Berkeley
Power Amplifiers for Communications
Prof. Ali M. Niknejad
U.C. Berkeley
c 2014 by Ali M. Niknejad
Copyright Niknejad
Advanced IC’s for Comm
PA System Level Specifications
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Advanced IC’s for Comm
Power Amplifier Specifications
Peak Output Power
Efficiency
Pdc
Power Gain
Amplifier Linearity
Pin
Stability over VSWR
Ability to transmit into an
unknown/varying load
Power Control
Zin = 50 Ω
Step size,
range
High efficiency at back-off
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Advanced IC’s for Comm
Zout != 50 Ω
PL
Heat
Peak Output Power
The peak output power determines the range for two-way
communications. When we hit sensitivity limits, the only way
to increase the range is to increase the Tx power.
The peak power is specified at the 1-dB compression point or
the maximum output power – the “clipping” point (makes a
big difference).
∼1W for cellular handsets ( 1 km distance)
∼100mW for W-LAN (100 m)
∼10mW for W-PAN (Bluetooth) (1-10 m)
∼1mW for body area networks.
In practice, the average power transmitted may be much lower
than the peak output power due to “back-off”, to obtain
linearity for the amplitude modulation (fast time scale) or for
power control (slow time scale)
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Peak Efficiency
Power Added Efficiency (PAE) is a popular metric.
Pout is the output power, Pin is the input power,
and Pdc is the DC power consumption of the PA
For high power gain systems (Gp ), the efficiency
approaches the drain
ηPAE =
drain efficiency (ηd ), or for a BJT, the “collector”
efficiency, or simply the efficiency of the last stage
ηPAE =
The efficiency of the PA is an important measure of
the battery life of the wireless transceiver. Since the
PA power dwarfs the power consumption in the
receiver, it is usually the most important
specifications.
For lower power systems (below 10mW), the power
of the entire transmitter chain is important and
should be taken into consideration.
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Pout − Pin
Pdc
Pout −
=
Pout
Gp
Pdc
Pout
(1 − Gp−1 )
Pdc
ηPAE = ηc (1 − Gp−1 )
≈ ηc
Average Efficiency with Power Control
For a constant envelope signal (phase/frequency modulation),
the average efficiency is equal to the average efficiency at
peak power.
Due to power control, though, we must take into account the
statistics of the transmitted signal. Modern systems use
power control to minimize the impact of a transmitter on
nearby systems (interference) and hence only use as much
power as needed to achieve low error communication with the
base station.
Thus the actual average efficiency depends on how the
efficiency varies with output power
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Power Statistics
Z
∞
ηav =
η(P)g (p)dP
−∞
Given the distribution of power levels, or the PDF g (P), we
can calculate the expected value of the efficiency
Unfortunately, for most PAs the efficiency drops at low power.
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Envelope Statistics
For signals with amplitude modulation, the average efficiency
depends not only on the desired power level, but also on the
statistics of the envelope.
The amount of power variation is usually captured by the
PAR, or the Peak-to-Average Radio.
The PAR is a strong function of the type of modulation.
Systems with the highest PAR are OFDM systems employing
multiple carriers.
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Linearity
dB
transmit filter mask
IM5
3f1-2f2
IM3
f1 f2
2f1-f2 2f2-f1
HD2
2f1 2f2
HD3
f
3f2-2f1
The traditional way to characterize narrowband system
linearity is with IM3. Since the system may be driven into a
strongly non-linear regime, all odd order harmonics should be
carefully taken into account to ensure that excessive spectral
leakage does not occur.
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Sources of Non-Linearity
PAs exhibit nonlinear distortion in amplitude and phase. For a
mulated signal, both sources of distortion are significant.
The dominant sources are AM-to-AM and AM-to-PM.
Amplitude distortion: AM-to-AM conversion
Phase distortion: AM-to-PM conversion
For input: x(t) = A(t) cos(ωt + φ(t))
Corresponding output:
y (t) = g [A(t)] cos(ωt + φ(t) + ψ[A(t)])
AM-to-AM conversion dominated by gm non-linearity before
clipping
AM-to-PM conversion dominated by non-linear capacitors
(phase delay)
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AM-AM and AM-PM Non-Linearity
For a narrowband signal, we can partition the non-linearity
into an amplitude-amplitude (AM-AM) component and an
amplitude-phase (AM-PM) component
This behavioral model can be used to run system level
simulations to see the effect of non-linearity on a modulated
waveform
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Adjacent Channel Power (ACP)
1.25MHz
P1
P2
30 kHz
885kHz
ACP=P1(dBm)-P2(dBm)
For modern communication systems, the IM specifications
leave a lot to be desired since they are only based on two-tone
excitation. Increasingly, the actual modulation waveform
needs to be tested with the PA.
To ensure proper etiquette, the amount of power leaking into
an adjacent channel is carefully specified.
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Unwanted emissions are caused by a number of factors including non-linearity in the
28
Chapter 2 Transmitter Fundamentals
Transmit Mask Specs
system, noise resulting from interference with other circuits or spurious tones created by
mask. Non-idealities in the transmitter, which distort the signal, will bring the modulated
clocks or frequency synthesizers. Because these non-idealities affect the in-band signal
signal higher and thus closer to violating the spectral mask. These non-idealities will be
they can also have an effect on the modulation accuracy. The in-band spectral mask
discussed in more detail in the following section.
requirement for DCS1800 is illustrated in Figure 2.6.
10
0
0
-20
-20
-30
PSD
Relative Power (dBc)
-10
-40
-40
-50
-60
-60
-70
-80
0
1
2
3
4
5
6
7
-400
Frequency O ffset (MH z)
-200
0
200
400
Frequency (KHz)
Figure 2.6 GSM in-band spectral mask requirement.
Figure 2.7 GSM spectral mask at low offset frequencies with GMSK modulated signal.
Every standard therefore has a transmit
mask specification
In addition to in-band requirements, out-of-band spectral emissions requirements
that must be met. This limits spectral regrowth, noise, and
other spurious transmissions in the band and in nearby bands.
Above examples are for GSM.
Typically, one of the most difficult portions of the spectral mask requirement is
close to the carrier at. For example, Figure 2.7 shows the same GSM spectral mask at
also have an affect on transmitter design. While the out-of-band emissions don't affect
relatively low offset frequencies. In addition a GMSK modulated signal has also been
the modulation accuracy or other transmitters in the same band, the limits are often lower
illustrated in the same plot. Notice that the modulated signal is always below the spectral
than the in-band limits, making them more difficult to satisfy. For example, one of the
most difficult requirements in the DCS1800 standard is the emissions requirement that
falls in the DCS1800 receive band.
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FCC Limts
EIRP Spectral Density (dBm/MHz)
-40
-45
-50
-55
Frequency
(MHz)
960-1610
1610-1990
1990-3100
3100-10600
> 10600
-60
-65
-70
-75
-80
1
1.5
2
3
4
5
Frequency (GHz)
EIRP
(dBm/MHz)
-75.3
-53.3
-51.3
-41.3
-51.3
6
7
8 9 10
While the transmit mask is standard specific, every transmitter
must comply with FCC limits (in the US). The above mask is
for an unlicensed device meeting part 15 requirements.
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Error Vector Magnitude
Chapter 2 Transmitter Fundamentals
26
Chapter 2 Transmitter Fundamentals
cos( /4- /2)
Ideal
Phase Error
Q
Actual Signal
Error Vector
sin( /4+ /2)
Ideal Signal
sin( /4- /2)
/2
/2
Phase Error
/4
Ideal Phase
I
cos( /4+ /2
Figure 2.5 Graphical representation of the error vector and phase error.
Figure 2.10 Quadrature phase error constellation.
While the ACP is a good way to measure how much the PA
signal will deteriorate a neighboring channel signal, the EVM
Generally, phase
modulation
are constant
envelope,
isandafrequency
measure
ofschemes,
howwhichmuch
the
PA
with
baseband
butuse
ininterferes
this test, the baseband
inputs itself.
are low frequency sinusoids given by
For certain types of modulation the difference is measured by the error
vector common and useful graphical representation of accuracy of a trans
Another
magnitude (EVM) while in other systems only the phase error is thesignal
critical
metric.altering the baseband input. Typically modulated data is applied
involves
phase error as the metric and NCE modulation schemes use EVM. For example for
The EVM measures the systematic deviation Iof
(t ) the
cos(2 f t )
constellation points from the ideal positions due to amplifier
phase error of less than 5 degrees.
and
non-linearity.
bb
GMSK modulation used in DCS1800 cellular phones, the standard requires an RMS
Q (t )
2.4.2 Spectral Emissions
sin(2 f bb t ) .
It might be expected that the up-converted spectrum would contain two frequenci
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IC’s for Comm
In addition to modulation accuracy, it is critical that a transmitter
only emit a Advanced
specified
Example: GSM Transmitter Non-Ideality
42Fundamentals
Chapter 2 Transmitter
Chapter 2 Transmitter Fundamentals
Q
1
1
0.5
0.5
0
Q
-0.5
-1
0
-0.5
-1
Ideal
Phase noise
-1
-0.5
0
0.5
1
-1
I
Figure 2.20 GMSK constellation diagram with phase noise.
Ideal
Therm al Noise
-0.5
0
0.5
1
I
Figure 2.18 GMSK constellation diagram with thermal noise.
The first example shows the impact of phase noise whereas
In addition to affecting the modulation accuracy, phase noise also In
changes
theto thermal noise, flicker noise can also negatively impact a tran
addition
the second plot shows the impact of noise (both amplitude
with thermal
modulated spectrum as illustrated in Figure 2.21. Much like the case signal.
Flickernoise,
noise is present in CMOS transistors and the mean-square drain cu
and phase).
phase noise can raise cause spectral mask violations. In this examplegiven
the phase
by noise is
large enough to cause spectral mask violations close to the carrier. However, unlike
thermal noise, phase noise decreases as the frequency of interest moves away from the
LO frequency. Even with this property, phase noise is still a major
concern at large
offset IC’s for Comm
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Advanced
id2
K
I Da
f
f
above approximately 300 kHz and below -300 kHz, the noise is clearly evident leading to
Transmitter
a spectralNoise
mask violations.
0
PSD (dB)
-20
-40
W ith noise
-60
W ithout noise
-80
-600
-400
-200
0
200
400
600
Frequency offset (KHz)
Figure 2.17 GMSK signal with thermal noise.
Transmitter noise is important for two reasons. First the noise
The effects
of thermal
noise on the constellation
a GMSK
signal are shown
level should
not
significantly
impact ofthe
EVM/BER
of inthe
will clearly
cause error in the
magnitude
phase of the
Figure 2.18. Thermal
transmitter
itself.noise
More
importantly,
the
noiseandleaking
into
other
bands
must meet specs. This is especially problematic
transmitted
signal.
in FDD systems that transmit and receive at the same time.
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Digital and Analog Modulation
Both digital and analog modulation schemes involve amplitude
and/or phase modulation: Vo (t) = A(t) cos(ωt + φ(t))
Linearity specs of PA determined by envelope variation
Most spectrally efficient modulation schemes have large
envelope variations
Analog FM (AMPS) uses constant envelope ⇒ can use
efficient non-linear power amplifiers (60%-70%)
GMSK (GSM) uses constant envelope as well
π/4 DQPSK (IS54/136) has 3dB peak to average ratio (PAR)
QPSK (CDMA base station) has 10dB of PAR, OQPSK
(CDMA handset) has 3dB of PAR
802.11g OFDM at 54 Mbps (52 sub-carriers) has about 6-8
dB of PAR
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Modulation Schemes
Key specifications are the peak-to-average radio, the peak
power, and the power control range.
Constant modulation schemes much easier (GSM, AMPS).
WiFi uses OFDM, which is the hardest ! LTE up-link uses a
single carrier to ease the PA back-off requirements.
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Switching versus “Linear” PA
Two general classes of PA: Linear and
Non-Linear
“Linear PAs” preserve amplitude and
phase information while “Non-linear PAs”
only preserve phase mod. Typically (not
strictly), linear PAs employ transistors as
current sources (high Z), non-linear PAs
employs transistors as switches (low Z)
Linear PAs can drive both broadband and
narrowband loads.
Non-linear PA usually drive a tuned
circuit load
Amplitude information in a non-linear PA
can be recovered by:
Oversampling, duty cycling, or varying
the supply voltage
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vi (t)
io (t)
vi (t) = A(t) cos(ωt + φ(t))
io (t) = Gm A(t) cos(ωt + φ(t))
VDD
−A0 cos(ω0 t + φ(t))
A0 cos(ω0 t + φ(t))
Clipping: A “Linear” PA is Impossible
All amplifiers eventually clip,
that is the output cannot be
higher than some multiple of
the power supply. Note that
the peak amplifier output
can be arbitrarily large, but
the average output power
will limit.
If we “back-off” sufficiently
from the peak so that the
amplifier never clips, then we
compromise the efficiency.
Limited Output
Ideal Clipping
Point
We can generally make a
compromise and choose
sufficient back-off to meet
the EVM specs.
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Power Back-off
In applications requiring a linear PA due to PAR, we must
back-off from the peak power point to avoid clipping the
waveform.
For 10 dB of PAR that means operating the PA at 10 dB
lower power (or power back-off).
An OFDM 802.11g system that needs 20 dBm at antenna and
has a PAR of about 17 dB. That means to transmit 20 dBm
average power, the PA should be capable of transmitting 37
dBm !!!
In practice the peak amplitude is a rare event and the PA
should be allowed to clip. A 6-7dB back-off is typical.
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Power Control
Most wireless systems have power control. Power control is
important to limit transmit power to the lowest possible
setting. This saves battery power and limits the amount of
interference to other nearby users.
There are two power control loops to consider: (1) Mobile
power control loop and (2) Basestation to mobile power
control loop
The mobile unit must transmit a given output power with a
certain resolution. In GSM the output power can be off by
±2dB.
in CDMA systems, the noise level is actually set by this
interference so all users are required to back-off to make the
system as a whole more efficient.
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Closed Loop Power Control
The mobile power control loop can be a closed loop or open
loop system. In an open loop system, the power of the output
power of the hand-held is measured and calibrated for each
DAC setting. Then an open loop system is used estimated
based on a one-time calibration.
In a closed-loop system, the output power is estimated using a
directional coupler, a voltage measurement, or a current
measurement.
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Stability over VSWR
45
50
1.0
0.9
55
1.4
0.8
0.7
0.6 60
1.6
1.8
2.0
06
0.5
0.
44
0.
70
5
0.4
20
3.0
0.6
VSWR = 3 Circle
0.3
0.8
4.0
5.0
1.0
0.2
IND
UCT
IVE
15
0.28
1.0
0.22
RE
AC
TA
75
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EC
OM
PO
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EN
T
6
0.18
0.32
25
10
8
0.25
0.26
0.24
0.27
0.23
0.25
0.24
0.26
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0.27
REFLECTION COEFFICIENT IN DEG
REES
LE OF
ANG
ISSION COEFFICIENT IN
TRANSM
DEGR
LE OF
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ANG
0.
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7
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0.43
Advanced IC’s for Comm
0.18
-80
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4
3.0
4
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9
5
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10
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0.4
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RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo)
50
0.0 —> WAVELE
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GEN
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SWR −1 ≤ |x + jy | ≤ SWR +1
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0.37
0.
On the Smith Chart any load lying
on a constant VSWR circle is a
valid load, or any impedance such
that
0.13
0.38
19
A system with a VSWR of 3:1 can
drive any load with magnitude as
large as 3 × 50Ω or as small as
50Ω ÷ 3 = 17Ω.
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jB/
E (+
NC
TA
EP
SC
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The PA generally must be able to
drive a varying load. The ability to
drive a given range of loads is
specified as the VSWR, e.g. a
VSWR of 3:1
Power Control Loops
Power Amplifier
Filter
V+
Directional Coupler
≈V+
V−
Analog
Power
Control
V +
Z0
V −
Power Detector
Reference
A directional coupler is one of the more accurate methods to
measure the power delivered to the load (antenna). The
power reflected from the antenna due to a mismatch is not
computed. But the directionality of the coupler is key.
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Power Devices
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RF Power Transistors
In a BJT there is a direct trade-off between the breakdown
voltage and the
fT of the device. Some people define this as a metric for the
transistor.
Thus we should employ the lowest tolerable fT device for our
PA. That’s because such a device can swing the largest
voltage.
Unfortunately, the trend in technology is the opposite, mostly
for digital and RF applications, giving us over 100 GHz fT and
only 1-2 volts to operate with. This is good for digital.
CMOS devices also require a large Cox (small Tox )
(gate control) for short channels, and thus gate oxide
breakdown is a big issue. Punchthrough is another breakdown
mechanism.
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BJT Device Power Gain
For 300mA of current need
20,000 mm2 of Si area
Operating at frequencies
∼ fT /10 (say 2.5 GHz in 25
GHz process)
Device parasitics dominate
impedance levels. Don’t
forget Temp!!!
Package parasitics limit gain
by providing feedback
Gain determined by (M.
Versleijen et. al.):
GP =
ωT 2
ω
1
C
Cjc
B
RC
Ccs
rb
Typical Terminal
Impedance Levels (CE)
Zin = 40 + j -50
Zout = 3 + j -30
RE
RS
LE
LG
E
1
RL
≈
int )
+ ωT RL CBC ωT LE + RE + RB (1 + ωT RL CBC
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S
1
CBC LE
ω2
CMOS Device Losses
For each finger of the CMOS device, we must carefully extract
the capacitance and resistive parasitics. The layout will have a
large impact on these parasitics.
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Power FET Layout
DRAIN
d1
source
source
drain
g3
d2
g3
d2
g3
g2
d3
g2
d3
g2
g1
gate
d1
g1
g1
GATE
Power FETs are typically very large (millimeter size) and the
layout is broken into sub-cells. Each sub-cell is broken into
multi-fingered transistors and the gate/drain lines are delay
equalized.
The layout metals introduce significant resistance and
capacitance, which needs to be carefully modeled.
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CMOS Device Power Gain
G
Need a good model of device, especially
resistive paraistics. To predict power gain
need to know the gate/source/drain
resistance, including the interconnect
parasitics (vias, metal, poly), and the gate
induced parasitics.
S
R3
Cjd
R2
R5
B
Drain
Cgdext
Gate
Note that the point the device looks
distributed depends on the slow wave
velocity due to large gate and drain cap.
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R1
R4
Substrate parasitics will also limit power
gain and needs to be extracted accurately.
Include inductance for high frequencies or
very large “distributed” device.
D
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Lg
Rg
Cgsext
Ld
Cdb
Rd
Rdb
Rbb
Rs
Rsb
Csb
Ls
Source
Bulk
Substrate Parasitics
perpenicular body contact
R⊥ ∝ W
W
R ∝
1
W
v
e
r
t
i
c
a
l
c
o
n
t
a
c
t
Notice that the placement of the substrate contacts has an
important impact on the overall substrate loss (the output
impedance of the device). Increasing the device finger width
W will decrease the “parallel resistance” component but
increase the “perpendicular resistance”.
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MOS CV Curve
Cgg
Classical
Cox
Quantum
Poly Depletion
VF B
VT H
VGB
Capacitors need to be large signal accurate over voltage
swing. Voltage swings maybe negative and move device into
accumulation (due to inductors).
Make sure the CV model is accurate (BSIM capmod=2).
Include quantum effects and poly depletion.
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Typical Multi-Stage PA
Off-chip Output
Matching Networks
On-chip Matching Networks
Power Loss ~ 2dB
10mW
100mW
1W
1mW
Po~1W=30dBm
Power Gain ~ 13dB
Need 1W at antenna and about 30 dB of power gain
Each amplifier stage has about 13 dB of power gain
Interstage matching networks have an insertion loss of about
2 dB
If you cannot afford loss at output stage, must off-chip
components (preferably in the package to keep them close and
parasitics minimal).
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Count your dBs
In RF receiver design we throw around a lot
of dB’s without giving it much thought. For
instance, you may put in a margin 3dB in
your design. But for a PA, this is not so easy
! 3dB is a factor of 2 in power !!
Likewise, any loss in the signal path can hurt
the PA efficiency considerably.
Consider designing a 1W PA with an
efficiency of 65%. But due to a customer
demand, you have to budget up to 1dB of
extra loss at the output.
That means your PA efficiency can
potentially drop to 52%!
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dB
Linear
0.1
0.3
0.5
0.7
0.9
1.0
1.2
1.4
1.6
1.8
2.0
0.977
0.933
0.891
0.851
0.813
0.794
0.759
0.724
0.692
0.661
0.631
PA Package and Interface Issues
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Diode/Switch
RX
DCS/PCS
BPF: DCS
Diplexer
RX
GSM
TX
BPF: PCS
chip boundary
What PA Needs to Drive
TX
BPF: GSM
chip boundary
LPF: PCS/DCS
Isolator
Coupler
LPF: GSM
Isolator
Coupler
Diode/Switch
Need SAW filter to eliminate out of band emissions.
Directional coupler measures output power.
In a half duplex system, a switch is used for RX and TX. In a
full duplex system, a duplexer is used to isolate the TX and
RX. In the extreme case, a circulator can be used as well.
Typical cell phone PA that needs to put out 0.5W to the
antenna (LTE). Due to loss in output matching network,
coupler, duplexer/diplexer, and SAW filter, need to put out an
additional 3 dB.
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Parasitic Coupling
Supply Bounce
VCC
Thermal feedback
substrate coupling
Signal through
ESD cap
Signal though
matching and
bias network
Self heating; Intrinsic
shunt/series FB
Package mutual
inductance/capacitance
Ground Bounce
Package: ESD, bias, pins, bond wires
Substrate: Devices (passive and active), thermal
Maximum safe power gain
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Ground Bounce
Since the load current is large (amps), and it flows out of the
chip to the external load, there is considerable “bounce” in
the ground and supply lines
Vbounce = L
dI
dt
Besides limiting the voltage swing (efficiency), for on-chip
signals referenced to the internal ground, this is not a big
issue. But for any external signals referenced to the clean
board ground, this ground bounce is a problem (it can
subtract or add from the input signal, for instance)
For this reason, the output stage ground is often separated to
mitigate this coupling effect.
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Packaging Issues
bond pad
bond wire
chip
package lead
package lead
bond wire
The emitter/source inductance is a major problem as it limits
the device swing, reducing the efficiency of the amplifier. It
also is a big source of ground bounce that can lead to
instability.
Use as many bondwires to reduce this inductance. If possible,
use a package with an exposed paddle to reduce the bondwire
length.
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Downbond
bond wire
down bond chip
package lead
exposed "paddle"
To reduce the inductance to “gnd”, we can use an exposed
paddle style package, where the chip is glued to a ground
plane which is directly soldered to the board.
The bond wire to ground is a downbond, and it is shorter and
thus the overall inductance for the ground can be reduced
substantially.
Leadless packages are also preferred (such as QFN).
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Flip-Chip Package
bump
chip
via
pads
Flip chip packages are more expensive but allow very low
inductance bumps (< 100pH) to the package ground. This
eliminates both the bond wire inductance and the package
lead inductance.
Another option, the entire PA can be constructed with
lumped components in the package by utilizing high quality
passives. This is more of a module than an integrated PA.
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Stability Issues
system
ground
ground
noise
common
mode
noise
local ground
The input signal comes from an off-chip source (driver amp or
VCO buffer). The local ground is bouncing due to the PA
output stage. To reduce the effects of this ground bounce, a
fully differential source can be employed. If not available, a
transformer can help isolate the two grounds.
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Balanced/Differential Operation
+ vn −
+ vn −
Go differential / balanced to reduce common mode coupling.
Transformer at input helps to isolate input/output.
Watch out for parasitic oscillations (see next slide).
Bypass capacitors (big and small) to cover multiple frequency
bands. Big caps are usually MOS varactors.
Plan the package layout early in design.
Spend at least as much time on ground/VDD/bypass issues
as the circuit design !!
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Parasitic Oscillator
Ld
Ld
iµ
Cgd
Zin
+
vi
−
Cgd
+
vo
−
Lg
Consider the medium frequency equivalent circuit for output
stage. Due to the large device size, the gate-to-drain
capacitance is substantial. The gate inductance is for biasing
or to tune out the input cap.
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Power Combining
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How Big?
The amount of power that we can extract from a PA device is
limited by the output impedance of the device. As the device
is made larger to handle a higher DC current (without
compromising the fT ), the lower the output impedance.
For a current source style of PA, eventually the device is so
large that power is lost in the device rather than the load.
This is the attraction of a switching PA.
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Power Combining (cont)
Lossy Power Combiner
But for a non-switching PA we must perform some power
combining to use more than one device. This way we can
transform the load into a higher impedance seen by each PA.
The power combining networks are lossy and large. We’ll
come back to them later.
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Can we “wire” PAs together?
Note that we cannot simply “wire” PAs together since the
impedance seen by each PA increases by N if we connect N in
parallel:
VL
RPA =
= NRL
IL /N
This means that each PA delivers less power for a fixed swing
PPA =
2
Vswing
2RPA
There is also “load pulling” effects if the sub-PAs are not
perfectly in phase
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