Download Course Outline Basic Concepts in RF Design Low

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.