae o ound enclosue do not suffe fom adiation loss, a solenoid ove gound is moe susceptible. The maximum size of shielded solenoids is limited by modeing in the enclosue. Additionally, the size and theefoe the unloaded Q is limited by the fact that the numbe of 38 APPLIED MIROWAVE & WIRELESS

6 Figue 9. Pototype 2nd-mode bandpass. 1.5 tun shunt-coupling inductos ae baely visible at the inside of the end walls of the cente section. tuns to achieve esonance deceases with an inceasing solenoid adius. As the numbe of tuns deceases below a few tuns, the phase shift between tuns becomes significant and the model fails. 2nd-mode oscillato The potential fo impoved unloaded Q of the new solenoid model suppots its use in oscillatos with lowe phase noise. Examination of the benchmak pape by Leeson [4] eveals that neithe a paticula oscillato topology no cicuit complexity is equied to achieve low phase noise. In fact, ovely complex designs isk additional esonances fom component and layout paasitics. Design elegance emedies these poblems. I stated with the selection of a Mini-icuits MAR-3 silicon MMI fo the sustaining amplifie. To illustate the application of basic design elements towad design elegance, let me descibe the thoughts that dove the development of the poposed oscillato in Figue 1. This design achieves a loaded Q of 59 using a 2nd-mode solenoid esonato. The natual input and output impedances of the MAR-3 ae nea 5 ohms. To achieve a loaded Q of 59 would equie a chaacteistic impedance of nealy 4 ohms in a seies connected 2nd-mode solenoid. This equies high inductance and low capacitance esulting in small wie and poo unloaded Q, theefo eesonato coupling will be used. Supply voltage to the MAR-3 is typically deliveed though the device output using a esisto o choke inducto. We will use this same inducto as a shuntcoupling element to effectively lowe the impedance of the MAR-3 pesented to the 1st-mode esonato, thus inceasing loaded Q. A shunt-coupling eacto is also equied at the output of the 2nd-mode esonato. Because the 2nd-mode solenoid is now at D supply potential, a coupling capacito is selected so as not to shot the supply. A coupling inducto would equie a bypass capacito. To fom oscillato feedback, the output of the esonato must be connected to the input of the MAR-3. This would shot the supply voltage to the MAR-3 input. A coupling capacito is equied hee. The gain, phase shift and input/output match of the amplifie-esonato cascade must be managed to satisfy Bakhousen s oscillation citeia. The values of all elements in the design ae adjusted by compute optimization in GENESYS [2] to satisfy these citeia and to achieve the desied loaded Q. The open loop (fom pot 1 to pot 2) gain and phase esponse of the design ae given in Figue 1. The phase shift is nea zeo degees at 1 MHz, and the gain magin is 5.53 db. To fom the oscillato, the output is connected to the input. Since the oscillato is self-teminating, fo the analysis to be accuate, the cascade input and output impedances should be appoximately matched: in this case 8 db o bette etun loss. The output powe is taken though a 2 pf coupling capacito at pot 3. It is not guaanteed that any chosen topology will satisfy all of these citeia simultaneously. In fact, the schematic in Figue 1 was not my fist attempt fo this design. Histoically, topologies that satisfied design citeia wee named afte thei discovee, fo example, Hatley. Design based on the fundamentals fees the designe fom the shackles of a paticula topology and often esults in a moe elegant, highe pefomance design. Fo a futhe desciption of design methods please efe to [9]. Limitations of the new model This pape illustates the usefulness of the new solenoid model. I found it compelling that inductance and capacitance measued at low fequency ae capable of pedicting solenoid behavio at high fequencies. Howeve, notice that the esonances in the measued solenoid data in Figue 3 ae not hamonically elated as they would be fo a simple tansmission line model. The eason is that popagation on a solenoid ove a gound plane is dispesive; it is not pue-tem mode. Only one month afte Rhea [1], Mezak [7] published a pape that pesents a solution to the solenoid that consides dispesion. Mezak s solution is moe mathematically involved than Rhea s solution, but Mezak s is moe accuate. Shielded solenoids ae less dispesive and theefoe the simple model is moe applicable. In Rhea [1], the measued data in Figue 16 of the helical coax unit shown in Figue 15 exhibits hamonically elated 1st (anti-esonant), 2nd (seies-esonant) and 3d (anti-esonant) modes. Dispesion is also low fo solenoids that ae small with espect to a fee-space wavelength and 4 APPLIED MIROWAVE & WIRELESS

7 Figue 1. The open loop tansmission gain and phase and loaded Q (left) of a 1 GHz 2nd-mode solenoid oscillato (lowe ight). The cascade input and output etun loss ae given on the Smith chat (uppe ight). that ae mounted close to gound planes. Dispesion is most sevee in lage solenoids with few tuns, such as the unit in Figue 18 of Rhea [1]. Refeences 1. A.G. Bogle, The Effective Inductance and Resistance of Sceened oils, Jou. IEEE, 194: GENESYS 7 Use s Guide, Eaglewae opoation, Nocoss, GA, M.A.R. Gunston, Micowave Tansmission-Line Impedance Data, Noble Publishing, Atlanta, D.B. Leeson, A Simple Model of Feedback Oscillato Noise Spectum, Poc. IEEE, Febuay 1966: G. Matthaei, L. Young and E.M.T. Jones, Micowave Filtes, Impedance-Matching Netwoks, and oupling Stuctues, Atech House, Dedham, Massachusetts, R.G. Medhust, H.F. Resistance and Self-apacitance of Single-Laye Solenoids, Wieless Enginee, Febuay 1947: 35 43; and Mach 1947: J.A. Mezak, Modeling Helical Ai oils fo Wieless and RF Applications, RF Design, Januay 1998: R.W. Rhea, HF Filte Design and ompute Simulation, Noble Publishing, Atlanta, R.W. Rhea, Oscillato Design and ompute Simulation, Noble Publishing, Atlanta, R.W. Rhea, A Multimode High-Fequency Inducto Model, Applied Micowave & Wieless, Novembe/Decembe 1997: S/FILTER Manual, Eaglewae opoation, Nocoss, GA, A.G. Webste, An Expeimental Detemination of the Peiod of Electical Oscillations, Physical Review, Vol. VI, 1898: H.A. Wheele, Inductance Fomulas fo icula and Squae oils, Poc. IEEE, Vol. 75, No. 2, 1982: Autho infomation Randy Rhea is the founde of Eaglewae opoation and Noble Publishing. He eceived his BSEE fom the Univesity of Illinois and his MSEE fom Aizona State Univesity. He has been a RF and micowave design enginee fo nealy 3 yeas. He can be eached at APPLIED MIROWAVE & WIRELESS