designing wide-band transformers for hf and vhf...

Mar/Apr 2005 3

Sonoran Radio ResearchPO Box 25240Tempe, AZ [email protected]

Designing Wide-bandTransformers for HF andVHF Power Amplifiers

By Chris Trask, N7ZWY

The author describes the alternatives available in the designof transformers for solid state RF amplifiers. The key

parameters of different construction techniquesare discussed with results shown for each.

IntroductionIn the design of RF power amplifiers,

wide-band transformers play animportant role in the quality of theamplifier as they are fundamental indetermining the input and outputimpedances, gain flatness, linearity,power efficiency and other performancecharacteristics. The three forms oftransformers that are encountered,unbalanced-to-unbalanced (unun),balanced-to-balanced (balbal), andbalanced-to-unbalanced (balun), areused in various combinations toaccomplish the desired goals.

Careful consideration needs to be

given when making choices of themagnetic materials (if any is to be used),the conductors, and the method ofconstruction, as the choices made weighsignificantly in the overall performanceof the transformer. The type and lengthof the conductors and the permeabilityof the magnetic material are theprimary factors that determine thecoupling, which in turn determines thetransmission loss and the low frequencycutoff. The type and length of conductorused and the loss characteristics of themagnetic material also affects thecoupling, and further influences theparasitic reactances that affect the highfrequency performance.

Parasitics and ModelsTransformers are not ideal compo-

nents, and their performance is highlydependent upon the materials used

and the manner in which they are con-structed. The transmission losses andthe low frequency cutoff are primarilydependent upon the method of con-struction, the choices of magnetic ma-terial and the number of turns on thewindings or length of the conductors.These choices further determine theparasitic reactances that affect thehigh frequency performance, whichinclude, but are not limited to, resis-tive losses, leakage inductance,interwinding capacitance and windingself capacitance. A complete equiva-lent model of a wide-band transformeris shown in Fig. 1.1 Here, the seriesresistances R1 and R2 represent thelosses associated with the conductorsin the primary and secondary wind-ings, respectively. These resistancesare nonlinear, increasing with1Notes appear on page 15.

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Fig 1—Complete wideband transformer model.

Fig 4—Equivalent circuit for determiningC'12 and C'22.

Fig 3—Equivalent circuit for determiningC'11.

Fig 2—Lossless widebandtransformer model.

frequency because of the skin effect ofthe wire itself.2 Since wide-band trans-formers using ferromagnetic cores havefairly short lengths of wire, the contri-bution of the resistive loss to the totalloss is small and is generally omitted.2

The shunt resistance RC represents thehysteresis and eddy current lossescaused by the ferromagnetic material,3which increases with ω2 or even ω3, andis significant in transformers that areoperated near the ferroresonance of thecore material.2 This is a serious con-sideration in the design of transform-ers used at HF and VHF frequencies,and therefore requires that proper con-sideration be given to the selection ofthe core material.

The low frequency performance isdetermined by the permeability of thecore material and the number of turnson the windings or length of theconductors. The mutual inductance Mof Fig 1 is a result of the flux in thetransformer core4 that links the twowindings. The high frequency perfor-mance is limited by the fact that notall of the flux produced in one windinglinks to the second winding, a deficiencyknown as leakage,5 which in turnresults in the primary and secondaryleakage inductances Ll1 and Ll2 ofFig 1. Since the leakage flux paths areprimarily in air, these leakage in-ductances are practically constant.6,7

The capacitances associated withwideband transformers are gener-ally understood to be distributed,but it is inconvenient to modeltransformers by way of distributedcapacitances per se, so lumped ca-

pacitances are used. In Fig 1, capaci-tor C11 represents the distributedprimary capacitance resulting fromthe shunt capacitance of the primarywinding. Likewise, C22 representsthe shunt capacitance of thesecondary winding. Capacitor C12 isreferred to as the interwinding ca-pacitance,8 and is also a distributedcapacitance. In transformers havinga significant amount of wire, the in-ter-winding capacitance can interactwith the transformer inductancesand create a transmission zero. Ingood quality audio transformers, theinter-winding capacitance is mini-mized by placing a grounded coppersheet between the windings, oftenreferred to as a Faraday Shield.

The complete model of the wide-bandtransformer shown in Fig 1 is wellsuited for rigorous designs in which thetransformer is used near the limits ofits performance. These and othermodels are well suited for use indetailed computer simulations.9 Ingeneral practice and in analyticalsolutions, it is more convenient toconsider the lossless wide-bandtransformer model shown in Fig 2. Thismodel has been reduced to the reactivecomponents and an ideal transformer.10

The three model capacitances of Fig 2are related to the model capacitancesof Fig 1 in the following manner:7

−+′ 1

n1

C n

C = C 122

2222 (Eq 3)

Using the lossless model of Fig 2, wecan devise two models that are propersubsets that can be used to measure thevarious reactive components. The modelof Fig 3 is used to visualize the mea-surement of the primary shunt capaci-tance C'11 and the primary referredequivalent series inductance LEQ1.10

Likewise, the model of Fig 4 is used tovisualize the measurement of thesecondary shunt capacitance C’22, theprimary referred equivalent series in-ductance LEQ2 and the interwinding ca-pacitance C’12.10 The turns ratio n of theideal transformer is the actual ratio ofthe physical number of turns betweenthe primary and secondary windings.The procedure for determining the val-ues of the parasitic reactances of Fig 2is as follows10:

1.With the secondary open, mea-sure the primary winding inductanceLP at a frequency well below the highfrequency cutoff of the transformer.

2.With the primary open, measurethe secondary winding inductance LS,also at a frequency well below the highfrequency cutoff of the transformer.

3.With the secondary open, apply asignal at an appropriate mid-bandfrequency (between the low and highfrequency cutoff frequencies) to theprimary winding and measure theinput and output voltages v1 and v2.

3. Calculate the coupling coefficientk using:

1 LL

vv

= kS

P ≤1

2(Eq 4)

−+′

n1

1C C = C 121111 (Eq 1)

nC

= C 1212′ (Eq 2)

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4. Calculate the mutual inductanceM using:

k L L = M SP (Eq 5)5. Calculate the primary leakage

inductance Ll1 using:

Mn L = L Pl −1 (Eq 6)6. Calculate the secondary leakage

inductance Ll2 using:

nM

L = L Sl −2 (Eq 7)

7. Calculate the primary referredequivalent inductance LEQ1 using:

12

21 l

l

l2

EQ L Ln ML Mn

= L ++

(Eq 8)

8. Calculate the secondary referredequivalent inductance LEQ2 using:

22

1

12 l

l

lEQ L n

L MnL Mn

= L ++ (Eq 9)

9. Referring to Fig 4, connect agenerator to the primary and an ap-propriate load across the secondary.Measure the transmission parallelresonant frequency f12, which is thefrequency at which the voltage acrossthe secondary is at a minimum.

10. Calculate C'12 using:

( )

f 2 L

1 = C

2EQ 122

12π

′ (Eq 10)

11. With the generator still con-nected to the primary and the second-ary open, measure the input seriesresonant frequency f22, which is thefrequency at which the voltage acrossthe primary is at a minimum.


( )( )2EQ

2EQ

f 2 L

f 2 L C 1 = C

222

2221222

π

π′−′ (Eq 11)

13. Referring to Fig 3, connect a gen-erator to the secondary and leave theprimary open. Measure the output se-ries resonant frequency f22, which is thefrequency at which the voltage acrossthe secondary is at a minumum.


( )( )2EQ

2EQ

f 2 L

f 2 L C 1 = C

111

1111211

π

π′−′ (Eq 12)

15. Calculate C12 using:

12C n = C12 ′ (Eq 13)16. Calculate C11 using:

−−′′

n1

1C C = C 121111 (Eq 14)

17. Calculate C22 using:

−−′ 1

n1

C C n = C 122

2222 (Eq 15)

MatchingOnce the values of the parasitic

reactive elements of the wide-bandtransformer model have been deter-mined, it is possible to design thetransformer into a matching networkthat not only absorbs them, but makesuse of them in forming a 3-pole πnetwork low-pass filter section.10,11,12

We begin by considering the fact that

in a properly designed transformer theequivalent series inductance will domi-nate and will determine the maximumfrequency for which a matching net-work can be realized. The input andoutput capacitances C11 and C22 are usu-ally much smaller than required forrealizing a π network low-pass filtersection, so additional padding capaci-tors will be required to properly designthe matching network. These threecomponents allow us to design 3-poleButterworth, Bessel, Gaussian, andTchebyschev filter sections with theequivalent series inductance dictatingthe cutoff frequency.

The presence of the interwindingcapacitance C12 suggests that a singleparallel-resonant transmission zero canbe included, which gives us further pos-sibilities of inverse Tchebyschev andElliptical (Cauer) filter sections. Sincethe interwinding capacitance is gener-ally small, it will also require an addi-tional padding capacitor to complete thedesign. Note, however, that adding atransmission zero to the matching net-Fig 5—Matching network components.

Table 1—Matching Section Prototype Values13

Filter Type C1norm C2norm Lnorm C3normButterworth 1.000 1.000 2.000Bessel 1.255 0.192 0.553Gaussian 2.196 0.967 0.336

Tchebyschev 0.1 dB 1.032 1.032 1.147 0.5 dB 1.596 1.596 1.097 1.0 dB 2.024 2.024 0.994

Inverse Tchebyschev 20 dB 1.172 1.172 2.343 0.320 30 dB 1.866 1.866 3.733 0.201 40 dB 2.838 2.838 5.677 0.132

Elliptical(0.1 dB Passband Ripple) 20 dB 0.850 0.850 0.871 0.290 25 dB 0.902 0.902 0.951 0.188 30 dB 0.941 0.941 1.012 0.125 35 dB 0.958 0.958 1.057 0.837 40 dB 0.988 0.988 1.081 0.057



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work is not practical for transformersthat go from balanced to unbalancedsources and loads (baluns) as theequivalent series inductances from theunbalanced port to the balanced portsare not identical.10

To begin the process of designing thewide-band transformer into a matchingnetwork, we must first decide what sortof passband performance is desired, andthen select the appropriate filterprototype values from Table 1. Now,with reference to the componentreference designators of Fig 5, thedesign process proceeds as follows:10

1. Calculate the maximum usablefrequency ωmax using:

EQ1

Snormmax L

R L = ω (Eq 16)

where RS is the source resistance2. Calculate the value for the input

matching capacitor C1 using:

11Smax

1norm1 C

R C

= C −ω (Eq 17)

3. Calculate the value for the out-put matching capacitor C2 using:

22Smax

2norm2

2 C R

C n = C −ω (Eq 18)

4. If required, calculate the valuefor the capacitor C3 using:

12Smax

3norm3 C

R C n

= C −ω (Eq 19)

Magnetic MaterialsThe first concern in the design of a

wide-band transformer is the choiceof the magnetic material. Both ferriteand powdered iron materials can beused, but ferrite is preferred overpowdered iron as the losses are lower.Powdered iron is lossier because of thedistributed air-gap nature of thematerial,13 and the excessive losses notonly result in decreased gain perfor-mance, but in power amplifierapplications they also result inexcessive heating that can damageinsulating and PC board materials.

There are three essential types offerrite materials that can be used forHF and VHF frequencies. These arelisted in Table 2. The first of these ismanganese-zinc (MnZn), which isgenerally suited for lower frequenciesand low power. Fair-Rite type 77material is an exception. It is availablein the form of E-I cores which can beused for high-power transformer coresat lower HF frequencies.

The second, which will be discussedlater, and undoubtedly most populartype of ferrite is nickel-zinc (NiZn). Ofthe three NiZn materials listed in

Table 2, the Fair-Rite types 43 and 61are by far the most widely because oftheir low loss, high saturation flux,and the wide variety of shapes andsizes that are available. They can bereadily used for both HF and VHFapplications, with the 61 materialbeing preferred for VHF. These twoferrites will be the focus of theapplications to be discussed later.

The third type of ferrite suitable forHF and VHF applications is cobalt-nickel-zinc (CoNiZn), available fromFerronics as types K and P. Theseferrites are available in a limitednumber of shapes and sizes. Toroidsmade from these materials can be usedto make transformer cores by string-ing them along a brass tube in a frame,as will be discussed later. The onedrawback to this material is that it canbe permanently damaged if it issubjected to excessively high fluxdensities.10

Transformer CoresThe ferrite materials mentioned in

the previous section are available in afairly wide variety of shapes such asrods, toroids, beads (or sleeves), E-I

Table 2—Commercial Ferrites Suitable for Power Amplifier Applications

Manganese-Zinc (MnZn) FerritesManufacturer Type Permeability Saturation Loss Usable Resonance

(ui) Flux (Gauss) Factor Frequency FrequencyFair-Rite 77 2000 4900 15 1 MHz 2MHz

Nickle-Zinc (NiZn) FerritesFair-Rite 43 850 2750 85 5 MHz 10MHzSteward 28 850 3250Fair-Rite 61 125 2350 32 25 MHz 50MHzSteward 25 125 3600 15 MHz 25MHz

Cobalt-Nickle-Zinc (CoNiZn) FerritesFerronics K 125 3200 50 MHz 60 MHzFerronics P 40 2150 80 MHz 100 MHz

Fig 7—High-power rftransformer core.

Fig 6—Binocular core.

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cores, and multi-aperture cores. Amongthe various multi-aperture coresavailable, there is one form, shown inFig 6, that is commonly referred to as a“binocular core” as the shape suggeststhat of a pair of field glasses. This shapeis available from numerous small sizessuitable for small-signal transformersto larger sizes suitable for poweramplifiers up to 5 W. Similar coresavailable from Fair-Rite having arectangular rather than an oval cross-section are available in larger sizessuitable for amplifiers of 25 W or more.

Higher-power amplifiers requirecores with larger cross-sections that canaccommodate the higher flux densitiesin the magnetic material. For theseapplications, it is more suitable toconstruct a transformer core usingferrite beads (or sleeves) supported bya frame made from brass tubing andPC board material, such as those thatare available from CommunicationsConcepts and made popular by thenumerous applications notes and otherpublications by Norman Dye and HelgeGranberg.14,15,16,17 An illustration of aCommunications Concepts RF600 coreassembly is shown in Fig 7. Notice thatthe left-hand endplate has two separateconductors while the right-hand endplate has a single conductor. This ishelpful in forming a center-tap groundconnection in some applications. In anapplication to be described later in thedesign of balun transformers, it will beseen that there are times when it isadvantageous to dispense with thiscommon connection.

The transformer core of Fig 7 canbe made for higher power levels byusing multiple ferrite beads along thesupporting tubes, such as the RF-2043assembly offered by CommunicationsConcepts. Such an assembly techniquecan also allow for the use of toroids toprovide a transformer core having alarger cross section or to provide ameans of using ferrite materials in theform of toroids when beads are notavailable as was mentioned earlier.

Conventional Wide-bandTransformers

The most common method used inthe design of power amplifiers for HFand VHF frequencies is shown inFig 8. Here, a 1:1 balun is made usingthe transformer core previously shownin Fig 7. The balanced side of thetrans-former is provided by the brasstubes that support the ferrite sleeveswith the center tap being provided bythe common connection foil of theright-hand endplate and the + and– terminals provided by the foil of theleft-hand endplate. The unbalanced

side of the transformer is provided byway of a piece of insulated wire thatis passed through the tubes.

There are at least two problems withtransformers constructed in thismanner, the first of which is the wirefor the unbalanced side of the circuitthat is exposed in the left-hand end ofthe assembly. The field created by thisexposed wire is not coupled to either thebrass tubes of the balanced side of thecircuit nor the ferrite material, and thisresults in excess leakage inductance.The second problem is that the couplingbetween the two sides of the circuit isnot uniform as the physical placementof the wire cannot be tightly controlled.This can lead to some small amount ofimbalance. Despite these problems, thisform of transformer remains verypopular in the design of amateur,commercial, and military HF and VHFpower amplifiers.

For demonstration purposes, a 1:1balun transformer was constructed,using a Communications Concepts RF-600 transformer core assembly, whichuses a pair of Fair-Rite 2643023402beads, made with type 43 material andhaving an inside diameter of 0.193 inch,an outside diameter of 0.275 inch, anda length of 0.750 inch.

The performance for this balun,shown in Fig 9, is marginal at best. Theaverage insertion loss for HFfrequencies is in the neighborhood of 2dB, and the cutoff frequency is around4 MHz. At higher frequencies, theinsertion loss improves to 1.2 dB, buteven this is of questionable value. Theslowly degrading return loss is more aresult of the increased losses caused bythe ferrite material, as was evidencedby the fact that adding matching capac-itors (see Fig 5) did little to improve theperformance. The increased trans-mission loss above 85 MHz is duemostly to the leakage inductancecaused by the exposed conductor on theleft-hand end of the assembly.

Transmission-Line TransformersThe leakage inductance of the balun

transformer of Fig 8, however small, isthe limiting factor for higher frequencyperformance. To fulfill the need forwide-band transformers at higherfrequencies and power, coaxial cable isoften employed as the conductors. Sincethe coupling takes place between theinner conductor and the outer shield,there is very little opportunity for anystray inductance. This means that wecan anticipate good performance at

Fig 9—Conventional 1:1 balun transformer performance.

Fig 8—Conventional1:1 baluntransformer.

8 Mar/Apr 2005

much higher frequencies, and it alsomeans that we can usually dispensewith the matching capacitors thatare often used with wide-band trans-formers.

In the design of transmission linetransformers, the cable should have acharacteristic impedance that is thegeometric mean of the source and loadimpedances:

LS0 ZZ = Z × (Eq 20)

In most cases, the use of coaxial cablehaving the exact impedance is simplynot possible as coaxial cable is generallyoffered in a limited number of imped-ances, such as 50 and 75 Ω. Otherimpedances such as 12.5, 16.7, 25, and100 Ω are available, but usually on alimited basis for use in military andcommercial applications. Low imped-ances such as 6.12 Ω are difficult toachieve, although it is possible toparallel two 12.5 Ω cables, which isstandard practice.16 The insertion losswill increase as the impedance of thecoaxial cable deviates from theoptimum impedance of Eq. 20. For mostapplications, the effects of using cablehaving a non-ideal characteristicimpedance is not great as long as theequivalent electrical length of the cableis less than λ/8. In general, the lineimpedance is not critical provided thatsome degree of performance degrad-ation is acceptable.16

The equivalent electrical length ofthe cable is actually longer than thephysical length due to the electricalproperties of the insulating materialbetween the inner and outer conduc-tors, and the relationship is:

rPE L L ε≅ (Eq 21)

where LE is the equivalent electricallength, LP is the actual physicallength, and εr is the relative dielectricconstant of the insulating material,typically 2.43 for PTFE. When thecable is inserted in a magneticmaterial, the equivalent electricallength is further lengthened by themagnetic properties of the material:

irPE L L µε≅ (Eq 22)

where µi is the relative permeabilityof the magnetic material. In general,a close approximation to the equiv-alent electrical length of the cable willbe a combination of Eq. 21 and Eq. 22,with the former applied to the lengthof cable that is outside the transformercore and the latter used for thatportion of the cable that is inside thetransformer core.

The 1:1 balun transformer of Fig 8is now modified by replacing theinsulated wire conductor with an

Fig 10—Transmission-line1:1 baluntransformer.

Fig 11—Conventional vs. Transmission-line 1:1 balun transformer performance.

Fig 12—Coaxialtransmission-line 1:1balun.

appropriate length of 0.141 inch OD50 Ω semi-flex coax cable, with a solder-filled braid outer conductor, as shownin Fig 10. Here, the cable is bent into aU shape and passed through the holesof the transformer core. The center tapfor the balanced side of the transformeris provided by soldering a wire to theouter conductor at the very center of thecurve. Because of the displacement ofthe center tap from the endplate, thecommon connection provided by thecopper foil on the right-hand endplate(see Fig 7) must be broken. Notice that

this design places the terminals for boththe balanced and unbalanced sides ofthe transformer on the same end ofthe core.

Semi-rigid coax is also availablewith the same 0.141 inch OD, but it isdifficult to use when small radii arerequired. The solid outer conductoroften splits or collapses if the bendingradius is too small. Semi-flex will bendto smaller radii, but will still splitwhen an excessively small radius isattempted. A mandrel, such as youwould use when bending copper

Mar/Apr 2005 9

tubing, should be used at all timeswhen bending these cables to thesmall radii required. A great deal ofcare must be exercised, which is bestdone by first bending the cable to alarger radius and then slowlydecreasing the radius until it issufficiently reduced so as to passthrough the two holes of the core withlittle effort. This method reduces therisk of splitting the outer conductor byway of distributing the mechanicalstresses over a longer length of thecable. For transformer cores havinglarger hole diameters, larger coaxialcables such as RG-58 and RG-59 canbe used, provided the outer vinyljacket is removed.

Fig 11 shows that the use of coaxialcable has done little to improve the lowfrequency characteristics of the 1:1balun transformer, however the highfrequency characteristics showsignificant improvement, especiallywith regard to the return loss. Withthe better coupling between the twocircuits, the losses induced by theferrite material have been reducedand a better match has been attained.Also, the lack of any appreciableincrease in the transmission lossabove 85 MHz indicates that theleakage inductance has been reduced,as was expected by using the coaxcable instead of wire for the conductor.

Transmission-Line BalunsReplacing the wire with coaxial

cable in the 1:1 balun transformer ofFig 7 and Fig 10 helps the highfrequency transmission loss andreturn loss performance to somedegree. It does not, however, improvethe low frequency performance nor thetransmission loss. This is due to thefact that the coupling coefficient of thetransmission line transformer ishighly dependent upon the length ofcable used.

Let’s take a broader look at the useof transmission line in the design of awide-band transformer. In this case,we’ll use a pair of 1:1 baluns as shownin Fig 12. We will use a length of 50-Ωsemi-flex cable as was used in theprevious example, but this timerequiring a tighter radius. The core forthe first of these transformers is a Fair-Rite 2843000102 binocular core, and forthe second a Fair-Rite 2861000102binocular core is used to demonstratethe differences in the performance ofthe two ferrite materials. Theperformance of these baluns is shownin Fig 13. It is immediately obvious thatthere is room for improvement. First,the transmission loss is 1.8 dB for thetransformer using the type 43 material

Fig 13—Coaxial transmission-line 1:1 balun performance (-0102 cores).

Fig 14—Coaxial transmission-line 1:1 balun performance (-6802 cores).

Fig 15—Extended coaxial transmission-line 1:1 balun using e-cores.

10 Mar/Apr 2005

and 1.5 dB for the type 61 material. Thecutoff frequency is 2.5 MHz for the type43 material and 11 MHz for the type61 material.

Another pair of transformers wereconstructed, this time using Fair-Rite2843006802 and 2861006802 binocularcores, approximately twice as long asthe previous -0102 cores. As shown inFig 14, this increase in the length oftransmission line improves thetransmission loss to about 1.1 dB forboth materials. As expected by virtueof the longer line length, the cutofffrequencies are significantly lower, lessthan 1 MHz for the type 43 materialand 4 MHz for the type 61 material.

Clearly, the longer length of coaxialcable has distinct advantages in termsof insertion loss and cutoff frequency.It would therefore appear obvious thatincreasing the length of the cable andferrite balun core further would resultin additional performance improve-ment. However, in the design of poweramplifiers we often encounter alimitation in terms of the amount ofphysical board space that is availablefor the various components.

A solution to increasing the lengthof the cable without sacrificing valuableboard space is to form the cable into aseries of two or more loops and embedit into an E-core, a two-turn version ofwhich is shown in Fig 15.16,17 Here, sixpieces of ferrite E-core have beencemented together to form a single pieceof ferrite. The method of constructionis to first cement two sets of three piecesof core material together to form theupper and lower halves of the to becompleted core. Next, the cable isformed to the shape necessary to fitwithin the channels of the core. Finally,the cable is placed inside the channelsof one core half and the second half iscemented in place.

The construction itself is fairlystraightforward, but implementing itonto a circuit board reveals a couple ofproblems, specifically the length of theleads for the two balanced ports areunequal and the coax loop on the right-hand side interferes with the unbal-anced and balanced positive terminals.At lower frequencies the inequality ofthe lead lengths will not presentsufficient imbalance in lead inductanceto create any problems, but withincreasing frequency the transformerwill become unbalanced and compen-sation will be required to offset theexcessive lead inductance, which will bedifficult to bring into balance. Anadditional constraint in the use of thisapproach is that the required E-coresare only available in the Fair-Rite type77 material, which is not well suited

Fig 16—Extended coaxial transmission-line 1:1 baluns using binocular cores.

Fig 17—Extended-length coaxial transmission-line 1:1 balun performance.

Fig 18—Coaxial transmission-line 4:1 balanced transformers.

Mar/Apr 2005 11

above lower HF frequencies. Even withthese shortcomings, the wide-bandtransformer approach of Fig 15 is wellworth consideration for applications atHF frequencies.

An alternative approach is shown inFig 16. A pair of ferrite binocular coreshave been used in place of the E-coresof Fig 15. Here, both ends of the cablehave equal lead lengths, and there isno mechanical interference to be dealtwith. The construction presents no moredifficulty than before. The cable is firstformed into a U shape, then passedthrough the holes of the first, or upperbinocular core. The free ends of thecable are then bent back over the firstcore, and subsequently passed throughthe holes of the second or lower core.The two cores may then be cementedtogether to make the assembly whole.A pair of endplates similar to thoseshown for the left-hand end of thetransformer-core assembly of Fig 7 maybe used to hold the two cores togetherand to ease mounting the transformeron the amplifier PC board.

A single example of the 1:1 baluntransformer of Fig 16 was constructed,using a pair of the longer Fair-Rite2843006802 binocular cores. The testresults shown in Fig 17 indicate thatfurther lengthening of the coaxial cablecontinues to improve the performance.A comparison of the three balunexamples using Fair-Rite type 43 ferritematerial is listed in Table 3, where thetransmission loss is as an average overwhat would be considered the usablefrequency range.

Even with the lower transmissionloss of the balun transformer of Fig 16,this performance of the transmission-line balun transformer remainssignificant. When used as an outputtransformer in a power amplifier, thisexcess loss degrades the powerefficiency, which should be taken intoconsideration in the overall design.

Other Transmission-LineTransformers

There are many possible impedanceratios that can be realized usingtransmission-line transformers.Fig 18 shows two methods for makingbalanced transformers having animpedance ratio of 4:1.15,16,17,18 The firstof these makes use of a singlebinocular core, and it should be

Fig 19—Coaxialtransmission line 9:1balancedtransformer.

Fig 20—Twisted-wire capacitances.

Fig 21—Two-conductor twisted-wire transformer configurations.

Fig 22—Three-conductor twisted-wire transformer configurations.

Table 3

Configuration Cutoff InsertionFrequency Loss

Double 2843006802 Core 1.1MHz 0.8dBSingle 2843006802 Core 3.9MHz 1.1dBSingle 2843000102 Core 11MHz 1.8dB

12 Mar/Apr 2005

obvious from the examples shown formaking balun transformers that thecore should be as long as possible. Thesecond method makes use of the samebent cable design used for making theearlier baluns of Fig 12,18 and theextended length design of Fig 16 canalso be used here to conserve boardspace, decrease the cutoff frequency,and decrease the transmission loss.

Fig 19 shows a method by which abalanced transformer having animpedance ratio of 9:1 can be realized,also using the bent cable designs ofFig 12 and Fig 16.16, 17,18

Numerous additional transformerscan be realized by way of a variety ofcombinations that will result ininteger impedance ratios.19

Twisted WireThe method of using coaxial cable

has been widely used in the design ofwide-band transformers for high powerand high frequencies, but the limita-tions imposed by low frequency perfor-mance and excessive transmission lossare a direct consequence of shortenedline lengths. These are often imposedon the designer due to limited physicalspace. We could make transformers atlower frequencies and with lowertransmission losses by extending theapproach of Fig 16 by using more cores,but this method would have its ownlimitations and can become fairlyexpensive to produce. An alternative to

Fig 25—Balun using monofilar primary and bifilar secondary.

Fig 23—Balanced-balanced transformer,4:1 transformation ratio.

Fig 24—Balanced-balancedtransformer, 4:1transformationratio.

Fig 26—Trifilar wound 1:1 balun.

the use of coaxial cable for the conduc-tors is twisted wire. It is by far moreflexible than cable, and with properattention to a few details it can yieldtransformers having exceptionalperformance in terms of bandwidth andtransmission losses.

It is first necessary to understandthat you cannot twist together anynumber of wires when making atransformer, and Fig 20 illustrates whynot. In the first case, a pair of wirestwisted together (bifilar) have a fairlyuniform distributed interconductorcapacitance. Both conductors will havea uniform and equal distributedinductance due to the uniformdistributed mutual inductance betweenthe two conductors, a characteristic thatallows twisted wire to be seen as a formof transmission line.20

In the second case, three wirestwisted together (trifilar) exhibit thesame properties. The distributedcapacitance is the same between allthree conductors, as is the distributedinductance. This implies that thecharacteristic impedance and propa-gation constant for all three wires arethe same.21

In the third case, four wires twistedtogether (quadrifilar) show dissimilarproperties. While the interconductorcapacitance between adjacent conduc-tors is the same, the capacitancebetween diagonally opposite conductorsis reduced by 1/√2, and the mutualinductance between these conductors isalso reduced. Now, the characteristicimpedances and propagation constantsfor the conductors are not identical, andthis results in unequal coupling as wellas phasing problems. This is more

noticeable as the amount of wire isincreased beyond λ/8. Because of this,it is best to use no more than threetwisted wires in the design of wide-bandtransformers.21

Another aspect of twisted wire thatneeds consideration is the insulation.The polyurethane nylon insulationused on magnet wire offers the bestin terms of dielectric properties andlosses, but is easily abraded whenpassed through the holes of a ferritebinocular core. Teflon and polyvinylchloride (PVC) insulated wire is moredurable, but the insulation tends tohave higher dielectric losses. In all,magnet wire is the better choiceprovided you exercise adequate carewhen assembling the transformer.

Twisted-Wire TransformersThe schematics of Fig 21 illustrate

a few of the transformers that can berealized using a pair of twisted wires.The 4:1 autotransformer and the 1:1unbalanced-to-unbalanced (unun)phase inverter are fairly obvious,whereas the 1:1 balun requires that aground reference be supplied elsewherein the circuit. This configuration isconvenient when used with an addi-tional transformer such as the 4:1balbal that will be discussed later inthis section.

The addition of a third wire increasesthe number of practical wide-bandtransformer configurations, as shown inthe schematics of Fig 22. The 9:4 and9:1 ununs and the 4:1 balun are allfairly obvious. Both the 4:1 and 9:1balbals require a ground referenceelsewhere in the circuit, just as with thebifilar 1:1 balun shown earlier. The 1:1

Mar/Apr 2005 13

Fig 27—Twisted-wire balun return loss.

balun offers the best possible perform-ance for this essential type of trans-former as will be shown later, and itprovides the center tap that is neededwhen using the 4:1 and 9:1 trifilarbalbals.

An additional wide-band trans-former that finds wide usage is the 4:1balbal shown schematically in Fig 23.It would appear at first that this wouldbe a suitable application for quadrifilartwisted wire, but due to the unequalcoupling considerations discussedearlier, an alternative approach is used.Shown in Fig 24, the transformer ofFig 23 is realized by winding a pair ofbifilar conductors around the outside ofthe balun transformer core. Windingthe wires along the outside results inan increased amount of leakage induc-tance, which is minimized by using thetwisted wire. The wires are twistedtogether as A/C and B/D pairs, as shownin the designations used in both Fig 23and Fig 24. This transformer isparticularly convenient as it providesa center tap that is needed when usingthe bifilar 1:1 balun and the 4:1 and9:1 trifilar balbals.

Twisted-Wire BalunsPerhaps the best way to illustrate

the distinct advantages of twisted wireis to examine the performance of acouple of variations of the 1:1 balun. Thenumber of configurations for thistransformer are too numerous tomention, and we are going to narrow itdown to just two forms. The first of theseis shown in Fig 25, and consists of amonofilar (single wire) primary and abifilar secondary. This form of construc-tion is very convenient as it places theunbalanced terminals on one end of thecore and the balanced terminals on theopposite. The example to be used hereconsists of two turns of #26 bifilar andfour turns of #26 monofilar wound on aFair-Rite 2843000102 binocular core. Apair of 47 pF mica capacitors were usedfor the matching capacitors C1 and C2(ref. Fig 5).

The second form to be evaluated isthe trifilar balun shown in Fig 26,which had earlier been shownschematically in Fig 22. Two of thesewere evaluated, each having threeturns of #26 trifilar wire, the first ona Fair-Rite 2843000102 core (as usedin the monofilar/bifilar balun) and thesecond on a 2861000102 core so as toagain illustrate the the difference inperformance tetween the two ferritematerials. The leakage inductances forboth of the trifilar baluns werenegligible, so no matching capacitorswere required.

As shown in Fig 27, all three Fig 28—Twisted-wire balun transmission loss.

transformers show good return loss(>15 dB) down to 1 MHz. At higherfrequencies, the monofilar/bifilarbalun of Fig 25 degrades beyond55 MHz while the trifilar baluns ofFig 26 continue to be usable beyond100 MHz. This performance isrepeated in the transmission lossesshown in Fig 28, where all threetransformers exhibit less than 0.25 dBloss for lower frequencies. Themonofilar/bifilar balun of Fig 25degrades rapidly beyond 55 MHzwhile the trifilar baluns of Fig 26 show

little sign of degradation up to100 MHz.

Twisted-Wire TransformerPower Amplifier Design

Let’s now take a look at a 1 W poweramplifier using twisted-wire wide-bandtransformers. The schematic of Fig 17is a push-pull augmented class-AB am-plifier.22,23,24 The transistors Q1 and Q2are 2N4427s, and diode D1 is a 1N4001.The emitter resistors R1 and R2 are6.0 Ω, each comprised of two 12 Ω 1206size SMT resistors in parallel. This

14 Mar/Apr 2005

Fig 29—Power amplifier (1 W) using twisted-wire transformers.

Fig 30—Return loss of 9:1 input balun.

Fig 31—Gain performance of 1 W amplifier.

assumes a 0.25 Ω emitter input resis-tance for the two transistors. The sup-ply voltage is 12 V, and resistor R3 isadjusted so that each transistor drawsa quiescent bias current of 10 mA.

T1 is a 1:1 balun transformer madewith three turns of #32 trifilar wirewound on a Fair-Rite 2843002402binocular core as shown in Fig 26. The4:1 balbal transformer T2 is made asshown in Fig 24, with three turns of#32 bifilar wire on each side of a Fair-Rite 2843002402 binocular core. Theaugmentation transformer T3 is alsomade on a Fair-Rite 2843002402binocular core with #32 bifilar wire,with one turn on the primary side andtwo turns on the secondary side. The4:1 output balun transformer T4 ismade of three turns of #26 trifilar wireon a Fair-Rite 2843000102 binocularcore, as shown in Fig 14.

A few words should be said here toexplain the theory of linearity aug-mentation, a new method for improv-ing the linearity of common-baseamplifiers. In the schematic of Fig 29,the signal voltages at the emitters oftransistors Q1 and Q2 are inverted,amplified, and coupled to the bases oftransistors Q1 and Q2. These emittersignal voltages are a result of the finitenonlinear emitter resistances of thetwo transistors, and are the principalsource of nonlinear distortion incommon-base amplifiers. By invertingthese voltages and coupling them tothe transistor bases, the signalvoltages at the emitters are reducedby as much as 85% for a transformerturns ratio of 2:1. As the turns ratio isincreased, the input current to thetransistor emitters becomes increas-ingly dependent upon the fixedresistors R1 and R2, thereby improvingthe linearity.

Mar/Apr 2005 15

In the push-pull amplifier, the actionof the augmentation transformerfurther causes the transistors to turnon and off faster during their respectivenegative and positive cycles, therebyimproving the collector efficiency to 95%or more.

The input return loss for thetransformers T1 and T2 is shown inFig 30. These tests were made with thetransformers terminated with a single12 Ω load resistor to make themeasurements easier and to show theperformance of the transformersthemselves. The matching capacitorswere 10 pF at the input and 68 pF atthe 12 Ω load resistor.

The gain and return loss of theamplifier is shown in Fig 31. Byapplying augmentation, the emitterinput resistances of the transistors arereduced to about 0.25 Ω, which with the6.0 Ω emitter input resistors providesan almost perfect match out to 95 MHzafter increasing the value of the secondmatching capacitor to 100 pF toaccommodate the inductive inputreactance of the two transistors. Witha current gain of 2.0 at the input andan additional current gain of 2.0 at theoutput, the power gain should be in theorder of 12 dB minus 0.75 dB for thethree transformers, or 11.25 dB. Thepower gain is at the expected 11.25 dBfor most of the HF bands, and has a lowcutoff frequency of around 1 MHz whichcould be improved with more turns onthe three transformers. The gain dropsslightly with increased frequency dueto stray capacitance at the transistorcollectors.

AcknowledgementsI would like to thank Fair-Rite

Products Corporation for having beengenerous in supplying some of theferrite cores that were used in thepreparation of this article. Theparticipation of manufacturers isalways welcome when makingextensive evaluations such as has beenmade here, and especially when small

quantities of parts used in suchevaluations are not readily availablethrough distributors.

Notes1E. Snelling, Soft Ferrites: Properties and

Applications (2nd ed.), Butterworths, 1988.2N. Grossner, Transformers for Electronic

Circuits, McGraw-Hill, 1967, Chapter 8.3F. Terman, Radio Engineers’ Handbook,

McGraw-Hill, 1943.4 P. Gillette, K. Oshima, and R. Rowe, “Mea-

surement of Parameters Controlling PulseFront Response of Transformers,” IRETransactions on Component Parts, Vol. 3,No. 1, Mar 1956, pp 20-25.

5T. O’Meara, “Analysis and Synthesis withthe ‘Complete’ Equivalent Circuit for theWide-Band Transformer,” Transactions ofthe AIEE, Part I, Mar 1962, pp 55-62.

6H. Lord, “The Design of Broad-Band Trans-formers for Linear Electronic Circuits,”Transactions of the AIEE, Part 1, Vol. 69,1950, pp 1005-1010.

7K. Clarke and D. Hess, Communication Cir-cuits: Analysis and Design, Addison-Wesley, 1971.

8Nordenberg, Harold M., Electronic Trans-formers, Reinhold, 1964.

9N. Hamilton, “RF Auto-Transformers-Transmission Line Devices Modelled Us-ing Spice,” Electronics World, Nov 2002,pp 52-56 (Part 1), Dec 2002, pp 20-26(Part 2).

10C. Trask, “Wideband Transformers: An In-tuitive Approach to Models, Characteriza-tion and Design,” Applied Microwave &Wireless, Nov 2001, pp 30-41.

11A. Hilbers, “High-Frequency WidebandPower Transformers,” Electronic Applica-tions, Vol. 30, No. 2, Apr 1971, pp 64-73(later published as Philips ApplicationNote ECO6907, Design of HF WidebandPower Transformers, March 1998).

12A.Hilbers, “Design of High-FrequencyWideband Power Transformers,” Elec-tronic Appl. 64-73 (later published asPhilips Application Note ECO7213, De-sign of HF Wideband Power Transform-ers; Part II, March 1998).

13C.Trask, “Powdered Iron Magnetic Mate-rials,” 2002 IEEE MTT-S Workshop onPassive Components for RF Applications,Seattle, June 2002.

14H. Granberg, Get 600 Watts RF From FourPower FETs, Motorola Engineering Bulle-tin EB-104, 1983. !!

15H.Granberg, Broadband Transformers andPower Combining Techniques for RF,Motorola Applications Note AN-749, 1975.

16N.Dye and H. Granberg, Radio FrequencyTransistors: Principles and Practical Applica-tions (1st ed), Butterworth-Heinemann, 1993.

17N.Dye and H. Granberg, “Using RF Tran-sistors: Transforming Wideband Circuits,”Electronics World & Wireless World, July1994, pp. 606-612.

18R. Blocksome, “Practical Wideband RFPower Transfomers, Combiners, and Split-ters,” RF Expo West, 1986, pp 207-227.

19T. Gluszczak. and J.D. Harmer, “Trans-mission Line Transformers with Integer-Ratio Voltage Transformations,” IEEE1974 International Symposium on Circuitsand Systems, pp 368-370.

20P. Lefferson, “Twisted Magnet Wire Trans-mission Line,” IEEE Transactions on Parts,Hybrids, and Packaging, Vol. 7, No. 4, Dec1971, pp 148-154.

21G.Pandian, Broadband RF Transformersand Components Constructed withTwisted Multiwire Transmission Lines,PhD Thesis, Indian Institute of Technol-ogy, Delhi, 1983.

22C.Trask, “Common Base Amplifier Linear-ization Using Augmentation,” RF Design,Oct 1999, pp 30-34.

23C.Trask, “Common Base Transistor Am-plifiers With Linearity Augmentation,” USPatent 6,271,721, 7 Aug 2001.

24C.Trask, “High Efficiency Broadband Lin-ear Push-Pull Power Amplifiers UsingLinearity Augmentation,” IEEE 2002 Inter-national Symposium on Circuits and Sys-tems, Phoenix, AZ, May 2002, Vol. 2, pp432-435.

Chris Trask, N7ZWY, is the principalengineer of Sonoran Radio Research,where he actively pursues methods forlinearizing amplifiers and mixers. Hehas published numerous articles andpapers in hobby, trade and profes-sional publications and holds six pat-ents in the application of feedback tomixers and the linearization techniqueof augmentation. Chris received hisBSEE and MSEE degrees from thePennsylvania State University and isa senior member of the IEEE.

You may reach the author atSonoran Radio Research, P.O. Box25240, Tempe, AZ 85285-5240 [email protected].

designing wide-band transformers for hf and vhf...

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