Berkeley. Mixers: An Overview. Prof. Ali M. Niknejad. U.C. Berkeley Copyright c 2014 by Ali M. Niknejad

Transcription

1 Berkeley Mixers: An Overview Prof. Ali M. U.C. Berkeley Copyright c 2014 by Ali M.

2 Mixers Information PSD Mixer f c The Mixer is a critical component in communication circuits. It translates information content to a new frequency. On the transmitter, baseband data is up-converted to an RF carrier (shown above). In a receiver, the same information is ideally down-converted to baseband.

3 Mixers Specifications Conversion Gain: Ratio of voltage (power) at output frequency to input voltage (power) at input frequency Downconversion: RF power in, IF power out Up-conversion: IF power in, RF power out Noise Figure DSB versus SSB Linearity Image Rejection, Spurious Rejection LO Feedthrough Input Output RF Feedthrough

4 Mixer Implementation Two tones 1, 2 f(x) Non-linear nd order IM We know that any non-linear circuit acts like a mixer

5 Squarer Example x x 2 y Product component: y = A 2 cos 2 ω 1 t + B 2 cos 2 ω 2 t + 2AB cos ω 1 t cos ω 2 t What we would prefer: v RF = v RF cos ω 1 t v LO = v RF cos ω 1 t v IF = 2v LO v RF cos (ω 1 ± ω 2 ) t LO RF A true quadrant multiplier with good dynamic range is difficult to fabricate IF

6 LTV Mixer f LTI 1 + f 2 f 1 + f2 f 1 + f 2 LTV No new frequencies New tones in output No new frequencies for a Linear Time Invariant (LTI) system But a Linear Time-Varying (LTV) Mixer can act like a multiplier Example: Suppose the resistance of an element is modulated periodically v o (IF ) v LO R(t) i in(t)(rf ) v o = i in R (t) = I o cos (ω RF t) R o cos (ω LO t) = I or o 2 {cos (ω RF + ω LO ) t+cos (ω RF ω LO ) t}

7 Periodically Time Varying Systems In general, any periodically time varying system can achieve frequency translation p (t + T ) = p (t) = c n e jωont v i (t) v (t) = p (t) v i (t) c n = 1 T T 0 p (t) e jωont dt n= ( e jω 1 t + e jω ) 1t v i (t) = A (t) cos ω 1 t = A (t) 2 v o (t) = A (t) Consider n=1 plus n=-1 c n e j(ωont+ω 1t) + e +j(ωont ω 1t) 2

8 Desired Mixing Product c 1 = c 1 v o (t) = c 1 2 ej(ωot ω 1t) + c 1 2 e j(ωot+ω 1t) = c 1 cos (ω o t ω 1 t) Output contains desired signal (plus a lot of other signals). Must pre-filter the undesired signals (such as the image band).

9 Convolution in Frequency Ideal multiplier mixer: p(t) y(t) x(t) Y (f ) = = c n y (t) = p (t) x (t) Y (f ) = X (f ) P (f ) P (f ) = c n δ (f nf LO ) c n δ (σ nf LO )X (f σ) dσ δ (σ nf LO ) X (f σ) dσ = c n X (f nf LO )

10 Up-Conversion: n > 0 X(f) f LO f LO f Y (f) n = 2 n = 1 n = 1 n = 2 n = 3 2fLO f LO f LO 2f LO 3f LO f Input spectrum is translated into multiple points at the output at multiples of the LO frequency. Filtering is required to reject the undesired harmonics.

11 Down-Conversion: n < 0 X(f) f LO f LO f Y (f) n = 3 n = 2 n = 1 n = 3 n = 2 n = 1 2fLO f LO f LO 2f LO 3f LO f Output spectrum is translated from multiple sidebands (both sides image issue) into the same output. This is the origin of the lack of image and harmonic rejection in a basic mixer. We have already seen image reject architectures. Later we ll study harmonic rejection mixers.

12 Balanced Mixer Topologies

13 Balanced Mixer An unbalanced mixer has a transfer function: y(t) = x(t) s(t) = (1 + A(t) cos(ω RF t)) which contains both RF, LO, and IF { 0 LO < 0 1 LO > 0 For a single balanced mixer, the LO signal is balanced (bipolar) so we have { 1 LO < 0 y(t) = x(t) s(t) = (1 + A(t) cos(ω RF t)) +1 LO > 0 As a result, the output contains the LO but no RF component For a double balanced mixer, the LO and RF are balanced so there is no LO or RF leakage { 1 LO < 0 y(t) = x(t) s(t) = A(t) cos(ω RF t) +1 LO > 0

14 Current Commutating Mixers I o1 = I 1 I 2 = F (V LO (t), I B + i s ) LO I 1 I 2 M1 M2 Assume i s is small relative to I B and perform Taylor series expansion. RF M3 I B + i s I o1 F (V LO (t), I B ) + F I B (V LO (t), I B ) i s +... I o1 = P o (t) + P 1 (t) i s v x -v x +1 V LO ( t) P 1 ( t) All current through M1 M2 Both on

15 Current Commutating M1 M2 i 1 i 2 i s i 1 i s = 1 g m2 i 2 1 g m1 + 1 g m2 i s = 1 g m1 1 g m1 + 1 g m2 p 1 (t) = g ( m1 (t) g m2 (t) = i ) 1 i 2 g m1 (t) + g m2 (t) i s ( ) Note that with good device matching: p 1 (t) = p 1 t + T o 2 Expand p 1 (t) into a Fourier series: p 1,2k = 1 T LO T LO 0 p 1 (t) e j2π2kt/t LO dt = Only odd coefficients of p 1,n non-zero T LO /2 0 + T LO T LO /2 = 0

16 Single Balanced Mixer +LO RF Current + IF LO Switching Pair +RF Transconductance stage (gain) Assume LO signal strong so that current (RF) is alternatively sent to either M2 or M3. This is equivalent to multiplying i RF by ±1. v IF sign (V LO ) g m R L v RF = g (t) g m R L v RF g(t) periodic waveform with period = T LO

17 Current Commutating Mixer (2) g(t) = square wave = 4 π (cos ω LOt cos 3ω LO t +...) Let v RF = A cos ω RF t Gain: A v = ṽif A = π g mr L = 2 π g mr L LO-RF isolation good, but LO signal appears in output (just a diff pair amp). Strong LO might desensitize (limit) IF stage (even after filtering).

18 Double Balanced Mixer +LO Q 1 Q 2 Q 3 Q 4 +LO LO +RF Q 5 Q 6 RF R E R E Transconductance I EE LO signal is rejected up to matching constraints Differential output removes even order non-linearities Linearity is improved: Half of signal is processed by each side Noise higher than single balanced mixer since no cancellation occurs

19 Mixer Design Examples

20 Improved Linearity IF+ LO- IF- LO+ RF Bias Role of input stage is to provide V -to-i conversion. Degeneration helps to improve linearity. Inductive degeneration is preferred as there is no loss in headroom and it provides input matching. An LO trap also minimizes swing of LO at output which may cause premature compression.

21 Common Gate Input Stage I out V bias V in I out I out I out I out V bias V bias V bias V bias V in V in V in V in V mirr

22 Gilbert Micromixer IF LOAD LO DRIVER RF INPUT The LNA output is often single-ended. A good balanced RF signal is required to minimize the feedthrough to the output. LC bridge circuits can be used, but the bandwidth is limited. A transformer is a good choice for this, but bulky and bandwidth is still limited. A broadband single-ended to differential conversion stage can be used to generate highly balanced signals over very wide bandwidths. G m stage is Class AB.

23 Gilbert Micromixer Gm Stage I 1 I 3 I Z Q1 I RF + V RF Q2 Q3 QZ1 QZ2 Set I Z to bias for match: R IN = 2V t /I Z. Using trans-linear concept to derive currents by defining λ = I RF /2I Z IZ 2 = I 1I 2 = I 1 (I 1 + I RF ) ( ) I 1,2 = I Z λ λ I 1 I 3 = 2λI Z = I RF R IN (λ) = V t 1 2I Z λ 2 + 1

24 Active and Passive Balun I o(+) I o( ) IF V bias2 V in V bias1 V out(+) LO C L Z 2/2 RF V in L C Z 2/2 Bias Z 1 V out( ) For broadband applications, an active balun is a good solution but impacts linearity. A fully passive balun cap be designed with good bandwidth and balance, resulting in very good overall linearity. Watch out for common mode coupling in balun.

25 Bleeding the Switching Core IF+ LO- IF- LO+ RF Bias Large currents are good for the gm stage (noise, conversion gain), but require large devices in the switching core hard to switch due to capacitance and also requires a large LO (large Vgs Vt) A current source can be used to feed the G m stage with extra current.

26 Current Re-Use Gm Stage V bias,p V RF I o1 I o2 V LO (+) V LO ( ) Note that the PMOS currents are out of phase with the NMOS, and hence the inverted polarity for the LO is used to compensate. I o = I o1 I o2 V bias,n CMOS technology has fast PMOS devices which can be used to increase the effective G m of the transconductance stage without increasing the current by stacking.

27 Single, Dual, and Back Gate I o I o V RF V RF V LO V LO V bias I o V LO V RF V LO V RF V bias I o Note that for weak signals, the second-order non-linearity will mix the LO and RF signals. We prefer to apply a strong LO to periodically modulate the transconductance G m (t) of the transistor.

28 Ring or Gilbert Quad? +LO +IF M1 +LO LO M2 M1 +RF RF LO +IF IF +RF M2 M3 M4 RF M3 M4 LO IF +LO +LO Unfold the quad, we get a ring. They are the same! Notice that a mixer is just a circuit that flips the sign of the transfer function from ± every cycle of the LO. We can either steer currents or voltages. The switches can be actively biased and switched with an LO of approximately a few V gs V T (or several times kt /q for BJT), or they can be biased passively and fully switched on and off.

29 Passive Mixers (V ) LO +RF IF RF LO +RF R s /2 R IF L 1 Devices act as switches and just rewire the circuit so that the plus/minus voltage is connected to the output with ±1 polarity. Very linear but requires a high LO swing and larger LO power. LO LO C 3 L 3 IF RF R s /2 LO LO L 2

30 Passive Mixers (I ) +LO +RF RF LO +IF IF Commutate currents using passive switches. Very linear (assuming current does not have distortion) but requires higher LO drive. Good practice to DC bias switches at desired operating point and AC coupling the G m stage. Since we re processing a current, we use a TIA to present a low impedance and convert I -to-v +LO

31 Sub-Sampling Mixers Φ RF V RF X(f) f RF X(f) f IF V IF f f Note that sampling is equivalent to multiplication by an impulse train (spacing of T LO ), which in the frequency domain is the convolution of another impulse train (spacing of 1/T LO ). This means that we can sample or even sub-sample the signal. The problem is that noise from all harmonics of the LO folds to common IF. Always true, but especially problematic for sub-sampling.

32 Rudell CMOS Mixer V LO(+) V CM,out V LO( ) M16 V bias V bias I gain I CM V in(+) V in ( ) Gain programmed using current through M16 (set by resistance of triode region devices) Common mode feedback to set output point Cascode improves isolation (LO to RF)

33 Power Spectral Density

34 Review of Linear Systems and PSD Average response of LTI system: = y 1 (t) = H 1 [x (t)] = h 1 (t) x (t τ) dτ = lim T y 1 (t) = lim 1 2T T T T 1 2T T T y 1 (t) dt h 1 (τ) x (t τ) dτ dt 1 T lim T x (t τ) dt h 1 (τ) dt 2T T }{{} x(t)

35 Average Value Property y 1 (t) = x (t) h 1 (t) dt H 1 (jω) = h 1 (t) e jωt dt y 1 (t) = x (t)h 1 (0)

36 Output RMS Statistics y 2 1 T 1 (t) = lim h 1 (τ 1 ) x (t τ 1 ) dτ 1 h 1 (τ 2 ) x (t τ 2 ) dτ 2 dt T 2T T 1 T = h 1 (τ 1 ) h 1 (τ 2 ) lim T x (t τ 1 ) x (t τ 2 ) dt dτ 1 dτ 2 2π T Recall the definition for the autocorrelation function ϕ xx (t) = x (t) x (t + τ) = lim T 1 2T T T x (t) x (t + τ) dt

37 Autocorrelation Function y 2 1 (t) = h 1 (τ 1 ) h 2 (τ 2 )ϕ xx (τ 1 τ 2 ) dτ 1 dτ 2 ϕ xx (jω) = ϕ xx (τ) e jωτ dτ ϕ xx (τ) = 1 2π ϕ xx (jω) e jωτ dω ϕ xx (jω) is a real and even function of ω since ϕ xx (t) is a real and even function of t

38 Autocorrelation Function (2) y 2 1 (t) = h 1 (τ 1 ) h 1 (τ 2 ) 1 ϕ xx (jω) e jω(τ 1 τ 2 ) dωdτ 1 dτ 2 2π = 1 ϕ xx (jω) h 1 (τ 1 ) h 1 (τ 2 ) e jω(τ 1 τ 2 ) dτ 1 dτ 2 2π = 1 ϕ xx (jω) h 1 (τ 1 ) e +jωτ 1 dτ 1 h 1 (τ 2 ) e jωτ 2 dτ 2 dω 2π y 2 1 (t) = 1 ϕ xx (jω) H 1 (jω) H 1 (jω) dω 2π = 1 ϕ xx (jω) H 1 (jω) 2 dω 2π

39 Average Power in X(t) Consider x(t) as a voltage waveform with total average power x 2 (t). Lets measure the power in x(t) in the band 0 < f < f 1 + x ( t) Ideal LPF 1 + y ( t) The average power in the frequency range 0 < f < f 1 is now y 2 1 (t) = 1 2π ϕ xx (jω) H 1 (jω) 2 dω ω 1 = 1 ϕ xx (jω) dω = 2π ω 1 f 1 f 1 ϕ xx (j2πf ) df

40 Average Power in X(t) (2) = 2 f 1 0 ϕ xx (j2πf ) df Generalize: To measure the power in any frequency range, apply an ideal bandpass filter with passband f 1 < f < f 2 y 1 2 (t) = 2 f 2 f 1 ϕ xx (j2πf ) df The interpretation of ϕ xx as the the power spectral density (PSD) is clear.

41 Spectrum Analyzer A spectrum analyzer measures the PSD of a signal Poor Man s spectrum analyzer: Wide dynamic range mixer Sharp filter vertical VCO Linear wide tuning range Sweep generation horiz. f1 f2 CRT