Designing Wide-Band Transformers for HF and VHF Power Amplifiers
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Designing Wide-band Transformers for HF and VHF Power Amplifiers The author describes the alternatives available in the design of transformers for solid state RF amplifiers. The key parameters of different construction techniques are discussed with results shown for each. By Chris Trask, N7ZWY Introduction given when making choices of the and the manner in which they are con- In the design of RF power amplifiers, magnetic materials (if any is to be used), structed. The transmission losses and wide-band transformers play an the conductors, and the method of the low frequency cutoff are primarily important role in the quality of the construction, as the choices made weigh dependent upon the method of con- amplifier as they are fundamental in significantly in the overall performance struction, the choices of magnetic ma- determining the input and output of the transformer. The type and length terial and the number of turns on the impedances, gain flatness, linearity, of the conductors and the permeability windings or length of the conductors. power efficiency and other performance of the magnetic material are the These choices further determine the characteristics. The three forms of primary factors that determine the parasitic reactances that affect the transformers that are encountered, coupling, which in turn determines the high frequency performance, which unbalanced-to-unbalanced (unun), transmission loss and the low frequency include, but are not limited to, resis- balanced-to-balanced (balbal), and cutoff. The type and length of conductor tive losses, leakage inductance, balanced-to-unbalanced (balun), are used and the loss characteristics of the interwinding capacitance and winding used in various combinations to magnetic material also affects the self capacitance. A complete equiva- coupling, and further influences the lent model of a wide-band transformer accomplish the desired goals. 1 Careful consideration needs to be parasitic reactances that affect the high is shown in Fig. 1. Here, the series frequency performance. resistances R1 and R2 represent the losses associated with the conductors Sonoran Radio Research Parasitics and Models in the primary and secondary wind- PO Box 25240 Transformers are not ideal compo- ings, respectively. These resistances Tempe, AZ 85285-5240 nents, and their performance is highly are nonlinear, increasing with [email protected] dependent upon the materials used 1Notes appear on page 15. Mar/Apr 2005 3 frequency because of the skin effect of pacitances are used. In Fig 1, capaci- C22 1 the wire itself.2 Since wide-band trans- tor C represents the distributed C22′ = + C12 − 1 (Eq 3) 11 2 n formers using ferromagnetic cores have primary capacitance resulting from n fairly short lengths of wire, the contri- the shunt capacitance of the primary Using the lossless model of Fig 2, we can devise two models that are proper bution of the resistive loss to the total winding. Likewise, C22 represents loss is small and is generally omitted.2 the shunt capacitance of the subsets that can be used to measure the various reactive components. The model The shunt resistance RC represents the secondary winding. Capacitor C12 is hysteresis and eddy current losses referred to as the interwinding ca- of Fig 3 is used to visualize the mea- caused by the ferromagnetic material,3 pacitance,8 and is also a distributed surement of the primary shunt capaci- 2 3 which increases with ω or even ω , and capacitance. In transformers having tance C'11 and the primary referred 10 is significant in transformers that are a significant amount of wire, the in- equivalent series inductance LEQ1. operated near the ferroresonance of the ter-winding capacitance can interact Likewise, the model of Fig 4 is used to core material.2 This is a serious con- with the transformer inductances visualize the measurement of the sideration in the design of transform- and create a transmission zero. In secondary shunt capacitance C’22, the ers used at HF and VHF frequencies, good quality audio transformers, the primary referred equivalent series in- and therefore requires that proper con- inter-winding capacitance is mini- ductance LEQ2 and the interwinding ca- 10 sideration be given to the selection of mized by placing a grounded copper pacitance C’12. The turns ratio n of the the core material. sheet between the windings, often ideal transformer is the actual ratio of The low frequency performance is referred to as a Faraday Shield. the physical number of turns between determined by the permeability of the The complete model of the wide-band the primary and secondary windings. core material and the number of turns transformer shown in Fig 1 is well The procedure for determining the val- on the windings or length of the suited for rigorous designs in which the ues of the parasitic reactances of Fig 2 conductors. The mutual inductance M transformer is used near the limits of is as follows10: of Fig 1 is a result of the flux in the its performance. These and other 1.With the secondary open, mea- transformer core4 that links the two models are well suited for use in sure the primary winding inductance 9 windings. The high frequency perfor- detailed computer simulations. In LP at a frequency well below the high mance is limited by the fact that not general practice and in analytical frequency cutoff of the transformer. all of the flux produced in one winding solutions, it is more convenient to 2.With the primary open, measure links to the second winding, a deficiency consider the lossless wide-band the secondary winding inductance LS, known as leakage,5 which in turn transformer model shown in Fig 2. This also at a frequency well below the high results in the primary and secondary model has been reduced to the reactive frequency cutoff of the transformer. 10 3.With the secondary open, apply a leakage inductances Ll1 and Ll2 of components and an ideal transformer. Fig 1. Since the leakage flux paths are The three model capacitances of Fig 2 signal at an appropriate mid-band primarily in air, these leakage in- are related to the model capacitances frequency (between the low and high ductances are practically constant.6,7 of Fig 1 in the following manner:7 frequency cutoff frequencies) to the primary winding and measure the The capacitances associated with 1 wideband transformers are gener- C11′ = C11 + C121 − (Eq 1) input and output voltages v1 and v2. ally understood to be distributed, n 3. Calculate the coupling coefficient but it is inconvenient to model k using: transformers by way of distributed C12 C12′ = (Eq 2) capacitances per se, so lumped ca- n v2 LP k = ≤ 1 (Eq 4) v1 LS Fig 3—Equivalent circuit for determining C'11. Fig 1—Complete wideband transformer model. Fig 2—Lossless wideband transformer model. Fig 4—Equivalent circuit for determining C'12 and C'22. 4 Mar/Apr 2005 4. Calculate the mutual inductance 2 1 in a properly designed transformer the M using: C22 = n C22′ − C12 − 1 (Eq 15) equivalent series inductance will domi- n nate and will determine the maximum M = LP LS k (Eq 5) frequency for which a matching net- 5. Calculate the primary leakage Matching work can be realized. The input and inductance L using: Once the values of the parasitic output capacitances C11 and C22 are usu- l1 ally much smaller than required for L = L − n M (Eq 6) reactive elements of the wide-band l1 P transformer model have been deter- realizing a π network low-pass filter 6. Calculate the secondary leakage mined, it is possible to design the section, so additional padding capaci- inductance Ll2 using: transformer into a matching network tors will be required to properly design the matching network. These three M that not only absorbs them, but makes Ll2 = LS − (Eq 7) use of them in forming a 3-pole π components allow us to design 3-pole n network low-pass filter section.10,11,12 Butterworth, Bessel, Gaussian, and 7. Calculate the primary referred We begin by considering the fact that Tchebyschev filter sections with the equivalent series inductance dictating equivalent inductance LEQ1 using: the cutoff frequency. 2 n M Ll2 The presence of the interwinding LEQ1 = + Ll1 (Eq 8) capacitance C suggests that a single M + n L 12 l2 parallel-resonant transmission zero can 8. Calculate the secondary referred be included, which gives us further pos- equivalent inductance LEQ2 using: sibilities of inverse Tchebyschev and Elliptical (Cauer) filter sections. Since n M Ll1 2 LEQ2 = + n Ll2 (Eq 9) the interwinding capacitance is gener- n M + Ll1 ally small, it will also require an addi- 9. Referring to Fig 4, connect a tional padding capacitor to complete the generator to the primary and an ap- design. Note, however, that adding a propriate load across the secondary. Fig 5—Matching network components. transmission zero to the matching net- Measure the transmission parallel resonant frequency f12, which is the frequency at which the voltage across Table 1—Matching Section Prototype Values13 the secondary is at a minimum. 10. Calculate C'12 using: Filter Type C1norm C2norm Lnorm C3norm 1 Butterworth 1.000 1.000 2.000 C′ = Bessel 1.255 0.192 0.553 12 2 (Eq 10) LEQ2 ()2 π f12 Gaussian 2.196 0.967 0.336 11. With the generator still con- Tchebyschev nected to the primary and the second- 0.1 dB 1.032 1.032 1.147 ary open, measure the input series 0.5 dB 1.596 1.596 1.097 1.0 dB 2.024 2.024 0.994 resonant frequency f22, which is the frequency at which the voltage across Inverse Tchebyschev the primary is at a minimum.