60 4. Single Band Power Amplifiers On the other hand, a Class-E PA has a more relaxed condition since it requires one termination for all harmonics. This makes the harmonics overlapping of the frequencies inside the targeted band not of any importance. Hence, designing broadband Class-E PA requires that:
f1>1
2f2 (4.3)
The Bandwidth and centre frequency are defined as:
BW = f2− f1 (4.4)
f0= f2+f1
2 (4.5)
Another factor which gives a better meaning for the bandwidth and centre frequency is the bandwidth factor (defined as geometrical bandwidth in filter design). The higher the value of this factor, the more complex filter design is needed. The bandwidth factor is given by:
BWf = BW
f0 =2f2−f1
f2+f1 (4.6)
Table 4.2 presents the calculated parameters, using the minimum value of the condition (4.1), for Class-F/F−1and Class-E PA. It is clearly seen that a Class-E power amplifier has a higher bandwidth than Class-F/F−1. This is because the condition (4.1) requires higher value to ensure the non-overlapping between the second and third harmonic signal.
Table 4.2:Frequency Parameters for Class-F/F−1and Class-E PA for minimal Broadband condition value design.
PA Classes Condition (4.1) BW f0 BWf
Class-E f1>1
2f2 1
2f2 3
4f2 2 3
Class-F/F−1 f1>2 3
1 3f2 5
6f2 2 5
5 | VHF and UHF Broadband PA
The design of highly efficient broadband power amplifiers is discussed in this chapter and the next. A low frequency broadband power amplifier will be introduced in this chapter. Broadband matching is defined as designing a lossless reciprocal two port network that can transfer the power from one side to another (namely from source to load) with minimum loss in a predefined range of frequencies [46].
The broadband matching problem is classified into three categories [46]:
Resistive Matching: The goal in this problem is to design a matching network that can transfer the power from a resistive source to a resistive load, Fig 5.1.a. This kind of problem can be solved using filter matching technique (insertion loss method). The achievable insertion loss in this method can be zero if an infinite number of reactive elements are used.
Single Matching: In this problem, the source impedance is resistive and the load impedance is complex, i.e., frequency dependent, Fig. 5.1.b. The achievable insertion loss is not zero due to the complex impedance which preserves some power.
Double Matching: In this problem, both impedances are complex impedances, Fig. 5.1.c. The insertion loss is smaller than the one in single matching techniques due to the two complex impedances of both sides of the network. It usually represents the interstage matching problem of a cascaded power amplifier.
Most of the power amplifier matching problems encounter the single matching technique. This is clear because the transistor output impedance can be modelled, in most cases, as an RC network, where the other side is terminated with 50Ω. The double matching technique is implemented for interstage matching network in power amplifiers as both sides have complex impedances.
5.1 Gain-Bandwidth Limit
Bode [47] and later Fano [48] have shown the minimum return loss that can be achieved from a matching network over a wide frequency range. If the output impedance of the transistor can be
62 5. VHF and UHF Broadband PA
− VS
+ RS
MN RL
(a)
− VS
+ RS
MN ZL
(b)
− VS
+ ZS
MN ZL
(c)
Figure 5.1:Circuit diagrams of broadband matching problems for a) resistive matching, b)single matching, c) double matching.
modelled as a parallel RC network, Fig. 5.2.a., the reflection coefficient is constructed by solving the following equation:
∫ ∞
0
ln
⏐
⏐
⏐
⏐ 1 Γ(ω)
⏐
⏐
⏐
⏐
dω≤ π
RC (5.1)
Assuming constant reflection coefficient over the frequency as in Fig. 5.3, equation (5.1) shows that the minimum return loss,Γm, is limited to the equivalent circuit value within a certain bandwidth given as (5.2). Outside this bandwidth, the matching network has to give a full reflection (Γ=1)
ln 1
Γm ≤ π
RC(ω2−ω1) = πω0
Qc(ω2−ω1) (5.2)
whereQcrepresents the quality factor of the parallel RC circuit, i.e.,Qc=2πf0RC
From (5.2) very important facts can be concluded. First, the bandwidth increases at the expense of a higher reflection coefficient. However, the minimum reflection coefficient can not be zero unless the bandwidth (ω2−ω1) is zero. Finally, (5.2) can be rewritten as (5.3), which clearly states that reducing the quality factor of the device output circuit increases the bandwidth
MN C R
Γ
(a)
MN
L
C R
Γ
(b)
Figure 5.2:Different load problem for Bode-Fano limit with passive lossless matching network (MN) and a) resistive load and shunt capacitor, b) resistive load and shunt capacitor series inductor.
5.1 Gain-Bandwidth Limit 63
|Γ| 1
Γm
ω1 ω2
BW
Freq.
Figure 5.3:Bode-Fano limit criterion for a constant reflection coefficient.
and hence the reflection coefficient increases.
△ f
f0 ≤ π Qcln(Γ1
m) (5.3)
The bandwidth limit for different reflection coefficients and quality factors is shown in Fig. 5.4.
A more general case, which is very common in PA design, is shown in Fig. 5.2.b. Fano [48]
extended the Bode theorem, including a more general low-pass load equivalent circuit. Usually, a low-pass to band-pass transformation, i.e.,ω→ωx(ω/ωx−ωx/ω)is used [23]. However, this transformation can not be implemented in the circuit of Fig. 5.2.b because the capacitor can not be accessed directly from the equivalent circuit terminals. The modified procedure of Fano is applied for this circuit in [49] using a load resistance equal to 1Ω. The circuit in Fig. 5.2.b has two zeros ats=∞, which produces two equations [48]:
∫ ∞
0
ln
⏐
⏐
⏐
⏐ 1 Γm
⏐
⏐
⏐
⏐
dω= π
2(A1−2
∑
λri) (5.4)∫ ∞
0
ω2ln
⏐
⏐
⏐
⏐ 1 Γm
⏐
⏐
⏐
⏐
dω=−π
2(A3−2
3
∑
λ3ri) (5.5)whereA1and A3are the Taylor coefficients of ln
⏐
⏐
⏐
1 Γm
⏐
⏐
⏐and equal to C2 and 23L−3C
LC3 , respectively.
λri=σri+jωriare the zeros of the reflection coefficient of the equivalent circuit and the matching
△f/f0
0 0.2 0.4 0.6 0.8 1
Γm
0 0.2 0.4 0.6 0.8 1
Qc
=10
Qc
=20
Qc
=50
Qc
=100 Qc=150
Qc=200 Qc=400
Figure 5.4: Fractional bandwidth limit according to Bode-Fano limit with different reflection coefficient and different quality factor assuming constant Γm.
64 5. VHF and UHF Broadband PA
RS MN
L
C RL
Γ2 Γ1
Figure 5.5: Reflection coefficient definition for resistive load network with shunt capacitor and series inductor including a passive lossless matching net-work.
network in Fig. 5.5. These zeros lie on the right half plane of the s-plane.
To minimize the reflection coefficient Γm, the matching network must have zeros which maximize the termΣλ3riand minimize the termΣλri. This can only be achieved by using single real zero, i.e.,σr [48]. Substituting in (5.4) and eliminatingσr will result:
ϒ(ω32−ω31) +3A3−1
4[(A1)−ϒ(ω2−ω1)]3=0 (5.6) whereϒ= 2
πln
⏐
⏐
⏐
⏐ 1 Γm
⏐
⏐
⏐
⏐ .
With some mathematical manipulation and substituting A2and A3 with their values, (5.6) becomes:
ϒ3− 6XC
BWfϒ2+ XC2 BWf2[12
XC2+12+BWf2
XC2 ]ϒ− 24XC2
BWf3XL =0 (5.7)
whereBWf =△ f/f0 is the fractional bandwidth defined as the ratio of the bandwidth to the centre frequency.
Equation (5.7) is a general case for any matching network, which is called a non-degenerated case. For the degenerated case, where the matching network is a mirrored of the equivalent load circuit in which it starts with seriesLand then shuntC, only (5.4) is then required. (5.4) becomes identical to (5.2):
ϒ= 2XC
BWf (5.8)
Equation (5.7) can be solved easily in MATLAB forϒand the reflection coefficient|Γm|with the corresponding loss, 1/(1− |Γm|2). These results are shown with differentXC andXL in Fig 5.6 and Fig. 5.7, respectively. In Fig. 5.6, the reflection coefficient is calculated with fixedXL value, i.e., equal to 2, and differentXCvalues as shown in the figure. Fig. 5.7 shows the same parameters withXL as variables andXC equal to 2. The dotted line on both figures is a solution for (5.8) with XC equal to 2. Finally, it is worth to note that these figures are valid for inductance values greater than a certain level [48]. On the other hand, (5.7) and (5.8) are shown in the figures with unity resistanceRL, but they can be implemented with any value using impedance scaling provided from (5.9) and (5.10):
XL =ω0L
R (5.9)
5.1 Gain-Bandwidth Limit 65
|Γ|
0 0.2 0.4 0.6 0.8 1
0.01 0.1 1 10
BWf 16
8 4
2 1
0.5
(a)
1/[1−|Γ|2 ]
0 5 10 15 20 25
0.01 0.1 1 10
BWf
16
8
4
2 1 0.5
(b)
Figure 5.6:Evaluation of Bode-Fano limit for the circuit in Fig. 5.5 withXCas a parameter andXL=2 showing a) reflection coefficient and b) return loss from the reflection coefficient; the dashed line shows the case where the inductor is not present.
XC= 1
ω0CR (5.10)
whereω0is the centre radian frequency for the targeted band.
5.1.1 The Real Frequency Technique
To this point, the discussed matching network technique is implemented via circuit approximation, which works well for simple load impedance. However, Carlin’s technique [50, 51], which called as real frequency technique (RFT), utilizes measured data for the required impedances over the targeted band. Further, it requires rational polynomial function with numerical optimization to synthesis the a matching network impedance. Finally, extraction of the matching network elements is the last step in this technique.
|Γ|
0 0.2 0.4 0.6 0.8 1
0.01 0.1 1 10
BWf
16 8
4 2
1 0.5
(a)
1/[1−|Γ|2 ]
0 5 10 15 20
0,01 0,1 1 10
BWf
16 8 4 2 1 0.5
(b)
Figure 5.7:Evaluation of Bode-Fano limit for the circuit in Fig. 5.5 withXLas a parameter andXC=2 showing a) reflection coefficient and b) return loss from the reflection coefficient; the dashed line shows the case where the inductor is not present.
66 5. VHF and UHF Broadband PA
5.1.2 Implemented Matching Technique and Design Steps
Several other approaches for broadband matching design and synthesis are presented in [52]. All of these approaches require numerical technique to give a proper matching network. It includes optimization needs to be applied that produce final solutions for the required impedance matching network.
In this work, real frequency techniques are implemented from the extracted optimum load and source impedances. These impedances are taken from a load/source pull simulation. Inspecting the required impedance on the Smith chart and applying curve fitting for impedances with polynomial equations are very important steps. The goal of applying curve fitting is to include the impedances for those frequencies which are not included in the source/load-pull simulation.
Synthesizing matching networks, using ideal lumped elements, is the major step toward finalizing the matching network. This step requires deep understanding of the influence of the lumped elements on the impedance and how they rotate in the Smith chart, Fig. 5.8. The arrows in this figure represent the increment of the lumped element value, short-stub(SC)/open-stub(OS) length, transmission lines (TL) and transformer ratio (Tra). It is clearly seen from this figure that the parallel lumped elements and the stub’s moves along an admittance circle. The transmission line length rotates the impedance over a constant VSWR as shown.
Matching network topology is constrained by three factors; first, the operating bandwidth that determines the number of lumped elements. As the bandwidth increases the required number of lumped elements for matching increases. The second factor is the distance between the stop-band, i.e., second harmonic, and the pass-band, i.e., fundamental frequencies. This factor mainly determines the matching topology between low-pass and band-pass. The last factor is the power amplifier type, which determines the kind of zero transmission in the stop-band. Usually, an inductor has a zero transmission at infinity. This zero could be an open circuit zero, if connected
series L
series Cseries
R
parallel L
series C series
R series
TL shunt
SC shunt
OC
Tra>1 Tra<1
Figure 5.8:Different matching component behaviour in a matching network, the black dot is the default load impedance and the arrow present the increasing value/length of the component and transmission line TL.
5.2 VHF Broadband Class-E PA 67