mixers: an overviewrfic.eecs.berkeley.edu/ee242/pdf/module_5_1_mixeroverview.pdf · 1 2 1 2 2nd...

Berkeley

Mixers: An Overview

Prof. Ali M. Niknejad

U.C. BerkeleyCopyright c© 2014 by Ali M. Niknejad

Niknejad Advanced IC’s for Comm

Mixers

cf

Information PSD Mixer

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 RFcarrier (shown above).

In a receiver, the same information is ideally down-convertedto baseband.


Mixers Specifications

Conversion Gain: Ratio of voltage (power) at outputfrequency to input voltage (power) at input frequency

Downconversion: RF power in, IF power outUp-conversion: IF power in, RF power out

Noise Figure

DSB versus SSB

Linearity

Image Rejection, Spurious Rejection

LO Feedthrough

InputOutput

RF Feedthrough


Mixer Implementation

f(x)

Non-linear Two tones

21

21

+ 2nd order IM 1, 2

We know that any non-linear circuit acts like a mixer


Squarer Example

x yx2

Product component:

y = A2cos2ω1t + B2cos2ω2t + 2AB cosω1t cosω2t

What we would prefer:

vRF = vRF cosω1t

vLO = vRF cosω1t

vIF = 2vLO · vRF cos (ω1 ± ω2) t

LO

RF

IF

A true quadrant multiplier with good dynamic range isdifficult to fabricate


LTV Mixer

LTI 21 ff + 21 ff + No new frequencies

LTV 21 ff + New tones in output

No new frequencies for a Linear Time Invariant (LTI) system

But a Linear Time-Varying (LTV) “Mixer” can act like amultiplier

Example: Suppose the resistance of an element is modulatedperiodically

vLO

R(t) iin(t)(RF )

vo(IF ) vo = iin · R (t)

= Io cos (ωRF t) · Ro cos (ωLOt)

=IoRo

2{cos (ωRF + ωLO) t+cos (ωRF − ωLO) t}


Periodically Time Varying Systems

In general, any periodically time varying system can achievefrequency translation

v (t) = p (t) vi (t)p (t + T ) = p (t) =

∞∑n=−∞

cnejωontvi (t)

cn =1

T

T∫0

p (t) e−jωontdt

vi (t) = A (t) cosω1t = A (t)

(e jω1t + e−jω1t

2

)

vo (t) = A (t)∞∑−∞

cne j(ωont+ω1t) + e+j(ωont−ω1t)

2

Consider n=1 plus n=-1


Desired Mixing Product

c1 = c−1

vo (t) =c12e j(ωot−ω1t) +

c−12

e−j(ωot+ω1t)

= c1 cos (ωot − ω1t)

Output contains desired signal (plus a lot of other signals).Must pre-filter the undesired signals (such as the image band).


Convolution in Frequency

Ideal multiplier mixer:

p(t)

x(t) y(t)

y (t) = p (t) x (t)

Y (f ) = X (f ) ∗ P (f )

P (f ) =∞∑−∞

cnδ (f − nfLO)

Y (f ) =

∞∫−∞

∞∑−∞

cnδ (σ − nfLO)X (f − σ) dσ

=∞∑−∞

cn

∞∫−∞

δ (σ − nfLO)X (f − σ) dσ

=∞∑−∞

cnX (f − nfLO)


Up-Conversion: n > 0

X(f)

fLO−fLO

f

fLO

n = 1

n = 2n = 3

f

Y (f)

n = −1

n = −2

2fLO 3fLO−fLO−2fLO

Input spectrum is translated into multiple points at the outputat multiples of the LO frequency.

Filtering is required to reject the undesired harmonics.


Down-Conversion: n < 0X(f)

fLO−fLO

f

fLO

n = 1n = 2

n = 3

f

Y (f)

n = −1

n = −2

2fLO 3fLO−fLO−2fLO

n = −3

Output spectrum is translated from multiple sidebands (bothsides – image issue) into the same output.

This is the origin of the lack of image and harmonic rejectionin a basic mixer. We have already seen image rejectarchitectures. Later we’ll study harmonic rejection mixers.


Balanced Mixer Topologies


Balanced Mixer

An unbalanced mixer has a transfer function:

y(t) = x(t)× s(t) = (1 + A(t) cos(ωRF t))×{

0 LO < 01 LO > 0

which contains both RF, LO, and IF

For a single balanced mixer, the LO signal is “balanced”(bipolar) so we have

y(t) = x(t)× s(t) = (1 + A(t) cos(ωRF t))×{−1 LO < 0+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 sothere is no LO or RF leakage

y(t) = x(t)× s(t) = A(t) cos(ωRF t)×{−1 LO < 0+1 LO > 0


Current Commutating Mixers

I1 I2

M1 M2

RF

LO

M3 sB iI +

Io1 = I1 − I2 = F (VLO (t) , IB + is)

Assume is is small relative to IB and performTaylor series expansion.

Io1 ≈ F (VLO (t) , IB) +∂F

∂IB(VLO (t) , IB) · is + ...

Io1 = Po (t) + P1 (t) · is

+1

All current through M1

M2 Both on

( )tVLO

( )tP1

vx

-vx


Current Commutating

i1 i2

M1 M2

is

i1is

=

1gm2

1gm1

+ 1gm2

i2is

=

1gm1

1gm1

+ 1gm2

p1 (t) =gm1 (t)− gm2 (t)

gm1 (t) + gm2 (t)

(=

i1 − i2is

)Note that with good device matching: p1 (t) = −p1

(t + To

2

)Expand p1(t) into a Fourier series:

p1,2k =1

TLO

TLO∫0

p1 (t) e−j2π2kt/TLOdt =

TLO/2∫0

+

TLO∫TLO/2

= 0

Only odd coefficients of p1,n non-zero


Single Balanced Mixer

+LO −LO

+RF

IF+ −

Switching Pair

Transconductance stage (gain)

RF Current

Assume LO signal strong so that current (RF) is alternativelysent to either M2 or M3. This is equivalent to multiplying iRFby ±1.

vIF ≈ sign (VLO) gmRLvRF = g (t) gmRLvRF

g(t) periodic waveform with period = TLO


Current Commutating Mixer (2)

g(t) = square wave =4

π(cosωLOt − cos 3ωLOt + ...)

Let vRF = A cosωRF t

Gain:

Av =vIFA

=1

2

4

πgmRL =

2

πgmRL

LO-RF isolation good, but LO signal appears in output (just adiff pair amp).

Strong LO might desensitize (limit) IF stage (even afterfiltering).


Double Balanced Mixer

IEE

+LO

−LO

Q1 Q2 Q3 Q4

Q5 Q6

+LO

+RF −RF

RE RE Transconductance

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 cancellationoccurs


Mixer Design Examples


Improved Linearity

Bias

RF

LO+ LO-

IF+ IF-

Role of input stage is to provide V -to-I conversion.Degeneration helps to improve linearity. Inductivedegeneration is preferred as there is no loss in headroom andit provides input matching.

An LO trap also minimizes swing of LO at output which maycause premature compression.


Common Gate Input Stage

Iout

Vin

Vbias

Iout

Vin

Vbias

Iout

Vin

Vbias

Iout

Vin

Vbias

Vmirr

Iout

Vin

Vbias


Gilbert Micromixer

LODRIVER

IF LOAD

RF INPUT

The LNA output is often single-ended. A good balanced RFsignal 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 andbandwidth is still limited.A broadband single-ended to differential conversion stage canbe used to generate highly balanced signals over very widebandwidths. Gm stage is Class AB.


Gilbert Micromixer Gm StageI1 I3

IZ

IRF

+VRF

−

Q1

Q2

Q3QZ1

QZ2

Set IZ to bias for match: RIN = 2Vt/IZ .

Using “trans-linear concept” to derive currents by definingλ = IRF/2IZ

I 2Z = I1I2 = I1(I1 + IRF )

I1,2 = IZ

(√λ2 + 1∓ λ

)I1 − I3 = 2λIZ = IRF

RIN(λ) =Vt

2IZ

1√λ2 + 1


Active and Passive Balun

Io(+) Io(−)

Vin

Vbias

Vbias

Vin

Vout(+)

Vout(−)

LC

Z1

Z2/2

L C Z2/2 Bias

RF

LO

IF

For broadband applications, an active balun is a good solutionbut impacts linearity. A fully passive balun cap be designedwith good bandwidth and balance, resulting in very goodoverall linearity. Watch out for common mode coupling inbalun.


Bleeding the Switching Core

Bias

RF

LO+ LO-

IF+ IF-

Large currents are good for the gm stage (noise, conversiongain), but require large devices in the switching core ⇒ hardto switch due to capacitance and also requires a large LO(large Vgs − Vt)

A current source can be used to feed the Gm stage with extracurrent.


Current Re-Use Gm Stage

VRF

VLO(+)

VLO(−)

Vbias,n

Vbias,p

Io1 Io2

Io = Io1 − Io2

Note that thePMOS currentsare out of phasewith the NMOS,and hence theinverted polarityfor the LO is usedto compensate.

CMOS technology has fast PMOS devices which can be usedto increase the effective Gm of the transconductance stagewithout increasing the current by stacking.


Single, Dual, and Back Gate

VRF

VLO

Io

VRF

VLO

Io

Vbias

VRF VLO

Io

Vbias

Io

VRF

VLO

Note that for weak signals, the second-order non-linearity willmix the LO and RF signals.

We prefer to apply a strong LO to periodically modulate thetransconductance Gm(t) of the transistor.


Ring or Gilbert Quad?+LO

−LO

+LO

M1

M4

M3

M2

+LO

M1

−LO

M2

+LO

M4

−LO

M3

+RF

−RF

+RF −RF+IF

−IF

+IF

−IF

Unfold the quad, we get a ring. They are the same!

Notice that a mixer is just a circuit that flips the sign of thetransfer function from ± every cycle of the LO. We can eithersteer currents or voltages.

The switches can be actively biased and switched with an LOof approximately a few Vgs − VT (or several times kT/q forBJT), or they can be biased passively and fully switched onand off.


Passive Mixers (V )

LO

LO

+RF

−RF

IF

RIF

Devices act as switches andjust rewire the circuit sothat the plus/minus voltageis connected to the outputwith ±1 polarity.

Very linear but requires ahigh LO swing and largerLO power.

+RF

− RF

LO

LO

LO

LO

IF

L1

L2

C3 L3

Rs/2

Rs/2


Passive Mixers (I )

+LO

−LO

+IF

−IF+RF −RF

+LO

Commutate currents using passive switches. Very linear(assuming current does not have distortion) but requireshigher LO drive.

Good practice to DC bias switches at desired operating pointand AC coupling the Gm stage.

Since we’re processing a current, we use a TIA to present alow impedance and convert I -to-V


Sub-Sampling Mixers

ΦRF

VRF

VIF

X(f)

fRF

f

X(f)

fIF

f

Note that sampling isequivalent to multiplicationby an impulse train (spacingof TLO), which in thefrequency domain is theconvolution of anotherimpulse train (spacing of1/TLO).

This means that we cansample or even sub-samplethe signal. The problem isthat noise from allharmonics of the LO folds tocommon IF. Always true,but especially problematicfor sub-sampling.


Rudell CMOS Mixer

VLO(+)

VLO(−)

Vbias Vbias

Vin(+) Vin (−)

VCM,out

ICMIgain

M16

Gain programmed using current through M16 (set byresistance of triode region devices)

Common mode feedback to set output point

Cascode improves isolation (LO to RF)


Power Spectral Density


Review of Linear Systems and PSD

Average response of LTI system:

y1 (t) = H1[x (t)] =

∞∫−∞

h1 (t) x (t − τ) dτ

y1 (t) = limT→∞

1

2T

T∫−T

y1 (t) dt

= limT→∞

1

2T

T∫−T

∞∫−∞

h1 (τ) x (t − τ) dτ

dt

=

∞∫−∞

limT→∞1

2T

T∫−T

x (t − τ) dt

︸︷︷︸

x(t)

h1 (τ) dt


Average Value Property

y1 (t) = x (t)

∞∫−∞

h1 (t) dt

H1 (jω) =

∞∫−∞

h1 (t) e−jωtdt

y1 (t) = x (t)H1 (0)


Output RMS Statistics

y12 (t) = limT→∞

1

2T

T∫−T

∞∫−∞

h1 (τ1) x (t − τ1) dτ1

∞∫−∞

h1 (τ2) x (t − τ2) dτ2

dt

=

∞∫−∞

∞∫−∞

h1 (τ1) h1 (τ2)

limT→∞1

2π

T∫−T

x (t − τ1) x (t − τ2) dt

dτ1dτ2

Recall the definition for the autocorrelation function

ϕxx (t) = x (t) x (t + τ)

= limT→∞

1

2T

T∫−T

x (t) x (t + τ) dt


Autocorrelation Function

y12 (t) =

∞∫−∞

∞∫−∞

h1 (τ1) h2 (τ2)ϕxx (τ1 − τ2) dτ1dτ2

ϕxx (jω) =

∞∫−∞

ϕxx (τ) e−jωτdτ

ϕxx (τ) =1

2π

∞∫−∞

ϕxx (jω) e jωτdω

ϕxx(jω) is a real and even function of ω since ϕxx(t) is a realand even function of t


Autocorrelation Function (2)

y12 (t) =

∞∫−∞

∞∫−∞

h1 (τ1) h1 (τ2)1

2π

∞∫−∞

ϕxx (jω) ejω(τ1−τ2)dωdτ1dτ2

=1

2π

∞∫−∞

ϕxx (jω)

∞∫−∞

∞∫−∞

h1 (τ1) h1 (τ2) ejω(τ1−τ2)dτ1dτ2

=1

2π

∞∫−∞

ϕxx (jω)

∞∫−∞

h1 (τ1) e+jωτ1dτ1

∞∫−∞

h1 (τ2) e−jωτ2dτ2

dω

y12 (t) =1

2π

∞∫−∞

ϕxx (jω)H1 (jω)H1∗ (jω) dω

=1

2π

∞∫−∞

ϕxx (jω) |H1 (jω)|2dω


Average Power in X(t)

Consider x(t) as a voltage waveform with total average powerx2(t). Lets measure the power in x(t) in the band 0 < f < f1

Ideal LPF

1

1+

-

( )tx+

-

( )ty

The average power in the frequency range 0 < f < f1 is now

y12 (t) =1

2π

∞∫−∞

ϕxx (jω)|H1 (jω)|2dω

=1

2π

ω1∫−ω1

ϕxx (jω) dω =

f1∫−f1

ϕxx (j2πf ) df


Average Power in X(t) (2)

= 2

f1∫0

ϕxx (j2πf ) df

Generalize: To measure the power in any frequency range,apply an ideal bandpass filter with passband f1 < f < f2

y12 (t) = 2

f2∫f1

ϕxx (j2πf ) df

The interpretation of ϕxx as the the power spectral density(PSD) is clear.


Spectrum Analyzer

A spectrum analyzer measures the PSD of a signal

“Poor Man’s” spectrum analyzer:

VCO Sweep

generation 1f 2f

vertical

horiz. Linear wide tuning range

Wide dynamic range mixer

Sharp filter

CRT


mixers: an overviewrfic.eecs.berkeley.edu/ee242/pdf/module_5_1_mixeroverview.pdf · 1 2 1 2 2nd...

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