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Course Outline Basic Concepts in RF Design Low-Noise Amplifiers Mixers Oscillators Phase-Locked Loops General Considerations Mixers perform frequency translation by multiplying two waveforms While a decade ago, most mixers were realized as a Gilbert cell, many more variants have recently been introduced to satisfy the specific demands of different RX or TX architectures. A stand-alone mixer design is no longer meaningful because its ultimate performance heavily depends on the circuits surrounding it. We will focus on downconversion mixers. General Considerations 3 ports: RF, LO and IF (BB) Linear & system wrt VRF time-variant VRF is multiplied by a square wave toggling 1 & 0 The LO port of this mixer is very nonlinear producing at the output “mixing spurs”. The RF port must remain sufficiently linear to satisfy the compression and/or intermodulation requirements. As seen later, mixers suffer from a lower gain and higher noise as the switching in the LO port becomes less abrupt. General Considerations: Noise and Linearity, Gain Noise and Linearity: The design of downconversion mixers entails a compromise between the noise figure and the IP3 (or P1dB). In a receive chain, the input noise of the mixer following the LNA is divided by the LNA gain when referred to the RX input. Similarly, the IP3 of the mixer is scaled down by the LNA gain. Gain: mixer gain is critical in suppression of noise while retaining linearity. Downconversion mixers must provide sufficient gain to adequately suppress the noise contributed by subsequent stages. However, low supply voltages make it difficult to achieve a gain of more than roughly 10 dB while retaining linearity. General Considerations: Port-to-Port Feedthrough Due to device capacitances, mixers suffer from unwanted coupling (feedthrough) from one port to another. The gate-source and gate-drain capacitances create feedthrough from the LO port to the RF and IF ports. The effects of mixer port-to-port feedthrough on the performance depends on the architecture: direct-conversion or indirect (heterodyne) receiver. Port-to-Port Feedthrough in Direct-Conversion RX LO-IF feedthrough is heavily suppressed by the baseband low-pass filter(s). LO-RF feedthrough produces: LO radiation from the antenna offset in the baseband: suppose the RF input is a sinusoid having the same frequency as the LO. Each time the switch turns on, the same portion of VRF appears at the output, producing a not null DC component. Port-to-Port Feedthrough in Direct-Conversion RX RF-LO feedthrough: a large in-band interferer can couple to the LO and injection-pull it, thereby corrupting the LO spectrum (a buffer is typically interposed between the LO and the mixer). RF-IF feedthrough corrupts the baseband signal by the beat component resulting from even-order distortion in the RF path. Port-to-Port Feedthrough in Heterodyne RX Example: ωLO=ω ωRF/2=ω ωIF the LO-RF feedthrough is suppressed by the LNA selectivity the RF-LO feedthrough is not critical because in-band interferers are far from the LO frequency, creating little injection pulling the RF-IF feedthrough proves benign because low-frequency beat components appearing at the RF port can be removed by high-pass filtering. the LO-IF feedthrough, becomes serious if ωIF and ωLO are too close to allow filtering of the latter. Potential desensitization of IF mixers. General Considerations: Mixer NF in Heterodyne RX For simplicity, let us consider an ideal noiseless mixer with unity gain and flat response in the entire band. The output SNR is half the input SNR NF=2 (3 dB) This quantity is called single-sideband” (SSB) noise to indicate that the desired signal resides on only one side of the LO frequency General Considerations: Mixer NF in Direct-Conversion RX In this case, only the noise in the signal band is translated to the baseband, thereby yielding equal input and output SNRs if the mixer is noiseless. The noise figure is thus equal to 0 dB. This quantity is called the “double-sideband” (DSB) noise figure (the signal resides on both sides of ωLO). The SSB noise figure of a mixer is 3 dB higher than its DSB noise figure if the signal and image bands experience equal gains at the RF port of the mixer. Typical noise figure meters measure the DSB-NF and predict the SSB-NF simply adding 3 dB. Passive Mixers: Single-Ended Mixer The voltage conversion gain of the single-ended mixer is equal to 1/π (~ -10dB) for abrupt LO switching. We call this topology a “return-to-zero” (RZ) mixer because the output falls to zero when the switch turns off. Single-Ended Mixer: Noise Assume RL is noiseless, RL >> RS and the LO has a 50% duty cycle DC amplitude = 0.5 fLO amplitude = (1/π) 3fLO amplitude = 1/(3π) ………. sin (nπ / 2 ) An = nπ Single-Ended Mixer: Noise kTRS The output spectrum consists of (a) 2kTRS × 0.52, (b) 2kTRS shifted to the right and to the left by ± fLO and multiplied by (1/π)2, (c) 2kTRS shifted to the right and to the left by ± 3fLO and multiplied by [1/(3π)]2, etc. We therefore write It follows that the two-sided output spectrum is equal to kTRS and hence the one-sided spectrum is given by Single-Ended Mixer: Noise More generally, if white noise is turned on for ∆T seconds and off for T - ∆T seconds, then the resulting spectrum is still white and its power is scaled by ∆T I T (the duty cycle). 2 n , out V ∆T = 4kTRS T Single-Ended Mixer: Noise Figure The output noise is given by 4kT(Ron||RL) when S1 is on and by 4kTRL when it is off (Ron includes RS). On the average, If we select Ron << RL to minimize the conversion loss, Dividing the result by the power gain 1/π2 NF = 1 + Vn2,in 4kTRS ≈ 1+ 5 RL RS Ron << RL If Ron = 100 Ω and RL = 1 kΩ, determine the input-referred noise and the NF of the above RZ mixer. Solution: Ron << RL Vn2,in R NF = 1 + ≈ 1 + 5 L = 101( 20dB) 4kTRS RS Vn2,in = 9.1nV/Hz1/2 Passive Mixers: Single-Balanced Mixer The single-ended mixer operates with a single-ended RF input and a single-ended LO discarding the RF signal for half of the LO period. Single-Balanced Mixer: two switches are driven by differential LO phases, thus “commutating” the RF input to the two outputs. Passive Mixers: Single-Balanced Mixer the differential output contains twice the amplitude of each single-ended output the voltage conversion gain is 2/π (≈ -4 dB). the input-referred noise is half of the single-ended mixer (twice the output noise and twice the voltage gain). providing differential outputs and twice the gain, this circuit is superior to the singleended topology. suffers from significant LO-IF leakage (doubled wrt to single-ended mixer). Passive Mixers: Double-Balanced Mixer We connect two single-balanced mixers such that their output LO feedthrough cancel (appears as a common mode output) but their output signals do not. Called a “double-balanced” mixer, the circuit above operates with both balanced LO waveforms and balanced RF inputs. Double-Balanced Mixer: Gain RL RL Vout1 is equal to VRF+ for one half of the LO cycle and equal to VRF- for the other half, i.e, the load resistors can be omitted because the outputs do not “float.” Vout1-Vout2 can be decomposed into two RZ waveforms, each having a peak amplitude of 2V0 and IF amplitude of (1/π)2V0. Since they are 180 ° out of phase, the Vout1-Vout2 contains an IF amplitude of (1/π)(4V0). The peak differential input is equal to 2V0 the voltage conversion gain is 2/π, equal to that of the single-balanced counterpart. It is possible to apply a single-ended RF input grounding one input. Double Balanced Mixer: Noise the behavior of the circuit does not depend much on the absence or presence of load resistors. with abrupt LO edges, a resistance equal to R1 appears between one input and one output at any point in time Since the voltage conversion is equal to 2/π π Optimal LO Waveform The LO waveform must ideally be a square wave to ensure abrupt switching and hence maximum conversion gain. if VLO changes gradually, then the two phases remain approximately equal for a substantial fraction of the period (∆ ∆T). During this time, all four transistors are on, treating VRF as a common-mode input. That is, the input signal is ''wasted'' because it produces no differential component for roughly 2∆ ∆T seconds each period. Passive Mixers: Sampling Mixer If the resistor is replaced with a capacitor, such an arrangement operates as a sample-and-hold circuit and exhibits a higher gain because the output is held—rather than reset—when the switch turns off. Called a “Non-return-to-Zero” (NRZ) Mxer The output waveform can be decomposed into two waveforms return-to-zero output output stored on CL when the switch is open Sampling Mixer: Conversion Gain (Ⅰ Ⅰ) We first recall the following Fourier transform pairs: Since y1(t) is equal to x(t) multiplied by a square wave toggling between zero and 1, and since such a square wave is equal to the convolution of a square pulse and a train of impulses shown below, Sampling Mixer: Conversion Gain (Ⅱ Ⅱ) The component of interest in Y1(f) lies at the IF and is obtained by setting k to ± 1 As expected, the conversion gain from X(f) to Y1(f) is equal to 1/π, but with a phase shift of 90° °. Sampling Mixer: Conversion Gain (Ⅲ Ⅲ) The second output, y2(t), can be viewed as a train of impulses that sample the input and are subsequently convolved with a square pulse: Figure below depicts the spectrum, revealing that shifted replicas of X(f) are multiplied by a sinc envelope. Sampling Mixer: Conversion Gain (Ⅳ Ⅳ) The component of interest in Y2(f) is obtained by setting k to ± 1 (if the IF is much lower than 2fLO) The total IF output is therefore equal to the voltage conversion gain is 0.593 If realized as a single-balanced topology, the circuit provides a gain twice this value, 1.186~1.48 dB (5.5 dB higher than RZ single-balanced mixer). Though a passive circuit, the single-balanced sampling mixer actually has a voltage conversion gain greater than unity. The RZ mixer is rarely used in modern RF design. Double-Balanced Sampling Mixer: Gain the capacitors play no role here because each output is equal to one of the inputs at any given point in time. the conversion gain is therefore equal to 2/π, about 5.5 dB lower than that of the single-balanced topology. ruling out double-balanced mixers is not a serious limitation because most of RF designs incorporate single-ended LNAs. Output Current Combining of Two Single-Balanced Mixers If necessary, double-balanced operation can be realized through the use of two single-balanced mixers whose outputs are summed in the current domain M1, M2, M3, M4 provides voltage to current conversion summed currents produce the differential output voltage across the load resistors (current-to-voltage conversion). the conversion gain is still equal to 1.48 dB Sampling Mixers: Noise The total (one-sided) output noise of the singleended sampling mixer is The input-referred noise is obtained by dividing this result by the squared conversion gain 1/π2 + 1/4: Single-balanced passive (sampling) mixer: the differential output exhibits twice the noise, but the voltage conversion gain is also doubled Double-balanced passive (sampling) mixer (does not depend on the load) Noise of the Subsequent Stage The low gain of passive mixers makes the noise of the subsequent stage critical. A quasi-differential pair serves as an amplifier and its input capacitance holds the mixer output. Each common-source stage exhibits an input-referred noise voltage of This power should be doubled to account for the two halves of the circuit and added to the mixer output noise. The network consisting of RREF , MREF, and IREF defines the bias current of M1 and M2. RREF is chosen much greater of the output resistance of the preceding stage. Active Mixers Mixers can be realized so as to achieve conversion gain in one stage. Called active mixers, such topologies perform three functions: they convert the RF voltage to a current, “commutate” (steer) the RF current by the LO, and convert the IF current to voltage. Single-balanced topology (M2 and M3 are called the “switching pair”) Active Single-Balanced Mixer: Conversion Gain for R1 = R2 = RD The conversion gain is equal to 2/π, yielding an output given by Active Single-Balanced Mixer: Conversion Gain To maintain transistors in saturation, the voltage drop across load resistor should not exceed a maximum value so that RD should not exceeds a maximum value. The maximum voltage conversion gain is AV ,max = 2 π g m1 RD ,max Low VDD reduces VR,max gain/voltage headroom trade-off IP3 is proportional to gate overdrive Given a fixed power budget (IDD), gain/linearity trade off (gm inversely proportional to overdrive) IP3 requirements can be relaxed if the LNA gain can be reduced. This is possible if the mixer NF can be reduced. Active Single-Balanced Mixer: Conversion Gain As for passive mixers, the conversion gain may fall if the LO swing is lowered. While M2 and M3 are near equilibrium, the RF current produced by M1 is split approximately equally between them, thus appearing as a commonmode current and yielding little conversion gain for that period of time. Reduction of the LO swing tends to increase this time and lower the gain. Gain Degradation Due to Capacitance at Drain of Input Transistor With abrupt LO edges, M2 is on and M3 is off, yielding a total capacitance at node P equal to: C P = C DB1 + CGD1 + CGS 1 + CGS 2 + CSB 2 + C SB 3 The RF current produced by M1 is split between CP and the resistance seen at the source of M2, 1/gm2. The voltage conversion gain is modified as: AV = 2 π g m1 RD gm2 C P2ω 2 + g m2 2 Active Single-Balanced Mixer: Noise The noise components of interest lie in the RF range before downconversion and in the IF range after downconversion. The frequency translation of RF noise by the switching devices prohibits the direct use of small-signal ac and noise analysis in circuit simulators, necessitating simulations in the time domain. Moreover, the noise contributed by the switching devices exhibits timevarying statistics, complicating the analysis. Active Single-Balanced Mixer: Noise assume abrupt LO transitions and consider the representation in figure above for half of the LO cycle. In this phase, the circuit reduces to a cascode structure, with M2 contributing some noise because of the capacitance at node P. At frequencies well below fT , the output noise current generated by M2 is equal to ~Vn,M2CPs (gm2>>ω ωCP) This noise and the noise current of M1 (which is dominant) are multiplied by a square wave toggling between 0 and 1. Active Single-Balanced Mixer: Noise Assuming 50% duty cycle half of the noise powers (squared current quantities) of M1 and M2 is injected into node X, the total noise at node X is equal to Accounting for the noise at node Y and then divided by the square of the conversion gain If the effect of CP is negligible: Active Single-Balanced Mixer: Noise Consider a more realistic case where the LO transitions are not abrupt The circuit now resembles a differential pair near equilibrium, amplifying the noise of M2 and M3—while the noise of M1 has little effect on the output because it behaves as a common-mode disturbance. Active Single-Balanced Mixer: Noise ID1 (gm1) can be increased, to lower the noise, without changing voltage drops in the circuit by scalig the transistor widths and currents by a factor of α and the load resistors by a factor of 1/α. Unfortunately, this scaling also scales the capacitances seen at the RF and LO ports, making the design of the LNA and the LO buffer more difficult and/or more power-hungry. Active Single-Balanced Mixer: Flicker Noise The input refered flicker noise due to M2 is multiply by √2 to account for the noise of M3. it is desirable to minimize the bias current ISS (in passive mixers ISS=0) Vn2(f) is typically very large because M2 and M3 are relatively small Active Single-Balanced Mixer: Linearity The input transistor imposes a direct trade-off between nonlinearity and noise (at fixed power budget ID). Vn2,M1 The linearity of active mixers degrades if the switching transistors enter the triode region. Thus, the LO swings cannot be arbitrarily large. Active Mixers: Double-Balanced Topology VX1=VY2 (VY1=VX2) so that X1 (Y1) and Y2 (X2) can be shorted (VX1 - VY1)/VRF+ is equal to the conversion gain of a single-balanced mixer the differential gain of the double-balanced topology is therefore given by which is half of that of the single-balanced counterpart. Active Double-Balanced Mixer: Noise Assume the same bias drain currents (i.e. the bias current of double- balanced mixer is twice the current of single-balance mixer) If the total differential output noise current of the single-balanced topology is then that of the double-balanced circuit is equal to (double hardware with same noise of single-balanced topology) Active Double-Balanced Mixer: Noise determine the output noise voltages, bearing in mind that the load resistors differ by a factor of two Recall that the voltage conversion gain of the double-balanced mixer is half of that of the single-balanced topology. Thus, the input-referred noise voltages of the two circuits are related by Active Double-Balanced Mixer: LO Noise The LO noise voltage is converted to current by each switching pair and summed with opposite polarities. The double-balanced topology is much more immune to LO noise obtained at the cost of the higher noise, lower gain and higher power dissipation. Active Single-Balanced Mixer: Example A 6-GHz active mixer in 60-nm technology has a bias current of 2 mA from a 1.6 V supply. Assuming an overdrive of 300 mV for the input transistor, and of 150 mV for the switching transistor (for complete switching), RD=500Ω Ω, kn=500 µA/V2, γ=1, determine the voltage conversion gain, the SSB and DSB noise figures. Active Single-Balanced Mixer: Example The current and overdrives given above lead to W1 = 2.64 µm and W2,3 = 10.68 µm. We can now estimate the voltage conversion gain and the noise figure of the mixer. To compute the noise figure due to thermal noise, we first estimate the input-referred noise voltage as We now write the single-sideband NF with respect to RS = 50 Ω as: The double-sideband NF is 3 dB less. Improved Mixer Topologies: Active Mixers with Current-Source Helpers The principal difficulty in the design of active mixers stems from the conflicting requirements between the input transistor current (high enough to meet linearity and noise requirements) and the load resistor current (low enough, to allow large RD and gain). adding current sources (“helpers”) in parallel with the load resistors alleviates this conflict. V0: maximum allowable voltage drop across RD RD can be as large as V0/[(1-α α)I0] allowing higher gain Assuming M4,5 at the edge of saturation (VDS=VGS-VTH), the noise due to each currentsource helper and its corresponding load resistor is Active Mixers with Current-Source Helpers The voltage conversion gain is proportional to RD This noise rises with α, beginning from 4kTl0/V0 for α = 0 and reaching (4kTl0/V0)(2γγ) for α=1. Modeled by a gate-referred voltage (Vn,1/f), the flicker noise of each device is multiplied by g2m4,5R2D as it appears at the output. Normalizing by R2D: Since the voltage headroom V0 is typically limited to a few hundred millivolts, the helper transistors tend to contribute substantial 1/f noise to the output, a serious issue in direct-conversion receivers. The addition of the helpers also degrades the linearity because they work at the edge of saturation to minimize gm and the noise. Active Mixers with Enhanced Transconductance We can insert the current-source helper in the RF path rather than in the IF path. The above approach nonetheless faces two issues. First, transistor M4 contributes additional capacitance to node P reducing the gain. Second, the noise current of M4 directly adds to the RF signal. Use of Inductive Resonance at Tail with Helper Current Source In order to suppress CP, an inductor can be placed in series with the drain of M4, allowing the inductor to resonate with CP. C1 acts as a short at RF, shunting to ground the noise current of M4 The choice of the inductor is governed by the following conditions: CP.tot includes the capacitance of L1 in order to negligibly shunt RF current in order to not add noise Active Mixer Using Capacitive Coupling with Resonance Shown below is a topology wherein capacitive coupling permits independent bias currents for the input transistor and the switching pair. Here C1 acts as a short circuit at RF and L1 resonates with the parasitics at nodes P and N. Furthermore, the voltage headroom available to M1 is no longer constrained by (VGS - VTH)2,3 and the drop across the load resistors. Effect of Low-Frequency Beat in a Mixer Using Capacitive Coupling and Resonance The high-pass filter consisting of L1, C1, and the resistance seen at node P suppresses low-frequency beats generated by the even-order distortion in M1. At low frequencies, this result can be approximated as revealing a high attenuation.