Chapter 3 Receiver Circuit Design
3.4 Transformer Design
Between the LNA and the mixers, there is a 3-terminal transformer, as shown in Fig. 3.4. The 3-terminal transformer needs to be carefully designed to transfer the single-end RF current signal to differential form with the maximal current gain.
The 3-terminal transformer is composed of two coupled resonators, which is called resonator coupling network (RCN). Under the critical resonance condition, the RCN can provide the maximal current gain at resonance frequency, which is almost equivalent to an ideal transformer [6]. Fig. 3.5 (a) shows the transformer used in the receiver. It can be modeled into the resonator coupling network, as shown Fig. 3.5 (b).
In Fig. 3.5 (b), L1 is the primary coil of the transformer, L2 and L3 are two secondary coils of the transformer with the same value, and M is the mutual inductance between primary and secondary coils. C1 and L1 form a resonator which connects to the output node of the LNA, and L2, C2, L3, and C3 form two resonators which connect to the
Fig. 3.5 (a) Designed 3-terminal transformer, (b) Resonator coupling network
LNA Down-conversion
Fig. 3.4 Block diagram of the receiver
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input nodes of the mixers. The whole receiver is modeled into the equivalent network, as shown in Fig. 3.6. In Fig. 3.6, RLNA and CLNA model the output impedance of the LNA, L1 and Cp1 model the inductance and parasitic capacitor of the primary coil of the transformer, L2-3 and Cp2-p3 model the inductances and parasitic capacitors of the secondary coils of the transformer, and CMixer1-4 and RMixer1-4 model the input impedance of the mixers. The two parallel capacitors CLNA and Cp1 form the C1 of the RCN, and Cp2-p3 and CMixer1-4 form the C2 of the RCN.
The transformer connects two identical parallel double balance mixers, so these two mixers can be basically simplified into a mixer with double size. Moreover, because the transformer is symmetric to the virtual ground of secondary coil, the RCN can be analyzed by the two-port network, as shown in Fig. 3.7. Iin represents the RF current signal from the LNA, C1 includes the parasitic capacitors of the LNA and the primary coil of the transformer and C2 includes the parasitic capacitors of the mixers and the secondary coils of the transformer.
CLNA Cp1 L1
Fig. 3.6 Resonator coupling network
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The resonance frequencies of primary and secondary uncoupled resonators are defined as ω1 and ω2, and m is the ratio of these two resonance frequencies.
(3.3)
The mutual inductance between the primary and secondary coils is M, and the coupling coefficient is defined as
k
(3.4)
In the coupled network, the resonance frequencies would shift to other two frequencies, as shown in Fig. 3.8. These two resonance frequencies in coupled network can be expressed in term of m, k, and ω2, as shown in eq. (3.5) and (3.6) [6].
(3.5)
(3.6) At these two resonance frequencies ωH and ωL, the transformer passes the RF current signal from the LNA to the mixers in highest efficiency. Either resonance frequency can be chosen as the operating frequency.
ω
1ω
2ω
Lω
HFig. 3.8 Resonance frequencies of the RCN
C
1L
1-M L
2-M R
LNAI
inM C
2R
MixerZ
in1Z
in2Z
in3I
outFig. 3.7 Two-port network of the RCN
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In the two-port analysis, the transfer function from Iin to Iout at the resonance frequency is derived as
where
(3.7) The k is the coupling coefficient between the primary coil and the secondary coils, and n is the ratio of the inductance of the primary coil to the inductance of one secondary coil. The k and n of the maximal current gain condition can be found by the partial differential equation of eq. (3.7) for k and n. These two partial differential equations are
(3.8) These two partial differential equations lead the result as
(3.9) Eq. (3.9) means the impedance matching between the LNA and the transformer. The impedance matching also happens at the interface between the transformer and the mixers. Under the critical coupling condition, the RF current signal is coupled from the LNA to the mixer in the highest efficiency. From eq. (3.7) and (3.9), the maximal current gain under the critical coupling condition is
(3.10) Eq. (3.10) means that the maximal current gain of the transformer is determined by
and . The RCN gets the same result of the maximal current gain like an ideal transformer as k and n are chosen appropriately. eq. (3.10) also means that the maximal current gain is higher with higher RLNA and lower RMixer. This result corresponds to the simple circuit theory. Current is injected from high impedance to
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low impedance. With the same voltage drop between the high impedance node and the low impedance node, more current is injected when the high-impedance is higher and the low-impedance is lower.
The higher impedance ratio of RLNA/RMixer is design to get the higher maximal current gain under the critical coupling condition. Under the condition of that all performances of the receiver meet all specifications, the output impedance of the LNA and the input impedance of the mixer are designed to as higher and as lower as possible, respectively. In this receiver, is designed to be 552 Ω and is designed to be 15 Ω. Therefore, maximal current gain
is equal to about 3. After getting the value of maximal current gain of the RCN, the corresponding ratio of and the ratio of need to be chosen to realize the transformer with the maximal current gain at 1.4 GHz. In the current gain contour plots mapping k and r of n=1~4, as shown in Fig. 3.8, there are two areas of maximal current gain when and in each contour plot, that represents two resonance frequencies of the RCN ωH and ωL. The maximal current gain of operates at ωH and the other one operates at ωL. The area of the maximal current gain at ωL is bigger than the other one at ωH, so the RCN is designed at ωL can be more tolerant of the variation of r and k. Therefore, the RCN at ωL is easier to be realized and more tolerant of process variation.
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Here, n is chosen as 3. In Fig. 3.9 (c), the current gain degrades very little when k ranges from 0.4 to 0.6. The designed transformer is as shown in Fig. 3.10, in which W1 = 8 μm, W2 = 10 μm, OD = 400 μm. Table 3.1 shows each parameter of the RCN.
Because the designed RCN needs bigger capacitances C1 and C2 which are bigger than the total capacitance of the parasitic capacitances of the LNA and primary coils of the transformer and the parasitic capacitance of the secondary coils of the transformer and the mixers. Therefore, there are two additional capacitors C1’ and C2’
Table 3.1 The RCN realization
OD W2
Fig. 3.9 Current gain contour plots mapping k and r (a) n=1(b) n=2(c) n=3(d) n=4
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putted between the LNA and the transformer and between the transformer and the mixers in order to compensate the parasitic capacitances to realize the needed C1 and C2, as shown in Fig. 3.11.