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(e) FET Mixers : have conversion gain (not loss) Pozar (RF Ch 7) * R. A. Pucel, D. Masse, and R. Bera, "Performance of GaAs MESFET Mixers at X Band," IEEE Tram.Microwave Theory and Techniques, vol. MTT-24, pp. 351-360, June 1976. Single-Ended FET Mixer There are several FET parameters that offer nonlinearities used for mixing The strongest is the transconductance, gm, when the FET is operated in a common source configuration with a negative gate bias (Vgs ) When used as an amplifier, the gate bias voltage is near zero, or positive, so the transconductance is near its maximum value, and operates as a linear device. When the gate bias Vgs is near the pinch-off region: where the transconductance approaches zero a small variation of gate voltage causing a large change in transconductance leading to a nonlinear response. 7-2 Thus the LO voltage can be applied to the gate of the FET to pump the transconductance to switch the FET between high & low transconductance states => provide mixing as the switching model (see the Diode Large-Signal Model for Mixer) RF chokes are used to bias the gate at a negative voltage near pinch-off and to provide a positive bias for the drain of the FET. A bypass capacitor at the drain provides a return path for the LO signal, and a LPF provides the final IF output signal * Based on the standard unilateral equivalent circuit for a FET. vRF (t ) VRF cos RF t vLo (t ) VLO cos LO t Zg = Rg + jXg : Thevenin source impedance for the RF input port ZL = RL + jXL : Thevenin source impedance at the IF output port. LO port has a real generator impedance of ZO => since we are not concerned with maximum power transfer for the LO signal. The same as for the large-signal analysis of the diode mixer, the LO pumped FET transconductance is espressed as a Fourier series of harmonics of LO signal: g n cos n LO g (t ) g 0 2 n 1 2008 not having an explicit formula for the transconductance must rely on measurements for values of g n , in the switching model, the desired down-conversion is due to (n = 1) only need g1 coefficient & the typical measured value in the range of 10 mS H.-R. Chuang, EE NCKU 7-3 Conversion gain of the FET mixer can be found as (?) 2 P Gc IF avail PRF avail VDIF RL / Z L VRF 2 2 / 4 Rg 2 4 Rg RL VDIF *(see ch3, p.3-26, conjugate matching formula) 2 V RF Z L VDIF : IF drain voltage Zg & ZL chosen for maximum power transfer at the RF and IF ports. The RF signal across the gate-to-source capacitance is given as: VcRF & VRF VRF (*) jRF C gs [( Ri Z g ) (1 / jRF C gs )] 1 jRF C gs ( Ri Z g ) vcRF (t ) VcRF cos RF t From g (t ) g 0 2 g n cos(nLOt ) n 1 g m (t ) vcRF (t ) [ g 0 2 g n cos n(LOt )]VcRF cos(RF t ) n 1 g 0VcRF cos(RF t ) 2 g1VcRF cos( RF t ) cos( LOt ) The down-converted IF signal can be extracted from the second term by using the usual trigonometric identity: g m (t )[vcRF (t ) | IF ] g1 VcRF cos(IF t ) (?) Then the IF component of the drain voltage (in phasor form) is (by using (*) ) R Z VDIF ( g1 VcRF )Rd // Z L g1 VcRF d L Rd Z L Rd Z L VRF g1 1 j RF C gs ( Ri Z g ) Rd Z L 2008 H.-R. Chuang, EE NCKU 7-4 The conversion gain GC (before conjugate matching) is then from 4 Rg RL VDIF Gc 2 V RF Z L 2 & Rd Z L VRF VDIF g1 1 j RF C gs ( Ri Z g ) Rd Z L we have VRF ( Rd Z L ) g 1 4 Rg RL 1 jRF C gs ( Ri Z g ) Rd Z L Gc |not 2 VRF ZL matched 2 2 2 g1Rd Rg RL 2 2 RF C gs [( R R ) ( X 1 ) ] [( R R ) 2 X 2 ] i g g C d L L RF gs By conjugately matching the RF & IF ports: ( Rg Ri , X g 1/ RF Cgs , RL Rd , X L 0 ) 2 2 g1Rd Ri Rd 2 g12 Rd Gc RF C gs [( 2 R ) 2 (0) 2 ] [( 2 R ) 2 (0) 2 ] 4 C 2 R i d RF gs i * Practical mixer circuit generally use matching circuits to transform the FET impedance to 50 ohm for the RF, LO & IF ports. 2008 H.-R. Chuang, EE NCKU 7-5 Diode Large-Signal Model for Mixer (Pozar RF P233) I (V ) I s (eV 1) v(t ) VRF VLo Vr cos r t Vo cos o t For a diode small-signal approximation I Vo v(t ) I o vGd v 2 Gd 2 ................... Gd 2 G 2d v 2 Gd 2 Gd 4 Vr cosr Vo cosot 2 V 2 cos2 rt 2VrVo cosrt cosot Vo2 cos2 ot r V 2 V 2 V 2 cos 2 t V 2 cos 2 t 2V V cos t 2V V cos t r o r o o r o r o r o r o r de siredIF signal - Usually LO power (typical 5~10 dBm) will violate the small signal approximation of the diode - Large signal model is needed for fully nonlinear analysis 2008 H.-R. Chuang, EE NCKU 7-6 2008 H.-R. Chuang, EE NCKU 7-7 2008 H.-R. Chuang, EE NCKU 7-8 2008 H.-R. Chuang, EE NCKU 7-9 2008 H.-R. Chuang, EE NCKU 7-10 2008 H.-R. Chuang, EE NCKU 7-11 2008 H.-R. Chuang, EE NCKU 7-12 EXAMPLE 7.2 MIXER CONVERSION GAIN A single-ended FET mixer is to be designed A single-ended FET mixer is to be designed for a wireless local area network receiver operating at 2.4 GHz. The parameters of the FET are: Rd = 300 ?, R, = 100, Cgs =0.3 pF, and g\ = 10 mS. Calculate the maximum possible conversion gain. Solution This is a straightforward application of the formula for conversion gain given in (7.54): Note that this value does not include losses due to the necessary impedance matching networks. FIGURE 7.12 A dual-gate FET mixer. g (t ) g0 2 g1 cos LO t 2 g 2 cos 2LO t I RF g 0 g1 g 2 VRF I RF Rg I g g g I R 0 1 IF IF IF 1 I IM g 2 g1 g 0 I IM Rg g1VRF I SC I IF R 0 IF 1 g 0 Rg g 2 Rg VOC VIF I 0 IF g1VRF 2 g12 Rg g0 g 2 Rg g0 (1 g0 Rg ) 2 g12 Rg I SC GIF g0 VOC 1 g 0 Rg g 2 Rg PIF avail I SC 2 4GIF2 V PRF avail RF 4 Rg (1 g0 Rg g 2 Rg ) g0 (1 g0 Rg g 2 Rg ) 2 g12 Rg P Lc RF avail PIF avail g12 Rg Lc 2( x a)( x a b) bx xopt a ( a b) Lc min 2 a a ( a b) a b a ( a b) b a ( a b) g n I s I n (VLO ) I s 2008 eVLO 2VLO 2a a ( a b) b 2 2 1 1 b / a 1 1 b / a H.-R. Chuang, EE NCKU 7-13 b 2 g12 1 a g 0 ( g 0 g 2) Te nT ( Lc 1) 2 g (t ) 1 2 n sin cos n LO t 2 n 1 n 2 1 2 n i (t ) g (t )vRF (t ) cos RF t sin cos nLO t cos RF t n 2 n 1 2 1 2 n cos(RF RF )t cos(RF nLO )t VRF cos RF t sin 2 n 2 n 1 Mixer Type Diode Conversion Noise 1dB 3rd Order Gain Figure Compression Intercept -5 dB FET 5-7dB 6 dB -6 to –1 dBm 7-8dB 5 dBm 5 to 6 dBm 20 dBm g (t ) g 0 2 g n cos n LO n 1 2 VDIF RL P Gc IF avail PRF avail VcRF ZL 2 VRF 4 Rg 2 4 Rg RL VDIF 2 V RF Z L VRF j jRF C gs Ri Z g RF C gs 2 VRF 1 jRF C gs ( Ri Z g ) g m (t )vcRF (t ) g 0VcRF cos RF t 2 g1VcRF cos RF t cos LO t g m (t )vcRF (t ) |RF g1VcRF cos IF t R Z Rd Z L g1VRF VDIF g1VcRF d L R Z 1 j C ( R Z ) R Z L RF gs i g d L d 2008 H.-R. Chuang, EE NCKU 7-14 2 2 g1Rd Rg RL Gc |not RF C gs 2 R R 2 X 2 matched d L L 1 R R 2 X i g g RF C gs Gc g12 Rd 2 42RF C gs Ri Using the small-signal approximation of (7.20) gives the total diode current as The first term in (7.24) is the DC bias current, which will be blocked from the IF by the DC blocking capacitors. The second term is a replication of the RF and LO signals, which will be filtered out by the low-pass IF filter. This leaves the third term can be rewritten using trigonometric identities as This result is seen to contain several new signal components, only one of which produces the desired IF difference product. The two DC terms again will be blocked by the blocking capacitors, and the 2&>RF, 2&>Lo^ and &)RF + (ULO terms will be blocked by the low-pass filter. This leaves the IF output current as where (UIF = O)RF ?a LO is the IF frequency. The spectrum of the down- converting single-ended mixer is thus identical to that of the idealized mixer shown in Figure 7.1b. Large-Signal Model While the small-signal analysis of a mixer demonstrates the key process of frequency conversion, it is not accurate enough to provide a realistic result for conversion loss. This is primarily because the power supplied to the mixer LO port is usually large enough to violate the smallsignal approximation. Here we consider a fully nonlinear analysis of a resistive diode mixer [3][4], with the goal of deriving an expression for the conversion loss defined in (7.10). The term "resistive" in this context means that reactances associated with the diode junction and package are ignored, to simplify the analysis. Our results should be useful in understanding the nonlinear operation and losses of the diode mixer, but for actual design purposes modem computer-aided design (CAD) software is preferred [5].Such software can model the diode nonlinearity, as well as the effects of diode reactances and impedance matching networks. We again assume a diode I-V characteristic as given by (7.18), with a relatively low-level RF input voltage given by (7.22), and a much larger LO pump signal given by (7.23). A DC bias current may also be present, but will not directly enter into our analysis. As we have seen from the small-signal mixer analysis, these two AC input signals generate a multitude of harmonics and other frequency products: &)RF RF input signal (low power) &IIF = &IRF ?'"LO IF output signal (low power) (DIM = (ULO ?t^iF image signal (low power) 2008 H.-R. Chuang, EE NCKU 7-15 (ULO LO input signal (high power) ?>LO harmonics of LO (high power) racfLO ?'MIF harmonic sidebands of LO (low power) is leaves the IF output current as In a typical mixer, harmonics of the LO and the harmonic sidebands are terminated reactively, and therefore do not lead to much power loss. This leaves three signal frequencies of most importance: &IRF, a>w, and n)iw To evaluate conversion loss, we will find the available power of the RF input signal, the power of the IF output signal, and the power lost in the image signal. The image signal is important because it is relatively close in frequency to the RF signal, and thus sees essentially the same load. We will see that approximately half the input power gets converted to the image frequency. Note that the image term at frequency &>IM = WLO - "RF = ^LO - <^RF was not explicitly shown in the small-signal expansion of (7.24), since this product is generated by the u3 term of (7.20). Under the assumption that the RF input voltage is small, we can write the AC diode current as a Taylor series expansion about the LO voltage as This Taylor series is similar in form to (7.20), as used for the small-signal analysis, but with the important difference that the expansion point here is about the LO voltage, where as(7.20) was expanded about the DC bias point. The first term in (7.26) is due only to the LO input, and does not enter into the calculation of conversion loss. The second term is a function of the RF and LO input voltages, and will provide a good approximation for the three products at frequencies ww, w^, and &>IM, with a large LO pump signal. The coefficient of the second term has dimensions of conductance, so we can use (7.18) to write the differential conductance as Then for small input voltages v(t) we can write the resulting diode current as We see from (7.27) that g(t) is a real number (consistent with our description as a resistive mixer) and is a periodic function of the LO frequency. Thus, g(t) can be expressed as a Fourier cosine series in terms of harmonics of w^o'with Fourier coefficients given by where In(x) is the modified Bessel function of order n, defined in Appendix B. Now let the AC diode current consist of three components at the frequencies &)RF, (DIF, and where /pp, /IF, and /[M are the amplitudes of the RF, IF, and image signals to be determined, If the RF voltage of (7.22) is applied to the diode through a source resistance Rg, and the IF and image ports are terminated in load resistances /?ip and Rg, respectively, then the voltage FIGURE 7.4 Equivalent circuit for the large-signal model of the resistive diode mixer.across the diode can be written as (7.32) The equivalent circuit consists of a three-port network, with one port for each of the frequency components at &)RF, a>ip, and a>m[, as shown in Figure 7.4. We assume the terminations for the RF and image ports are identical, because aiRp is very close to u>wi, while the termination for the IF port may be different. 2008 H.-R. Chuang, EE NCKU 7-16 Using the first three terms of the Fourier series of (7.29) for the diode differential conductance gives (7.33) Multiplying the voltage of (7.32) by the conductance in (7.33), and matching like frequency terms with the current of (7.31) gives a system of three equations for the unknown port currents: (7.34) where VRF is the source voltage, and the gn 's are defined in (7.30). Note that multiplication of (7.32) by (7.33) creates several frequencies in addition to oipp, (UIF, and (UIM, but we assume these frequencies to be reactively terminated so that they do not lead to significant power dissipation. The easiest way to find the available power from the IF port is to first find the Norton equivalent source for the IF port. As shown in Figure 7.5, this consists of a current source equal to /sc, the short-circuit current at the IF port, and Gip, the conductance seen looking into the IF port. This conductance can be found as GIF = /sc/ Voc, where Voc is the open-circuit voltage of the IF port. The short-circuit IF port current can be found by setting R^p == 0 in (7.34) and solving for /ip. After some straightforward algebra, we obtain FIGURE7.5 Norton equivalent circuit for the IF port of the large-signal model of the resistive diode mixer. The open-circuit IF port voltage is found by setting /ip = 0, and solving (7.34) for Vy. Then the Norton conductance of the IF port is The available output power at the IF port is and the available input power from the RF source is So from (7.10) the conversion loss is (not in dB) Note that the conversion loss does not depend on the IF port termination, /?ip, because of the use of available powers. It does depend on Rg, the RF and image port terminations, so it is possible to minimize the conversion loss by properly selecting Rg. If we let x = I//?., a = go + g2?and b = 2g^/go, then (7.39) can be rewritten as (7.40) Differentiating with respect to x and setting the result to zero gives the optimum value of x as for which the minimum value of conversion loss is (7.41) We can evaluate this result by approximating values for go, gi, and g^. For an LO input power of 10 mW, VLO is about 0.707 V rms, and a = 1/28 mV, so aV^n, the argument of the modified Bessel functions for gn given in (7.30), is approximately 25. Thus the modified Bessel functions can be approximated asymptotically using the large-argument formula given in Appendix B, and the gn's simplified as 2008 H.-R. Chuang, EE NCKU 7-17 and the minimum conversion loss of (7.42) reduces to L(. = 2, or 3 dB. This means that half the RF input power is converted to IF power, and half is converted to power at the image frequency. In principle this result could be improved by terminating the image port with a reactive load, but it is usually difficult in practice to separate the image termination from the RF termination. Also, this result is highly idealized in that it assumes no power loss at higher harmonic frequencies, and it ignores diode reactances. This same model can be used to derive the SSB noise temperature for the resistive mixer as (7.45) where n is the diode ideality factor and T is the physical temperature of the diode [3]. Switching Model The large-signal model suggests that the diode mixer can be viewed as a switch. As the LO voltage cycles between positive and negative values of cosecant, the diode becomes conducting or nonconducting, respectively. Thus, the diode conductance (the ratio of diode current to diode voltage) switches between large values and zero at the same rate as the LO voltage. Figure 7.6 shows a typical diode conductance waveform, where T = ITCJW^O is the period of the LO waveform. The conductance waveform of Figure 7.6 can be calculated directly from the diode V-I characteristic of (7.18), or from the Fourier series representation of (7.29). But since a conductance greater than a few Siemens is essentially a short circuit, we can approximate the diode conductance as the square wave shown in Figure 7.7. This square wave has a Fourier transform given by (7.46) which is similar in form to the Fourier series of (7.29). An equivalent circuit of the diode mixer then consists of the RF input voltage applied across a load resistor in series with an ideal switch, as shown in Figure 7.8. The time-varying FIGURE 7.6 Conductance waveform of a mixer diode pumped with a large-signal FIGU運7.7 LO voltage waveform and idealized square-wave diode conductance waveform for the switching model of a diode mixer. switch conductance is given by (7.46). The diode current can be found by multiplying the RF input voltage of (7.22) by the conductance of (7.46): Filtering all but the lowest-frequency component for the n = 1 term of the summation gives the desired IF output as The switching model is useful for mixers of any type, including the FET mixer discussed in the following section. Note that the switching model of a mixer can be considered as a linear, but time-varying, circuit. FIGURE 7.8 Equivalent circuit for the switching model of the diode mixer. 7.3 - FET MIXERS 2008 H.-R. Chuang, EE NCKU 7-18 Mixers can also be implemented by using the nonlinear properties of transistors. FETs, in particular, offer low noise characteristics and easy integration with other circuitry such as switches and low-noise amplifiers. Transistor mixers can provide conversion gain, but their noise figure is generally not as good as can be obtained with diode mixers. PET mixers also offer higher dynamic range. The following table compares the characteristics of typical diode and FET mixers. Because a FET mixer has conversion gain, but usually worse noise figure, the proper comparison with a diode mixer should include the cascade effect of adjacent stages. In this section we will analyze the single-ended FET mixer, and derive an expression for its conversion gain. We will also discuss a few other popular FET mixer configurations. Single-Ended FET Mixer There are several FET parameters that offer nonlinearities that can be used for mixing but the strongest is the transconductance, ^, when the FET is operated in a common source configuration with a negative gate bias. Figure 7.9 shows the variation of transconductance with gate bias for a typical FET. When used as an amplifier, the gate bias voltage is near zero, or positive, so the transconductance is near its maximum value, and the transistor operates as a linear device. When the gate bias is near the pinch-off region, where the transconductance approaches zero, a small variation of gate voltage can cause a large change in transconductance, leading to a nonlinear response. Thus the LO voltage can be applied to the gate of the FET to pump the transconductance to switch the FET between high and low transconductance states, and provide mixing in much the same manner as the switching model discussed in the previous section. The circuit for a single-ended FET mixer is shown in Figure 7.10. A diplexing coupler is used to combine the RF and LO signals at the gate of the FET. An impedance matching net-work is also usually required between the inputs and the FET, which typically presents a verylow input impedance. RF chokes are used to bias the gate at a negative voltage near pinch-off and to provide a positive bias for the drain of the FET. A bypass capacitor at the drain pro-vides a return path for the LO signal, and a low-pass filter provides the final IF output signal FIGURE 7.9 Variation of FET transconductance versus gate-to-source voltage. FIGURE 7.10 Circuit for a single-ended FET mixer. Our analysis of the mixer of Figure 7.10 follows the original work described in reference [6]. The simplified equivalent circuit is shown in Figure 7.11 and is based on the standard unilateral equivalent circuit for a FET. The RF and LO input voltages are given in (7.22) and (7.23). Let Zg = Rg + jXg be the Thevenin source impedance for the RF input port, and ZL = RL + JX^ be the Thevenin source impedance at the IF output port. These impedances are complex to allow us to conjugately match the input and output ports for maximum power transfer. The LO port has a real generator impedance of Zo, since we are not concerned with maximum power transfer for the LO signal. As we did for the large-signal analysis of the diode mixer, we express the LO pumped FET transconductance as a Fourier series in terms of harmonics of the LO signal: Because we do not have an explicit formula for the transconductance, we cannot calculate directly the Fourier coefficients of (7.48), but must rely on measurements for these values, As in the case of the switching model, the desired down-conversion result is due to the n = 1 term of the Fourier series, so we only need the g\ coefficient. Measurements typically give a value in the range of 10 mS for g\. The conversion gain of the FET mixer can be found as 2008 H.-R. Chuang, EE NCKU 7-19 where V^f is the IF drain voltage, and the impedances Zg and Z^ are chosen for maximum power transfer at the RF and IF ports. The RF frequency component of the phasor voltage FIGURE 7.11 Equivalent circuit for the FET mixer for Figure 7.10 across the gate-to-source capacitance is given in terms of the voltage divider between Z ,R,SindCgs: Multiplying the transconductance of (7.48) by ^(r) = V^COSCURF; gives terms oftr form The down-converted IF frequency component can be extracted from the second ten of (7.51) using the usual trigonometric identity: Then the IF component of the drain voltage is, in phasor form, where (7.50) has been used. Using this result in (7.49) gives the conversion gain (before conjugate matching) as We must now conjugately match the RF and IF ports. Thus we let Rg = Ri,X^ = l/oiRpCg. RL = Rd, and XL = 0, which reduces the above result to The quantities gt, R^, R,, and Cg, are all parameters of the FET. Practical mixer circuit generally use matching circuits to transform the FET impedance to 50 ^ for the RF, LC and IF ports. EXAMPLE 7.2 MIXER CONVERSION GAIN A single-ended FET mixer is to be designed A single-ended FET mixer is to be designed for a wireless local area network receiver operating at 2.4 GHz. The parameters of the FET are: Rd = 300 ?, R, = 100, Cgs =0.3 pF, and g\ = 10 mS. Calculate the maximum possible conversion gain. Solution This is a straightforward application of the formula for conversion gain given in (7.54): Note that this value does not include losses due to the necessary impedance matching networks. FIGURE 7.12 A dual-gate FET mixer. Other FET Mixers There are several variations of mixer circuits that can be implemented using FET. Figure 7.12 shows a single-ended mixer using a dual-gate FET, where the RF and LO inputs are applied to separate gates of the PET. This provides a high degree of RF-LO isolation generally an inferior noise figure relative to the transconductance mixer of Figure 7.10. Another configuration is shown in Figure 7.13, using two FETs in a differential amplifier configuration. The balun (balanced-to-unbalanced) networks on the LO and IP ports provide a transition between a two-wire line that is balanced with respect to ground and a single line that is unbalanced relative to ground. Baluns may be implemented with center-tapped transformers, or with 180 hybrid junctions. The differential mixer operates as an altenating switch, with the LO turning the top two FETs on and off on alternate cycles of the LO. These PETs are biased slightly above pinch-off so each PET will be conducting for slightly more than half of each LO cycle: Thus, one of the upper 2008 H.-R. Chuang, EE NCKU 7-20 FETs is always conducting, and the lower FET will remain in saturation The RF and LO ports should each be impedance matched. The IF output circuit must provide a return path to ground for the LO signal. An extension of the differential FET mixer is the Gilbert cel1 mixer show in Figure 7.14 .This mixer us differential FET mixer stages to form a double balanced mixer. This circuit achieves high RF-LO isolation and a high dynamic range. It also cancels all even-order intermodulation products. This circuit is very popular for wireless integrated circuits. FIGURE 7.13 A differential FET mixer. FIGURE 7.14 A Gilbert cell mixer. 7.4 OTHER MIXER CIRCUITS The single-ended diode and FET mixers discussed above provide frequency conversion, but often have poor RF input matching and RF-LO isolation. This reduces the performance of wireless systems, but fortunately it is possible to improve these characteristics by combining two or more single-ended mixers with hybrid junctions. Balanced Mixers RF input matching and RF-LO isolation can be improved through the use of a balanced mixer, which consists of two single-ended mixers combined with a hybrid junction. Figure 7.15 shows the basic configuration, with either a 90?hybrid (Figure 7.15a), or a 180?hybrid (Figure 7.15b). As we will see, a balanced mixer using a 90?hybrid junction will ideally lead to a perfect input match at the RF port over a wide frequency range, while the use of a 180?hybrid will ideally lead to perfect RF-LO isolation over wide frequency range. In addition, both mixers will reject all even-order intermodulation products. Microwave quadrature or ring hybrids [1] can be used to implement balanced mixers, but at lower frequencies a center-tapped transformer can be used. As shown in Figure 7.16, the secondary of the transformer provides outputs with a 180?phase shift to the two mixer diodes. The LO signal is applied to the center tap of the secondary. The double-balanced mixer of Figure 7.17 uses two hybrid junctions or transformers, and provides good isolation between all three ports, as well as rejection of all even harmonics of the RP and LO signals. This leads to very good conversion loss, but less than ideal input matching at the RF port. The double-balanced mixer also provides a higher third-order intercept point than either a single-ended mixer or a balanced mixer. FIGURE 7.15 Balanced mixer circuits, (a) Using a 90?hybrid, (b) Using a 180 hybrid. FIGURE 7.16 Balanced mixer using a hybrid transformer. FIGURE 7.17 Double-balanced mixer circuit The following table summarizes the characteristics of several types of mixers Small-Signal Analysis of the Balanced Mixer We can analyze the performance of a balanced mixer using the small-signal approachthat was used in Section 7.2. Here we will concentrate on the balanced mixer with a 90 hybrid, shown in Figure 7.15a, and leave the 180?hybrid case as a problem. As usual, let the RF and LO voltages be defined as The scattering matrix for the 90?hybrid junction is [1] 2008 H.-R. Chuang, EE NCKU 7-21 where the ports are numbered as shown in Figure 7.15a. Then the total RF and LO voltages applied to the two diodes can be written as Using only the quadratic term from the small-signal diode approximation of (7.20) gives the diode currents as where the negative sign on ('2 accounts for the reversed diode polarity, and K is a constantfor the quadratic term of the diode response. Adding these two currents at the input to the low-pass filter gives where the usual trigonometric identities have been used, and &IIF = WRF ?(ULO is the IF frequency. Note that the DC components of the diode currents cancel upon combining. After low-pass filtering, the IF output is as desired. We can also calculate the input match at the RF port, and the coupling between the RF and LO ports. If we assume the diodes are matched, and each exhibits a voltage reflection coefficient F at the RF frequency, then the phasor expression for the reflected RF voltages at the diodes will be and (7.61a) These reflected voltages appear at ports 2 and 3 of the hybrid, respectively, and combine to form the following outputs at the RF and LO ports: Thus we see that the phase characteristics of the 90?hybrid lead to perfect cancellation of reflections at the RF port. The isolation between the RF and LO ports, however, is dependent on the matching of the diodes, which may be difficult to maintain over a reasonable frequency range. Image Reject Mixer We have already discussed the fact that two distinct RF input signals at frequencies &IRF == &>LO ?^IF will down-convert to the same IF frequency when mixed with iMLo. These two frequencies are the upper and lower sidebands of a double-sideband signal. The desired response can be arbitrarily selected as either the LSB (&JLO ?^w) or the USB ((ULO + 展F) assuming a positive IF frequency. The image reject mixer, shown in Figure 7.18, can be used to isolate these two responses into separate output signals. The same circuit can also be used for up-conversion, in which case it is usually called a single-sideband modulator. In this case, the IF input signal is delivered to either the LSB or the USB port of the IF hybrid, and the associated single sideband signal is produced at the RF port of the mixer. We can analyze the image reject mixer using the small-signal approximation. Let the RF input signal be expressed as FIGURE 7.18 Circuit for an image reject mixer. where Vu and VL represent the amplitudes of the upper and lower sidebands, respectively. Using the S-matrix given in (7.57) for the 90?hybrid gives the RF voltages at the diodes as After mixing with the LO signal given in (7.56) and low-pass filtering, the IF inputs to the IF hybrid are where K is the mixer constant for the squared term of the diode response. The phasor 2008 H.-R. Chuang, EE NCKU 7-22 representation of the IF signals of (7.65) is Combining these voltages in the IF hybrid gives the following outputs which we see are the separate sidebands of the downconverted input signal of (7.63). These outputs can be expressed in time-domain form as which clearly shows the presence of a 90?phase shift between the two sidebands. Aso note that the image rejection mixer does not incur any additional losses beyond the usual conversion losses of the single rOection mixer. A practical difficulty with image rejection mixers is in fabricating a good hybrid at the relatively low IF frequency. Losses, and hence noise figure, are also usu3ly greater than for a simpler mixer. REFERENCES [1] D. M. Pozar, Microwave Engineering, 2nd edition. Wiley, New York, 1998. [2] S. Y. Yngvesson, Microwave Semiconductor Devices, Kluwer Academic Publishers, 1991. [3] K. Chang, Handbook of Microwave and Optical Components, vol. 2, Chapter 2, "Mixers and Detectors," by E. L. Kollberg, Wiley InterScience, New York, 1990. [4] C. T. Torrey and C. A. Whitmer, Crystal Rectifiers, MIT Radiation Laboratory Series, vol. 14, McGraw-Hill, New York, 1948. [5] S. A. Maas, Microwave Mixers, 2nd edition, Artech House, Dedham, MA, 1993. [6] R. A. Pucel, D. Masse, and R. Bera, "Performance of GaAs MESFET Mixers at X Band," IEEE Tram.Microwave Theory and Techniques, vol. MTT-24, pp. 351-360, June 1976. 2008 H.-R. Chuang, EE NCKU 7-23 2008 H.-R. Chuang, EE NCKU 7-24 3.3 5.7GHz CMOS雙端平衡式混波器設計與製作 為了減少共模雜訊以及clock feed-through透過基板寄生耦合電容對電路的干擾,且增 加LO-IF、LO-RF隔離度、抑制RF和LO訊號的偶次項諧波,故選擇double-balanced mixer 為主要架構(圖3.10)。Double-balanced mixer 架構較single-balanced mixer架構複雜在於混波 器的輸入端需接上一個單端轉雙端的balun,增加電路面積與功率消耗。由於使用在外差 式系統中,LO頻率(5.465~5.525GHz)遠高於IF(280MHz)的情形下,LO洩漏至IF輸出端之 訊號可以輕易被中頻濾波器所衰減,更提高了LO-IF隔離度。 VDD M7 M8 RL VIF- VLO- RL M3 M4 M5 M6 VIF+ VLO- VLO+ VRF+ M1 M2 VRF- 圖3.10 double-balanced mixer電路架構圖 在設計double-balanced mixer時首先決定轉導電晶體大小及偏壓。M1、M2為轉導電晶體, 負責將RF電壓訊號轉為電流訊號,因此其轉導大小為首要考量,轉導要大,偏壓電流勢必要 增大,需要較大的功率消耗,然而較大的直流偏壓卻可獲得較佳的雜訊表現。所以首先考量 轉導電晶體M1 、M2 在偏壓電流各為1mA的條件下,參考[8],找出最佳雜訊表現之電晶體寬 度,決定轉導電晶體M1、M2大小為各為20m/0.18m。 本地振盪信號VLO+、VLO-為大訊號,輸入M3 、M4、M5、M6閘極端形成開關電晶體負責 切換RF電流訊號,達到混波目的。開關電晶體大小決定了開關特性,通常電晶體寬度越大開 關特性越佳,但是其source端較大的雜散電容容易使RF電流訊號衰減,因此需審慎的選擇 M3~M6 的大小,經過模擬微調後,選擇各為40m/0.18m。已知訊號產生器輸出阻抗為50歐 姆,為了達到最大的功率傳輸,利用on chip電阻與電容再加上bondwire等效電感,將開關電 晶體閘極輸出阻抗匹配至50歐姆。由於本地振盪信號為雙端差模輸入,故利用一rat race ring 將原本單端本地振盪訊號分為雙端差動訊號。 混波器負載部分利用RL 電阻接成diode connected PMOS型式,負載大小由電阻RL 與 PMOS汲極與遠源級間的阻抗所構成。M7、M8電晶體寬度大小取決於負載壓降與負載阻抗, 在相同偏壓電流情況下,為了達到最小壓降,可增大M7、M8電晶體寬度,此時電晶體較容易 操作在低飽和區,易造成線性度因非線性負載而下降。增加M7、M8電晶體寬度的結果也增加 了電晶體本身之寄身效應,導致整體負載阻抗因寄生效應而變小,造成轉換增益下降,取決 於線性度與轉換增益之考量下,選擇M7 、M8電晶體寬度為160m/0.18m。考量量測時,所 使用的量測儀器皆是50歐姆輸入阻抗之負載,為了避免嚴重之負載效應,在混波器核心負載 2008 H.-R. Chuang, EE NCKU 7-25 輸出端與50歐姆負載端間置入一緩衝放大器,此緩衝放大器由一共源極電晶體所組成,因此 對混波訊號也有放大之功能。 Double-balanced mixer 之電路與電路佈局圖如圖 3.11 所示,輸出為雙端輸出,因 為 mixer 後級接單端中頻濾波器,因此加上 2:1 的 balun 作為雙端轉單端電路[10]。IF 端 balun 使用 TOKO 616PT-1039,insertion loss 為 3dB。Balun 單端輸出為 480MHz 中 頻訊號,利用 off-chip 的晶片電感 5.6nH、並聯電容 7pF 匹配至 50 歐姆。 5.6nH MLIN MLIN IF + 1.8V IF + 7pF 1.8V 0.8V IF_OUT IF - M8 M7 M10 M9 4.5K IF - 4.5K 1.1V M1&M2 ; gate width = 20um/0.18um LO+ LO+ M3 LO - M4 M5 M6 M3&M4&M5&M6 ; gate width = 40um/0.18um M7&M8 ; gate width = 160um/0.18um M9&M10 ; gate width = 15um/0.18um 0.7V M1 M2 pad equivalent circuit 0.065pF 625Ω bondwire equivalent circuit 2nH 0.65Ω 圖3.11 5.7GHz CMOS double-balanced mixer電路及晶片佈局圖 2008 H.-R. Chuang, EE NCKU 7-26 3.4 模擬與量測結果 Double-balanced mixer設計電路的RF輸入範圍為5.725~5.825GHz,中頻輸出為480MHz頻寬 20MHz,mixer核心部分直流偏壓為1.8V/1.8mA,緩衝放大器部分為各1.8V/1.4mA。在IF balun 插入損耗為 3dB代入轉換損代入模擬結果,轉換增益約12.76dB,input P1dB 約15dBm,IIP3約-6.9dBm,LO-RF的隔離度皆在42dB左右,LO-IF的隔離度約100dB。 量測上利用FR-4製作測試基板來量測晶片,mixer核心部分直流偏壓量測為1.8V/2mA,緩 衝放大器部分為各1.8V/1mA。RF輸入訊號端與本地振盪埠(LO)的輸入訊號由訊號產生器 經由rat race ring產生差動訊號饋入。混波器量測特性結果為轉換增益約11.06dB,input P1dB 約-16.4dBm,IIP3約-7.5dBm, LO-RF的隔離度皆在19dB左右,LO-IF的隔離度約 50dB。模擬與量測結果比較如圖3.12、表3.1所示。晶片照片圖、測試板佈局圖及測試板 照片,如圖3.13所示。 3.5 結果與討論 量測結果除了LO-RF隔離度外,在輸入/輸出匹配、轉換增益、P1dB、OIP3與模擬大 致吻合。而在LO-IF隔離度方面,設計之時因為balun沒有model可以模擬,因此利用一個 理想balun加上3dB的insertion loss代入模擬軟體,由於量測使用之IF balun的高頻響應對LO 有額外的衰減因此得到比模擬更佳的LO-IF隔離度。LO-RF隔離度方面模擬、量測差距較 大,理想上single-balanced的架構LO-RF隔離度應該是相當大。原因可能是因為製程偏移 使得M2及M3電晶體些微不對稱或是substrate coupling,造成LO洩漏至RF端。改進之處為 佈局方面可以再將M2及M3電晶體對稱性提高及將substrate coupling問題列入佈局考量。 0 0 5.7GHz double-balanced mixer measurement simulation -5 -5 -10 S22(dB) S11(dB) -10 -15 -20 -15 -20 -25 -25 -30 -30 -35 5.5 5.55 5.6 5.65 5.7 5.75 5.8 5.85 5.9 5.95 6 -35 300 5.7GHz double-balanced mixer measurement simulation 350 400 RF Frequency (GHz) 450 500 550 600 650 700 IF Frequency (MHz) (a) RF input return loss (b) IF output return loss 25 52 LO-IF isolation(dB) LO-RF isolation (dB) 23 21 19 51 50 49 17 5.7GHz double-balanced mixer LO-RF isolation 15 5245 5265 5285 5305 5325 5345 5365 LO Frequency (MHz) (c) LO-RF isolation 2008 5.7GHz double-balanced mixer LO-IF isolation 48 5245 5265 5285 5305 5325 5345 5365 LO Frequency (MHz) (d) LO-IF isolation H.-R. Chuang, EE NCKU 16 15 14 14 Noise Figure(dB) Conversion Gain (dB) 7-27 12 10 5.7GHz double balanced mixer measurement simulation 8 6 -50 -45 -40 -35 -30 -25 -20 -15 13 12 -10 10 470 475 480 485 490 IF Channel Frequency (MHz) RF Input Power (dBm) (e)轉換增益及input P1dB (f)雜訊指數 m1 0 m7 -50 dBm(OUT) 5.7GHz double-balanced mixer measurement simulation 11 -100 m1 freq=480.5MHz dBm(OUT)=-16.921 -150 m7 freq=481.5MHz dBm(OUT)=-63.246 -200 -250 477 479 481 483 freq, MHz (g) Two tone test / OIP3模擬 (h) Two tone test / OIP3量測 圖3.12 5.7GHz CMOS double-balanced mixer模擬/量測結果 (a) (b) (c) 圖3.13 5.7GHz CMOS double-balanced mixer: (a)晶片照片圖 (b)測試板照片圖 (c)測試板佈局圖 2008 H.-R. Chuang, EE NCKU 7-28 表3.1 5.7GHz CMOS double-balanced mixer模擬/量測特性表 5.7GHz CMOS Double-Balanced Mixer (TSMC 0.18m) RF Frequency Range 5.725~5.825GHz IF Frequency 480MHz LO Frequency 5.265~5.325GHz Vdd 1.8V Simulation Measurement LO Power -3dBm -3dBm Core / Each Buffer Current 1.8/1.4mA 2/1mA Conversion Gain 12.76dB 11.06dB RF Input Return Loss >21dB >18dB Output Return Loss 15dB@280MHz 20dB@280MHz LO-RF Isolation(LO=-3dBm) 42dB 19dB LO-IF Isolation(LO=-3dBm) >100dB >50dB RF-IF Isolation(RF=-30dBm) >200dB Noise Figure 12.4dB 12.8dB IIP3 -6.9dBm@RF=-28dBm -7.5dBm@RF=-28dBm Input P1dB -15dBm -16.4dBm Die size 0.627 x 0.649 mm2 2008 H.-R. Chuang, EE NCKU 7-29 A Desired interfers Image Channel Band Select Filter Response BPF1 r im B Band Select Filter Desired Channel BPF1 Desired Channel LO1 Image im Desired Channel LO 2 Desired Channel IF 2 2008 f Channel Select Filter Response: BPF3 IF1 f E IF1 im D Desired Channel Desired Channel Image r f C r Band Select Filter Response BPF2 f F Channel Select Filter Response: BPF4 IF 2 f Desired Channel G f H IF Amp f IF 2 f H.-R. Chuang, EE NCKU 7-30 Stage 1 Stage 2 Stage 3 Stage 4 Av 4 15 dB Stage 5 Stage 6 A p 4 5 dB A B Duplexer L1 2 dB IF Amplifier NF4 12 dB LNA C Image-Reject Filter Av 2 15 dB NF2 12 dB D L3 6 dB E IF F Filter L5 5 dB LO G NF6 10 dB Figure 6.35 Calculation of noise figure in a cascade of stages Duplexer LNA A B Stage Gain (dB) -2 Voltage -2 Power Cumulative Voltage Gain (dB) Stage NF (dB) Cumulative NF (dB) Stage IP3 Cumulative IP3 Image-RejectMixer IF Filter Filter C D E 15 15 -6 -6 -2 13 2 2 6 8.79 6.79 20.1 +100 dBm 12 dBm +100 dBm -10.6 dBm -12.6 dBm +11 dBm 15 5 7 F -5 22 12 14.1 IF Amplifier 17 5 10 15 10 +5 dBm 1000 Vrms 700mVrms +5 dBm 700mVrms 22.1Vrms Figure 6.36 Level diagranm corresponding to the cascade of Fig. 6.35 2008 H.-R. Chuang, EE NCKU 7-31 Av 15 dB VDD Ap 5 dB SSB NF =10 dB 500 R1 B NF=10dB AIF LO Figure 6.32 Cascade of a mixer and an IF amplifier Spectrum at X Signal Band Image Band Thermal Noise Rs X Y Vin Lo + Lo Spectrum at Y IF Figure 6.17 Folding of RF and image noise into the IF band 2008 H.-R. Chuang, EE NCKU 7-32 1st mixer Filter # 1 Filter # 2 RF amplifier 1st IF stages Injection filter 1st local oscillator 2nd mixer ~ 2nd IF stages Detector 1st IF amplifier ~ 2nd local oscillator ( Tx ) ( Rx ) For a transceiver, if transnitting and receiving use different frequencies (for example, celluLOr phone system) a duplexer is used to Separate Tx & Rx duplexer is a kind of BPF Hence a duplexer will also act as a BPF ( before RF stage ) for image rejection VRF t Vr cos r t V Lo t Vo cos o t V 1VRF 2V Lo depending on combiner Where to add D.C. bias ? lumped circuit form impedance matching is implemmted in the inductor-coie winding. 2008 H.-R. Chuang, EE NCKU Amplitude at mixer RF port 7-33 Desired Signal f LO Image noise fRF RF KTB IF Frequency f LO - IFf f LO LO + IFf RF IF Amplitude at mixer RF port LO f LO 2f LO 3f LO KTB Frequency fLO - fIF 2fLO - fIF 2fLO + fIF fLO + fIF 3f LO - fIF 3f LO+ fIF (e) Dual-Gate MESFET Mixer MESFET VDS 3.65V , I DS 2.5mA VG1S 0.78V , I G2 S 0.78V G 8dB output impedance = 980 j1k RF & LO signal are combined in dual-gate FET structure MODEL RR-12 LO 1 to 2 GHz RF 1 to 2 GHz IF 0 to 0.64 GHz Characteristic Min. Concersion 2008 Typ. Max. Test Condition H.-R. Chuang, EE NCKU 7-34 Loss. dB Phase Deviation From 90o Deg. Amplitude. Unbalance. dB lsolation, dB LO-RF LO-IF RF-IF VSWR RF LO IF 1dB Compression. dBm 3-rd order HP. dBm 5 6 4 0.5 18 20 20 25 25 25 1.5 1.5 1.5 2.0 2.0 2.0 All measurements made in 50 Ohm system. RF=1-2GHz LO=1-2GHz IF=0.03GHz PLO = 7 dBm PRF =10 dBm 5 15 <Conversion Loss of Mixer> In mixer design several frequencies (RF,LO,IF) and Their harmonics are involved. Impedance matching design at three ports (RF,LO,IF) is complicated by the above situation. Undesired harmonic signal can be dissipated in resistive termination. or blocked with reactive terminetion. increase mixer loss frequency dependent An important figure of merit of the mixer Conversion loss Lc 10 log avaialable RF input power IF output power typical Lc = 5 ~ 8 dB for passire mixers;a active mixer can have a gain High local oscillator signal (pump) power can rgeduce the mixer conversion loss 2008 H.-R. Chuang, EE NCKU 7-35 minimum Lc usually sccurs for 0 ~ 10 dBm LO powers But (0 ~ 10 dBm) LO power will violate the small signal approximation of the mixer diode. precise design requires numerical ssolution of the nonlinear equation theat describes the diode charactoristics. i : IF VRF Vr cos r t 2VrVo sin r o t 2VrVo sin i t let r o i r o i let r o i r o i VRF Vr cos r t 2VrVo sin r o t 2VrVo sin i t fr f LO f IF all produce a same IF frequency f IF and cause "image interference" fim f LO f IF Fig. 12 Measured characteristics of a 2.4 GHz bandpass filter (FDK 2450B). V A VU cos t VL cos t o i o i 2 2 r V V VrB U cos o i t 90 o L cos o i t 90 o 2 2 V A cos o t Vi A IF mixing rB Vr cos o t Vi B V1 k VU sin i t VL sin i t VU sin i t 180 o VL sin i t 2 2kVL sin i t Vi A kVU sin i t kVL sin i t B o o Vi kVU sin i t 90 kVL sin i t 90 2008 H.-R. Chuang, EE NCKU 7-36 Mixer Performance Characteristics Mixer Conversion Loss : Lc 10log RF f RF IF f f f RF f LO f RF f LO LO => stringly depend on LO power level f f LO available RF input power dB IF output power Mixer Noise Figure 混頻器除了射頻(RF)頻帶之雜訊混至中頻(IF)帶外,同時有image頻帶及 贅餘(spurious)頻帶之雜訊混入中頻 Signal Band Image Band Thermal Noise Lo f IF f LO f RF 2 f LO- f IF 2 f LO+ f IF Mixer noise 單旁波帶(SSB: single side band)雜訊指數與雙埠放大器之定義相同 測量上多採用雙旁波帶(DSB:double side band) 雜訊指數較方便¡ => 雙旁波帶雜訊指數為單旁波帶雜訊指數的兩倍 Noise Figure Meter Broad band Noise Source Mixer Pre Amp LO Signal Generator Mixer DSB noise figure measurement 混頻器為雙旁波帶雜訊指數測量 2008 H.-R. Chuang, EE NCKU Amplitude at mixer RF port 7-37 Desired Signal f LO fRF Image noise KTB Frequency fLO - f IF 2008 fLO + fIF H.-R. Chuang, EE NCKU 7-38 APPENDIX 3 : SOME APPLICATIONS OF MICROSTRIP CIRCUITS * Fooks "Microwave Engineering using Microstrip Circuits" (Ch 12) This part introduces selected practical microstrip circuits or subsystems. It brings together some of the combination of microwave passive components & active devices to produce functioning self-contained building blocks, that in turn may be a part of a complete microwave systems. FIGURE 11.16 Frequency conversion in a receiver and transmitter. (a) Down-conversion in a heterodyne. (b) Up-conversion in a transmitter. f IF f RF f Lo , and a much higher frequency signal, f RF f Lo (filtered out). 2-stage mixers for vetter image frequency rejetion. Portable communication Receiver BLOck Diagram <Mixer in Rransmitter> v1 t Vr cos r t 90 o Vo Vn cos o t 180 o Vr sin r t Vo Vn cos o t conside only quadratic term o o v2 t Vr cos r t 180 Vo Vn cos o t 90 of the diode which givers the desired mixer product Vr cos r t Vo Vn sin o t 混頻器的雜訊指數,根據IEEE的定義,單旁波帶(SSB:single side band)雜訊指數與雙埠放 大器之定義相同,如(2.4)式,但在實際測量上較為困難,若以濾波器阻隔假像頻率(image grequency)頻帶之雜訊,則濾波器將造成輸入阻抗改變,且增加溫度雜訊。因此在測量上多 採用雙旁波帶(DSB:double side band)雜訊指數較方便,一般的定義,雙旁波帶雜訊指數為 單旁波帶雜訊指數的兩倍。圖2.5說明混頻器雜訊之造成,混頻器除了射頻(RF)頻帶之雜訊混 至中頻(IF)帶外,同時也將假像(image)頻帶及贅餘(spurious)頻帶之雜訊混入中頻,因此混 頻器的雜訊指數將比放大器來得大。圖2.6為測量混頻器雜訊指數之架構,所得為雙旁波帶雜 訊指數。 2008 H.-R. Chuang, EE NCKU 7-39 f IF2 small & requiring higher Q of the IF filter => channel bandwidth IF 2008 H.-R. Chuang, EE NCKU 7-40 Gr Gt Pt Pr R t t RF DC f f RF f (a) Diode rectifier t t Modulated RF Modulation f f RF f fm (b) Diode detector RF IF f f RF f f RF f LO f RF f LO LO f LO f (c) Mixer (frequency conversion) RF f RF IF f f f RF f LO f RF f LO LO f LO f For a single-ended mixer, the noise terms will be 2008 1 2 V V t V t o 2 n n 1V V 2 o n t H.-R. Chuang, EE NCKU 7-41 Desired Channel interfers Image A r im im r f Desired Channel C Image Desired Channel B f D Image r im f F E IF1 IF 2 f f H G IF 2 2008 IF1 f f IF 2 f H.-R. Chuang, EE NCKU 7-42 Mixer Nonlinear Simulation by Libra Since I (V ) I s (e V 1) dI dV Vo d 2 I dV 2 I s e Vo I o I s Gd R j 1 Vo R j : junction resistance d dI dV / dV Vo dGd dV Vo 2 I s e Vo 2 I o I s Gd Gd dI dV V I s e Vo I o I s Gd R j 1 o R j : junction resistance Since I (V ) I s (e V 1) d 2 I dV 2 Vo d dI dV / dV Vo dGd dV Vo 2 I s e Vo 2 I o I s Gd Gd Baseband filter Power amplifier Mixer fM Antenna fLo + fM Up-conversion (for transmitting) f LO Local oscillator LNA fLo + fM fo = fLo - fM IF filter Mixer fIF + fM Local oscillator IF amplifier Detector fIF Down-conversion (receiving) The band-stop response of the BPF will determine the image-rejection ratio. 2008 H.-R. Chuang, EE NCKU 7-43 Band Select Filter A BPF1 Image reject Filter LNA BPF2 B Channel Select Filter 1st Mixer C D BPF3 Image Reject Filter Mixer RF IF cos0t E BPF4 F IF Amplifier G H LO 2 LO1 LNA Channel Select Filter 2nd Mixer Channel Select Filter LO RF input VRF Vr cos r t LO input VLo 2008 v1 Diode 1 i1 90o 3 dB hybrid Vo Vn t cosot + v2 Diode 2 i2 If output LP filters H.-R. Chuang, EE NCKU 7-44 VU cos(o i )t vr VL cos(o i )t Mixer A IF vrA RF LO RF input vr vrB RF 90 o hybrid Z0 Mixer B LO v1 LO input 3-dB power divider o v2 2008 USB 90 o hybrid IF VRF VRF VRF V1 2 j V2 2 VRF VLo 2 1 2 1 1 j VRF VLo 2 12 Vr j 12 VLo j j 12 Vr 12 VLo 2 2 jVLo no Vr reflection, but VLO signal appears at RF port VLO VLO VLO jVr 1 2 no VLo reflection, but Vr signal appears at LO port VRF VRF VRF V1 2 jV2 2 1 V1 jV2 1 2 2 1 1 Vr jVLO j 1 jVr VLO 2 2 2 12 Vr jVLO j jVr VLO jVLO no Vr reflection, but VLO signal appears at RF port LO LO LO V V1 V2 jVr no VLo reflection, but Vr signal appears at LO port monolithic quad DMOS FET for mixer application LSB IF out H.-R. Chuang, EE NCKU 7-45 FET mixer has a gain . (Diode mixer has no gain) TABLE 11.1 Frequencies and Relative Amplitudes of the Square-Law Output of a 2 Detected AM signal V t Frequency Relative Amplitude 1 m2 2 0 m 2 m 2m m2 2 2 1 m 2 m 2 o 2 o m 2 o m desired demodulated output 2 1 V 2 Gd 2m cos t Vo Gd m cos t m m 4 o 2 1 2 Vo 2 2 m 4 EXAMPLE 1.4 A diode in as axial-lead package has the following equivalent circuit parameters: Cp =0.10 pF, Lp=2.0nH, Cj=0.15pF, Rs=10, and I s=0.1A. Calculate and plot the impedance of this diode from 4 to 14 GHz, for a bias current Io=0 and Io=60A. Ignore the change in Cj with bias, and assume =1/25 mV. solution From (11.27) the junction resistance for the two bias states is for I o 0 , 1 mV 22.5 105 , 25 0 . 1 A I o I s 1 Rj 25mV 417. I o I s 60 0.1A Rj for I o 60 A , Then the input impedance can be calculated from the equivalent circuit of Figure 11.12; the result is plotted versus frequency on a 50 Smith chart in Figure 11.13. Diode impedance is frequency dependent! FIGURE 11.13 Impedance of the diode of Example 11.4 for to bias states, from 4 to 14 GHz. 1 (11.27) R j Gd1 I o I s 2008 H.-R. Chuang, EE NCKU 7-46 FM broadcasting IF = 10.7MHz Cellular phone IF = 45MHz IF = 455MHz DC bias Combiner RF vi cos(r o )t Matching network vr cos r t DC return LO vo cos ot r ,o LP filter r o ........ Diode 1 RF input i1 If output + LO input Diode 2 3 dB hybrid (90 o or 180 o ) i2 LP filters Fig. Band Select Filter Image reject Filter Channel Select Filter Channel Select Filter BPF1 BPF2 BPF3 BPF4 A C B 10.19 interfers interfers Image Desired Channel r im f A LO1 Band Select Filter Image reject Filter BPF1 BPF2 B E D F H G LO 2 Channel Select Filter Channel Select Filter BPF3 C IF Amplifier D LO1 E BPF4 F LO 2 IF Amplifier G H Desired Channel IF 2 Fig. 6 Measured characteristics of a 2.4-GHz single-ended resistive FET mixer. 2008 H.-R. Chuang, EE NCKU 7-47 Mixer Noise Figure 混頻器除了射頻(RF)頻帶之雜訊混至中頻(IF)帶外,同時有image頻帶及 贅餘(spurious)頻帶之雜訊混入中頻¡ f IF f LO f RF 2 f LO- f IF 2 f LO+ f IF Mixer noise 單旁波帶(SSB: single side band)雜訊指數與雙埠放大器之定義相同 測量上多採用雙旁波帶(DSB:double side band) 雜訊指數較方便¡ => 單旁波帶雜訊指數為單旁波帶雜訊指數的兩倍 Noise Figure Meter Broad band Noise Source Mixer Pre Amp LO Signal Generator Mixer DSB noise figure measurement 混頻器為雙旁波帶雜訊指數測量 2008 H.-R. Chuang, EE NCKU 7-48 2008 H.-R. Chuang, EE NCKU 7-49 VRF v1 (Vr V RF Vr cosr t 2 ) ( j VLo i1 Diode 1 Diode 2 LO input v 2 VLo Vo coso t (90 ) 2 ) ( j VLo 2) 2) LP filter i1 Diode 1 + Diode 2 LO input v 2 If output i IF 2 KV rVo sin i t i2 LP filter 3 dB hybrid (90o ) 2008 2 ) (V Lo v1 VLo Vo coso t i IF 2 KVrVo sin i t i2 v 2 ( j Vr v1 (Vr V RF Vr cosr t If output LP filter RF input VLO + 3 dB hybrid o VRF LP filter v1 RF input VLO 2) v 2 ( j Vr 2 ) (V Lo 2) H.-R. Chuang, EE NCKU 7-50 2008 H.-R. Chuang, EE NCKU 7-51 DC bias Combiner RF IF vi cos(r o )t Matching network DC return VRF vr cosr t VLO vo cosot LO r ,o r o LP filter ........ Single-ended mixer circuit v VRF VLo Vr cos r t Vo cos o t For a diode small-signal approximation I V I o vGd v 2 Gd 2 ................... Gd 2 G 2d G IF output 4d v 2 Gd 2 2008 (note : v VRF VLo Vr cosr t Vo cosot ) Vr cosr Vo cosot 2 V 2 cos2 rt 2VrVo cosrt cosot Vo2 cos2 ot ....... r 2VrVo cosr o t G2d VrVo cosIF t H.-R. Chuang, EE NCKU 7-52 2008 H.-R. Chuang, EE NCKU 7-53 (a) Single-ended Mixer Pozar (RF Ch 7) Mixer in a Transmitter In a transmitter, a mixer is used to mix with IF signal to up-convert the signal frequency for efficient radio-wave transmission from antenna. Baseband filter fM Power amplifier Mixer Antenna fLo + fM f LO Up-conversion (for transmitting) t Local oscillator f Lo f IF :Upper Sideband USB f Lo f IF :Lower Sideband LSB Double sideband (DSB)= USB + LSB => For a single sideband transmission (SSB) USB f Lo f IF or LSB f Lo f IF 2008 H.-R. Chuang, EE NCKU 7-54 We use a sideband filter or an image rejection mixer to remove a sideband signal. ( also celled single sideband modulator ) direct-conversion transmitter Baseband I PA cos c t Matching Network sin c t Duplexer Baseband Q drawback: leakage of PA output to LO I PA LO BPF LO 0 0 -5 -5 -10 -10 -15 -15 S22 (dB) S11 (dB) Q -20 -25 -30 -35 -20 -25 -30 -35 measurement simultaioon -40 5.6 5.65 5.7 5.75 5.8 5.85 5.9 Frequency (GHz) -40 400 measurement simultaioon 420 440 460 480 500 520 540 560 Frequency (MHz) (a) (b) 14 Conversion Gain (dB) 12 10 8 6 4 2 0 -40 measuemsnt simulation -35 -30 -25 -20 -15 -10 -5 0 input power (dBm) (c) 2008 (d) H.-R. Chuang, EE NCKU 7-55 (中正大學電機M.S. Thesis) f LO 2f LO 3f LO fLO - fIF 2fLO - fIF fLO + fIF 2fLO + fIF 3fLO - fIF 3fLO + fIF Spurious chart due to LO harmonics 2008 H.-R. Chuang, EE NCKU 7-56 2008 H.-R. Chuang, EE NCKU 7-57 image rejection filter The BPF band-stop response determines the image-rejection ratio Desired Band Image Reject Filter image r LNA Image Reject Filter Response A0 cos LO t image Desired Band r im 2 IF A 2.4 GHz bandpass filter =>passband = 100 MHz => insertion loss < 1 dB RF f RF IF f f f RF f LO LO f RF f LO f f LO (c) Mixer (frequency conversion) Down-conversion (for receiving) Mixer LNA IF amplifier Detector fIF f RF f LO f M f RF f o fo ( f LO f IF ) Local oscillator 2008 IF filter f LO f o f M fM t t f IF f M H.-R. Chuang, EE NCKU 7-58 t t RF DC 1st mixer Filter # 1 f f RF f (a) Diode rectifier Filter # 2 RF amplifier Injection filter 1st local oscillator t 2nd IF stages Detector 1st IF amplifier ~ ~ 2nd local oscillator t Modulated RF Modulation f f RF 2nd mixer 1st IF stages fm (b) Diode detector f Then the input to the two mixers through a o 90 hybrid is V A VU cos( )t VL cos( )t o IF o IF r 2 2 RF inputs V V VrB U cos[(o IF )t 90 o ] L cos[(o IF )t 90 o ] 2 2 V A r VrB VU V cos[(o IF )t 90 o ] L cos[(o IF )t 90 o ] 2 2 VU V sin( o IF )t L sin( o IF )t 2 2 VU V cos[(o IF )t 180 o ] L cos[(o IF )t 180 o ] 2 2 V V U cos(o IF )t L cos(o IF )t 2 2 After mixing with an LO signal of cos o t , the IF outputs of the mixers are Vi A kVU sin IF t kVL sin IF t IF outputs Vi B kVU sin( IF t 90 o ) kVL sin( IF t 90 o ) Combining these two signals in the 90o hybrid at the IF output gives V1 1 2 (kVU sin IF t kVL sin IF t ) o o o o [kVU sin( IF t 90 90 ) kVL sin( IF t 90 90 )] k [V 2 U sin IF t VL sin IF t VU sin( IF t 180 o ) VL sin IF t ] 2kVL sin IF t 2008 LSB component H.-R. Chuang, EE NCKU 7-59 Then the input to the two mixers through a 90o hybrid is V A VU cos( )t VL cos( )t o IF o IF r 2 2 RF inputs V V VrB U cos[(o IF )t 90 o ] L cos[(o IF )t 90 o ] 2 2 After mixing with an LO signal of cos o t & lowpass filtered, the IF outputs of the mixers are Vi A IF outputs B Vi 1 [ kV U 2 2 cos IF t kVL cos IF t ] 1 [ kV U 2 2 cos( IF t 90 o ) kVL cos( IF t 90 o )] Combining these two signals in the 90o IF hybrid (transformer) at the IF output gives 1 [kVU cos IF t kVL cos IF t ] 2 2 1 V1 2 1 [kVU cos( IF t 90 o 90 o ) kVL cos( IF t 90 o 90 o )] 2 2 k4 [VU cos IF t VU cos( IF t 180 o ) VL cos IF t VL cos IF t ] 0 k2 VL cos IF t LSB component 1 [ VU cos( IF t 90 o ) VL cos( IF t 90 o )] 2 2 k V2 2 1 [VU cos( IF t 90 o ) VL cos( IF t 90 o )] 2 2 k2 VU sin IF t 2008 USB component H.-R. Chuang, EE NCKU 7-60 Then the input to the two mixers through a 90o hybrid is A v RF RF inputs B v RF 1 2 1 2 1 2 1 2 V U cos(o t IF t 90 o ) VL cos(o t IF t 90 o ) VU sin( o IF )t VL sin( o IF )t V U cos(o t IF t 180 o ) VL cos(o t IF t 180 o ) VU cos(o IF )t VL cos(o IF )t After mixing with an LO signal of cos o t & lowpass filtered, the IF outputs of the mixers are (K = the mixer constant for the squared term of the diode) A v IF IF inputs (to IF hybrid) B v IF K 2 2 VLO [VU VL ] sin IF t K V [V 2 2 LO U VL ] cos IF t A jK VIF 2 2 VLO VU VL *Phasor representation K B VIF VLO VU VL 2 2 Combining these two signals in the 90o IF hybrid (transformer) at the IF output gives V1 j A VIF V B KVLOVL IF 2 2 2 KVLOVL cos IF t , 2 KVLOVU v2 ( t ) sin IF t , 2 v1 (t ) A B jKVLOVU VIF VIF V2 j 2 2 2 VU cos(o IF )t image vr ( vDesired vRF ) IM Band V cos( )t o IF L r im vr RF input Mixer A IF A vRF B vRF Z0 2008 Desired Band LO RF 2 IF 90o hybrid LO 3-dB power divider LO input LSB v1 r vLO VLO cos(ot ) RF Mixer B A vIF LPF IF v2 90o image USB IF hybrid (transformer) B vIF LPF IF out im H.-R. Chuang, EE NCKU 7-61 I V I o vGd v 2 Gd 2 ................... (note : v VRF VLo Vr cosr t Vo cosot ) Gd Vr cosr t Vo cosot 2 2 G 2d (V r2 cos2 r t 2VrVo cosr t cosot Vo2 cos2 ot ) G G IF output 4d 2VrVo cos r o t 2d VrVo cosIF t v 2 Gd 2 v1 (Vr VLO poor isolation v1 V RF Vr cos r t 2008 2) LP filter i1 RF Diode 1 LO Diode 2 v 2 V Lo Vo cos o t VRF 2 ) ( j VLo + i2 i IF 2 KVrVo sin i t LP filter 3 dB hybrid (90o ) IF output v 2 ( j Vr 2 ) (V Lo 2) H.-R. Chuang, EE NCKU