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ATLCE - B5 07/03/2016
© 2016 DDC 1
07/03/2016 - 1 ATLCE - B5 - © 2016 DDC
Politecnico di Torino - ICT School
Analog and Telecommunication Electronics
B5 - Multipliers/mixer circuits» Error taxonomy» Basic multiplier circuits» Gilbert cell» Bridge MOS and diode circuits» Balanced mixers
AY 2015-16
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Lesson B5: multipliers and mixers
• Analog multipliers– Parameters and errors– Transconductance multipliers, 1/2/4 quadrant – Gilbert cell – Diode bridge
• Mixer parameters – Balanced and I/Q mixers– Noise, gain
• References:– Elettronica per Telecom.: 2.2.4 Moltiplicatori Analogici– Design with Op Amp …: 13.2 Analog multipliers
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Frequency translation basics (beats)
• Werner’s relations– sinA x sinB = 0,5 [cos(A-B) - cos(A+B)]– sinA x cosB = 0,5 [sin(A-B) + sin(A+B)]– cosA x cosB = 0,5 [cos(A-B) + cos(A+B)]
• Telecom applications– Frequency translation: Heterodyne RX and TX
» I, II, … conversion, Image rejection mixers
– Mo-Demodulation» Standard AM mo-demod» Suppressed carrier (DSB), Single Side Band (SSB)» Digital AM (ASK. PAM)» Phase Detector» ….
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Mixers and Multipliers
• Mixers: – Frequency translation
» Frequency conversion in heterodyne receivers and transmitters– Phase detectors
» PLL and demodulators
• Multipliers:– Mo-demodulators
» AM and PAM modulation and demodulation– Variable gain amplifiers
– (analog computation) digital
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Mixers in the handset
I/Q Mixers(secondconversion)
RF Mixers(first conversion)
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Mixers in a GPS receiver
Image reject mixer Mixer II
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Many uses of multipliers: GP2015
Mixers Variable gain amplifier
Phase detectors
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Multiplier as mixer
• Mixing is achieved by multiplication
– senA x senB = 0,5 [cos(A-B) – cos(A+B)]
• With sine inputs, the output of a multiplier includes– Difference component– Sum component
– Other terms, caused by nonlinearity and errors
• Only one term is used (sum or difference beat)– Filters– Cancellation– …
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Ideal multipliers
• Ideal multiplier: Vo = Km Vx Vy
– Sine input signals, frequency Fx and Fy– Vo spectrum includes only Fx - Fy and Fx + Fy
ffx fy
ffy-fx fy+fx
fxfx
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Errors in multipliers: real circuits
• Ideal multiplier: Vo = Km Vx Vy– Sine input signals, frequency Fx and Fy– Vo spectrum includes only Fx - Fy and Fx + Fy
• Actual multiplier– Vo = Km (Vx + ΔVx) (Vy + ΔVy) + ΔVo– Vo = Km Vx Vy + ΔVy Vx + ΔVx Vy + Eo + other terms order >2– Vo = Km Vx Vy + Ex Vx + Ey Vy + Eo + …..
– Sine input signals, frequency Fx and Fy– Vo spectrum includes Fx-Fy, Fx+Fy, Fx, Fy, DC– + other higher order terms:
2Fx, 2Fx-Fy, 3Fx-2Fy, ...
Distortion
Feedthrough
Harmonics: M Fx ± N Fy
Offset
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Spectrum with real mixer
• Vo = Km Vx Vy + Ex Vx + Ey Vy + Eo + ….
f
fx fy
f
fx fy-fx fy fy+fxfo fa fb fd
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f
fx fy-fx fy fy+fxfo
Spurious outputs: feedthrough
• Input signals reach the output– Mainly due to mixer unbalance or signal DC:
Vo = Km Vx (Vy + ΔVy) = Km Vx Vy + Km Vx ΔVy
DC error on Vy (offset) ΔVy causesVx feedthrough
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f
fa fy-fx fc fy+fxfo fb fd
Nonlinearity
• Nonlinearity causes higher order terms
– 2fx, 2fy, 2fx+fy, 2fx-fy, 2fx+2fy, 3fx, 3fy, 3fx+fy, …..
– Can be removed with tuned circuits
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Lesson B5: multipliers and mixers
• Analog multipliers– Parameters and errors– Transconductance multipliers, – 1/2/4 quadrant – Balanced mixer, Gilbert cell– Diode bridge
• Mixer parameters– Noise, – gain, – intermodulation, IP
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Example of mixer with nonlinear circuit
• BJT amplifier with input Vx on B and Vy on E– Effective input voltage: VBE = Vx – Vy (or VBE = Vx + Vy)
– Nonlinearity makes Vx · Vy appear in Ic
• Tuned circuit isolates desired component
• Constraints on Vx and Vy:– VBE > 0, that is Vx > Vy– Need DC component high feedthrough errors
• Merged with input amplifier and local oscillator in (old) low-cost receivers
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Mixers with nonlinear networks
• Vi = Vx + Vy (frequency Fx and Fy)
– Vo = Fnonlin(Vi)
– With power series expansion …» Vo = A0 + A1(Vx + Vy) + A2(Vx+Vy)2 + …..
– Vo components» Vx, Vy frequency Fx e Fy» Vx • Vy frequency Fx - Fy and Fx + Fy» Vx2, Vy2 frequency 2Fx, 2Fy» Other terms frequency M Fx + N Fy» Order III intermodulation (2Fx - Fy)
• Useful component (diff or sum beat) isolated by a tuned circuit
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Transconductance multiplier
• For small-signal amplifier: Vo = - Vx gm Rc– Gain proportional to transconductance gm
– gm depends from Ic (Id)– Id is controlled by Vy: gm = K Vy
– Vo = K Rc Vx Vy
• Single transistor: Vx, Vy > 0: 1 quadrant– DC components high feedthrough
• Differential circuits: 2/4-quadrants– No DC, less feedthrough
• Limited to low-level signals
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Transconductance circuit - 1 quadrant
• VO = VX gm1 RC
• gm1 = IC/VT
• IC ≈ gm2 VY
• VO ≈ VX (gm2 VY /VT) gm1 RC
• VO ≈ K VX VY
(polarity !)
RC
IC
gm1
gm2
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Transconductance circuit – tuned load
• ZC(ω) to isolate desired component(sum or diff beat)
VO = K’ ZC(ω) VX VY
• VX and VY > 01 quadrant
– Feedtroughon X and Y!
• Can be extended to 2/4-quadrant
IC
ZC
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Transconductance circuit - 2 quadrant
• 2-quadrant: differential VX
Balanced mixer
• No feedthroughfrom VY
– If VX = 0,VO = 0 for any VY(VY seen as common mode)
• DC required on VY– Feedthrough from VX
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• Differential VX and VY: double balanced mixer• No feedthrough
on VX and VY
• Exploit gm
• Can use MOS or BJT
Transconductance circuit - 4 quadrant
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VDD
VX
I1 I2
VZ
MOS Gilbert cell
• Output voltage VZ depends on Drain current unbalance; – Drain currents depend on input voltage VX and
Source current I1 , I2– Source current unbalance must depend on input voltage VY
DG
S
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Multipliers with Gilbert cell
• V I Conversion– The differential V(I) is linear only for low V– Limited dynamic range for both inputs
» To limit spurious outputs, only small signals
• Corrective actions
– Linearize by negative feedback» Re pair in the differential amplifier
– Wide range V I converter
– Compensation of exponential nonlinearity» I = exp(log Vi) I = K Vi
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Linearized differential stage
• Emitter feedback
• Lower gain
• Wider inputdynamic range
• I1 - I2 = ΔI ≈ VX/(2RE)
• Needs matched RE(hard for ICs!)
RE RE
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Wide dynamic VI converter
• Differential VI converter
I1 - I2 = ΔI ≈ VX/RX
• Needs matched current sources(OK for ICs!)
• Used also for instrumentation amplifiers
• Errors from ΔVBE(IE)
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V I dynamic range limits
• Actual voltage drop on R is not Vx– VBE change with current unbalance
• Io matching
• Beta matching
• ….VBE1 VBE2
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Wide range multiplier: block diagram
I Vlog
V Iwide dynamic
V Iwide dynamic
Gilbert cell
Vx Vy
VzΔV = K’ log Vx
ΔIx = K Vx ΔIy = K Vy
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Complete wide range circuit
I1 I2 I3 I4
IA IB
7 85 6
Vo
V’X
YXYXA
CO
AX
X
X
XCO
VVRRI
R2V
I21
RV2
RV2RV
X
X12 R
V2II
Y
Y43 R
V2II
1
2
7
8V'V
VVV
6
5
II
IIe
eII
T
X
T
6BE5BE
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Diode Mixer: single-balanced
• Single diode– Single diode used as switch from Vx to GND– Input Vi = Vx + Vy– Small Vx, sign defined by Vy
» Diode acts as switch controlled by Vy» Output Vx/0» Multiplication by 0/1
• Diode half-bridge– Couple of diodes as switch from Vx to GND
» Diodes act as switches controlled by Vy» Output Vx/0» Multiplication by 0/1
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Diode Mixer: double-balanced
• Couple of diodes – Switches Vout between two opposite polarity Vx– Output +Vx/-Vx
• Diode bridge 1– inverts Vx towards the output
• Diode bridge 2– Sine on Vx , low level signal– Squarewave on Vy , large signal– Vx + Vy applied to a diagonal– Each diode is a switch controlled by Vy– Vx direct/inverter on the other diagonal
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VX
Vz
VY
Switch bridge Mixer
• Switch bridge (switches VX/-VX at output)– Command: VY
– Same as multiply by ±1– Strong nonlinearity
• Diode or MOS switches
• Double-balancedmixer
• Used for high frequencies
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VX
Vz = Vx
VY = H
VX
VY = L
Vz = -Vx
Switch Mixer
• VX analog
• VY digital
• Switches on linked side receive complementary commands
• The sign of the transfer function is controlled by VY
– VY = H VZ = + VX
– VY = L VZ = - VX
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Diode bridge circuits
• Vx comes from a transformer with center tap
• VY >> VX(VX has no effect on diode bias)
VX
VY
VZ
VX
VY
VZ
+
+
VX
VY
VZ
+
+
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MOS/BJT bridge circuits
• Same structure as diode bridge– Input X on diagonal H– Input Y as command (on B or G) – Output from diagonal V
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Lesson B5: multipliers and mixers
• Analog multipliers– Parameters and errors– Transconductance multipliers, – 1/2/4 quadrant – Balanced mixer, Gilbert cell– Diode bridges
• Mixer parameters– Noise, – gain, – intermodulation, IP
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Mixers and amplifiers
• Mixer seen as amplifier with variable gain (VGA)
• Constant input = fixed gain for other input – Constant Vy amplifier for Vx– Constant Vx amplifier for Vy
• Same requirements as amplifiers– No harmonics, no distortion– Low noise, wide dynamic
• Parameters as amplifier + additional– 1 dB compression, IP2, IP3– Insulation, reflection, …– Mixer-specific parameters
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Mixer parameters
• Conversion gain– IFrms/RFrms
• Isolation– Leakage in unwanted paths
• Noise figure
• Nonlinearity – Input dynamic range– Intermodulation– Compression level– Intercept Point (order 2, 3, …)
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Ideal multiplier linear mixer
• Only sum and difference spectral lines at output
VX (fX) , VY (fY)
VZ (fX - fY , fX + fY)VX
VYVZ
fX - fY , fX + fY
fX
fY
X
ffx-fy fy fx fy+fx
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Ideal Mixer output spectrum
• Sine VY (fY), Wideband VX (FA FB)– Output includes sum and difference beats– VX spectrum translated around 0 and 2 fY
f0 fy
fFb-fy fy fy+Fb2fy
Fa FbDifference beat Sum beat
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Mixer and nonlinearity
• Input nonlinearity generates harmonics
• Inputs to ideal mixer with order 2, 3 terms
• With multicomponentinput signals
Vx = Vxa(Fa) + Vxb(Fb),
possible intermodulation– Same problems as amplifiers
VZX
VX
VY
fY, 2fY , 3fY, ...fY
fX ± fY, 2fY ± fX , 3fY ± fX, ...
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Effects of mixer nonlinearity
• Input nonlinearity– Products among Vx, Vy signals
and their harmonics – Fx±Fy, 2Fx±Fy, 2Fy±Fx,
2Fy±2Fx …
• Output nonlinearity– Products among Vx, Vy signals– Harmonics of the product – Fx±Fy, 2(Fx±Fy), 3(Fy±Fx), …
• Inband terms more dangerous (intermodulation)
VZX
VX
VY
VZX
VX
VY
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Actual mixer real multiplier
nonlinearity harmonics
Harmonicsbeat and intermodulation
VZX
VX
VY
fX, 2fX , 3fX, ...fX
fX - fY , 2fX - fY , fX - 2fY, 3fX, ...
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Spectrum with nonlinearities
• Nonlinearity on Vy: components fY , 2fY, 3fY, ...– Multiple spectral translations: Vx to fY , 2fY, 3fY, …
f0 fy
fy-Fb fy
Fb 2fy
2fy-Fb 2fy+Fb
3fy
3fy-Fb fy+Fb
4fy-Fb
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Mixer vs. amplifiers
• Input signal: – Two sine signal, frequency f1 and f2
• Amplifier output:– Same frequency
• Mixer output:– Difference/
sum frequency
• From both:– Harmonics:
2f1, 2f2, 3f1, ...– III ord. beats
(intermodulation): 2f1-f2, 2f2-f1, ...
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Mixer vs. amplifiers
Good for an amplifier
Good for a mixer
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Lesson B5: final test
• Which are the techniques usable to build units with predefined nonlinearity?
• Which is the difference among 1/2/4 quadrant multipliers?
• Define feedthrough in a multiplier
• How can the Vx feedthrough error be compensated?
• Which is the main limit of transconductance multipliers?
• Draw the output spectrum for linear and nonlinear mixers with input signals: Vx: 2,3 + 2,5 MHz (2 components), Vy: 10 MHz
• An analog multiplier mixer receives on Vx a 100-120 MHz signal, and a pulse sequence (δ) a 25 MHz rate on Vy. Draw the complete output spectrum from 0 to 400 MHz at the output Vz (assume a fully linear multiplier).
