Chapter 3 Passive Quadrature Signal Generations and Their Applications on BJT-
3.1 Passive Quadrature Signal Generation
3.1.2 Distributed Quadrature Signal Generation Method(Quadrature Coupler) 31
the input port to the through port (S21) and to the coupled port (S31), respectively. By dividing the two outputs,
Thus, the phase difference is always 90 on a lossless condition; however, the amplitude is determined by the operating frequency (f) and the coupling factor
k=(Z
0eZ0o)/(Z0eZ0o), where Zoe and Zoo are the even-mode and odd-mode characteristic impedances, respectively. Thus, the equivalent error function is1
The quadrature phase maintains with arbitrary output loadings for C and T ports if and only if the characteristic impedance (Zo) of the coupler is 50 Ω=Zinc=Ziso.
A λ/4 coupled-line coupler with different coupling factor k has different bandwidth on the same image-rejection criterion (e.g., 40 dB IRR; ε=0.01). Since the
maximum amplitude imbalance occurs at the center frequency. Thus, the maximum ε equals to
2
max 2
1 1
k k
k k
, (3.10) and thus, for the criterion of εmax, the required coupling coefficient will be
max 2
1
2(1 )
k
. (3.11)
For example, a 3-dB coupler without amplitude imbalance (i.e., εmax=0), the kmax
should be 1 2 ( 0.707) but the perfect balanced amplitude and phase only occur at the center frequency. A higher coupling results in a wider bandwidth under a given tolerable εmax. To achieve a higher coupling, a broadside-coupled coupler or an Lange coupler can be utilized [30]. Moreover, multiple sections of /4 coupled-line couplers with a properly designed characteristic impedance of each section are cascaded to cover a much wider bandwidth [30]. However, the huge area arises much production cost for chip implementation.
Frequency
Fig. 3-2 Description of the S parameters for a 3-dB quadrature coupler and a high-coupling quadrature coupler.
As mentioned earlier, the conventional quadrature coupler is implemented by a
quarter-wavelength coupled line [68]. As a result, the 3-dB quadrature coupler is realized with a coupling factor of 1/ 2 and the description of the S parameters is shown in Fig. 3-2. To widen the bandwidth, the coupling factor can be chosen larger if the amplitude imbalance between the coupled and through ports is still tolerable. It is noteworthy that the phase difference between these two ports of a quarter-wavelength quadrature coupler under the lossless condition is always 90 [68].
However, another design concept is proposed for miniaturization. A high coupling factor (~0.85) results in a significant amplitude difference at the center frequency (f0) and the perfect balanced amplitudes occur at two frequencies, fbal1 and
f
bal2, as shown in Fig. 3-2. If the fbal1 is chosen for the usage frequency, the coupler can be very compact since the center frequency of the coupler is several times the operating frequency. However, the bandwidth is relatively narrow. The high coupling factor can be achieved by using a broadside-coupled quadrature coupler, consisting of two spiral inductors using metal 6 and metal 5 (in 0.18-μm CMOS technology). The dielectric thickness between metal 6 and metal 5 is 0.8 μm, which is much less than the 3-μm line spacing of the inductor; thus, the broadside-coupling is dominant.When considering metal loss or substrate loss, i.e., α=attenuation constant≠0 in practice, the perfectly quadrature phase no longer maintains and thus the amplitude/phase accuracy trade-offs increase. However, a BJT mixer can tolerate the LO amplitude imbalance and thus the lossy quadrature coupler can be designed for perfect quadrature phase, which will be introduced in Section 3.2.1.
3.1.3 Lumped Quadrature Signal Generation Methods
Instead of the microwave realizations, lumped quadrature signal generators are widely used at low frequencies. Traditionally, there are two primary kinds of methods for a quadrature signal generation:
1) constant quadrature phase architecture, 2) balanced amplitude architecture.
R
Fig. 3-3 (a) RC-CR phase shifter (b) LR-CR phase shifter (c) PPF with Q input shorted to ground (d) RC-CR all-pass filter (APF) topology (e) LR-CR APF topology and (f) a PPF with I/Q input connected together.
An RC-CR phase shifter [Fig. 3-3(a)], an LR-CR phase shifter [Fig. 3-3(b)], and a PPF with Q input shorted to ground [Fig. 3-3(c)] belong to the former type while the latter type includes an RC-CR all-pass filter (APF) topology [Fig. 3-3(d)], an LR-CR APF topology [Fig. 3-3(e)], and a PPF with I/Q input connected together [Fig. 3-3(f)].
An RC-CR phase shifter and an LR-CR phase shifter have the same quadrature output ratio The output phasor sequence belongs to a constant-quadrature-phase sequence,
illustrated in Fig. 3-1(b); thus, by Eqn. (3.2), the error function becomes
For an RC-CR phase shifter, the output quadrature phase maintains at every frequency with arbitrary loadings if the two loadings are the same, but the result holds for an LR-CR topology only when the load impedance equals L C.
The ratio of VQ (CR-path) and VI (LR-path) can be described as
Besides, the input impedance of the LR-CR quadrature generator is
//
1
(1 )//(1 )which is always equals to the load impedance (R) under the balanced condition.
Note that, no power loss is necessary for the LR-CR topology since all reactive components are used when compared with the lossy RC-CR phase shifter.
A single-stage PPF with Q input shorted to the ground is a differential variant of an RC-CR phase shifter; thus the same quadrature output ratio Q/I=j/0 and error function |=|(0)/(+0)| can be obtained straightforwardly.
On the other hand, the APF variant for an RC-CR topology and its differential type (PPF with I/Q input connected) have a quadrature output ratio of
1
where/2= tan-1(ω/ω0)−45.
Therefore, output phasor sequence has balanced amplitudes but a phase error the same as Fig. 3-1(c). By Eqn. (3.4), the error function becomes
0 0
0
( ) tan( 2)
. (3.17) By the same token, an LR-CR APF topology has the same quadrature output ratio and error function as the RC-CR APF topology on an additional condition that the source/load impedances are equal to L C.
Fig. 3-4 IRR with a pole frequency deviation due to a process variation.
Moreover, Fig. 3-4 shows the correspondent IRR (=−20log|εo|) of the lumped quadrature generator (e.g., PPF) with a center frequency variation due to process variation. If a center frequency varies 20%, the IRR is drastically degraded to only 20.8 dB. Therefore, the process variation should be taken into account when deciding a sufficient IRR bandwidth.
However, multiple PPFs can be cascaded and thus extend the bandwidth but the LR-CR topology can not. The followings are the discussion of multi-band and wide-band extensions by using a multi-stage PPF.
The transfer function of a single-stage PPF [Fig. 3-3(f)] with a balanced quadrature phasor sequence ( R
orL
Thus, the equivalent error function is defined as
0 For a quadrature signal generation, the input differential signal can be decomposed into R
and L
with the same amplitudes; thus, the output error function has the same meaning as the ratio of negative and positive signal gain, defined in Eqn.
(3.19). For a multi-stage PPF with a balanced quadrature phasor sequence ( R
orL
), the transfer function can be expressed as
1 and its correspondent error function can thus be expressed as
1
where
A
Voi( )
is the open-circuited voltage gain of the ith-stage PPF, defined by Eqn.(3.18); ZSi/ZLi is the equivalent source/load impedance of the ith stage.
The
A
Vo+ of each stage is larger than 1 and reaches the maximum value of 2 at the center frequency. Thus, the impedance ratio between each stage and the input/output impedance dominate the overall voltage gain/loss. On the other hand, the voltage gain of the AVo has a transmission zero and results in a narrow-band rejectionresponse.
With different locations of transmission zeros of a multi-stage PPF, there are three typical circumstances:
1) narrow-band (ωmax/ωmin→1), 2) wide-band (ωmax/ωmin>>1), and 3) multi-band applications.
(a) (b)
(c)
Fig. 3-5 Three-stage PPF on different pole locations (a) equal RC pole (b) unequal RC poles with an equiripple frequency response and (c) unequal RC poles for multi-band applications.
1) For a narrow-band application, all the transmission zeros are identical, ω0; thus, an
N-stage PPF provides a total error of ε=
0 , thus the error is reduced with increasing N
cascading stages as shown in Fig. 3-5(a). In other words, for a given tolerable IRR
(−20log|ε|), the ratio bandwidth becomes where ε is the target error function after the PPF, εo is the error function of one-stage PPF, and N is the number of stages.
Thus, for a target 40-dB IRR (ε=0.01) of a quadrature generator, the ratio bandwidth becomes 1.041, 1.494 and 2.4 for single-stage, two-stage and three-stage PPFs, respectively.
In addition, the voltage gain (or loss) of a multi-stage PPF with the same center frequency can be calculated. By Eqn. (3.20), the voltage gain of an N-stage PPF at the center frequency can be expressed as
0 ( 1)
Eqn. (3.23) shows that the increase of resistance lessens the voltage loss when the last stage is open-circuited [26]. However, in reality, the loading of the PPF is the gate (base) of the active mixer core with capacitive impedance of 1
sC . Thus, the
Linput impedance is typically open-circuited at low frequencies but degrades at high frequencies. The voltage gain should be modified as
0 0 0 2
Although the increase of resistance can increase AVo+, the small capacitance at the Nth stage (CN), resulting from a high resistance (RN), may degrade the overall voltage gain, especially at high frequencies. Thus, an optimum voltage gain always exists if the source impedance (ZS) and the output loading capacitance (CL) are given.
2) For a wide-band application, a multi-stage PPF with logarithmic increase of pole locations results in an equiripple image rejection frequency response as shown in Fig.
3-5(b). The derivation of optimum pole locations for a minimum IRR of an N-stage PPF had been proposed for a given ratio bandwidth ωmax/ωmin [26]. Thus, for a target IRR, the number of stages and the pole locations can be defined [26].
3) Finally, for a multi-band quadrature generation, each stage of a PPF is set at different desired frequencies as shown in Fig. 3-5(c). The image-rejection bandwidth becomes wider when compared to the individual response because the distant away transmission zeros still provide certain effect on a given band. Certainly, detailed information on gain and IRR bandwidth can be obtained by simulation tools.
The wideband and multi-band extensions are applied in Section 5.2 for further improvement of image rejection in a low-IF downconverter.