Fundamentals of
Integrated Transformers:
from Principles to Applications
Andrea Bevilacqua
University of Padova
andrea.bevilacqua@dei.unipd.it
© 2020 IEEE
International Solid-State Circuits Conference
Outline
 Introduction
 Physical principles


Self and mutual inductance
Integrated transformers
 Interim Q&A session
 Transformers in amplifiers



Baluns
Couplers and combiners
Matching networks
 Transformers in oscillators



Doubly-tuned resonators
Varactor coupling
Multiple resonance tanks
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© 2020 IEEE
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Fundamentals of Integrated Transformers:
from Principles to Applications
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Conventional Use of Transformers
 Electrical energy distribution network (grid)
 Conversion of electrical energy
 Power management
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Transformers in Integrated Circuits
20MHz power transfer
22-30GHz PA
[Wang19]
[Pellerano19]
2-3GHz DPLL
71-76GHz TRX
[Yue19]
[Liu19]
 Transformers are ubiquitous in analog, RF and mm-wave integrated circuits
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Fundamentals of Integrated Transformers:
from Principles to Applications
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What Can I Do with a Transformer?
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Galvanic isolation  ac coupling
Single-ended to differential conversion (balun)
Signal/power combining
Phase inversion  feedback networks
Passive voltage/current gain
Impedance transformation (matching network)
Higher-order resonator  useful for LC oscillators and filters



Multi-mode oscillator
Phase noise optimization
Implicit frequency multiplication
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© 2020 IEEE
International Solid-State Circuits Conference
Fundamentals of Integrated Transformers:
from Principles to Applications
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Working Framework of this Tutorial
 Quasi-static approximation

Lumped element devices
 We will not discuss higher frequency phenomena:


Distributed circuits (t-lines)
Radiative effects (antennas)
 Others topics we will not touch are:



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EM simulation techniques
Inductor/transformer compact modeling and circuit simulation
Design rule related issues (e.g. density rules-compliant layouts)
Measurement and characterization techniques
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© 2020 IEEE
International Solid-State Circuits Conference
Fundamentals of Integrated Transformers:
from Principles to Applications
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Ampère’s Law
Orientation of contour C and
direction of current I are
related by right-hand rule
A.-M. Ampère
 Relates magnetic field (H) to its sources (currents)
 It is a powerful tool to compute H in case of
cylindrical symmetry
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Fundamentals of Integrated Transformers:
from Principles to Applications
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Magnetic Field in a Solenoid
 Because of symmetry the field
lines are along the coil axis
 Assuming very long core can
show that H is approximately:


Uniform within the core
Null outside the core
 Ampère’s law yields:
Core with
permeability μ>μ0
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International Solid-State Circuits Conference
Fundamentals of Integrated Transformers:
from Principles to Applications
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Magnetic Flux and Inductance
 Magnetic flux is
 Flux density is B=μΗ, hence:
 Flux is proportional to current
Core with
permeability μ>μ0
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© 2020 IEEE
International Solid-State Circuits Conference
Inductance
Fundamentals of Integrated Transformers:
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Flux Linkage
Core with
permeability μ>μ0
 The flux produced by current I2 crossing surface S1 is
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© 2020 IEEE
International Solid-State Circuits Conference
Fundamentals of Integrated Transformers:
from Principles to Applications
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Mutual and Self Inductance
μ>μ0
 Swapping the roles of the coils:
 Mutual coupling is reciprocal:
Mutual
inductance
 Mutual (M) and self (L) inductance only depend on the geometry of the
system and the properties of the core material (μ)
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International Solid-State Circuits Conference
Fundamentals of Integrated Transformers:
from Principles to Applications
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Faraday’s Law
μ>μ0
Orientation of contour C
and surface S are related
by right-hand rule
M. Faraday
 Big discovery: crossed by a time-varying magnetic flux the coil behaves as a
generator!
 Induced emf creates a magnetic field opposing the flux that generated it
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International Solid-State Circuits Conference
Fundamentals of Integrated Transformers:
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Coupled Inductors
 Consider magnetic flux density
generated by both coils 1 and 2
 Conveniently define voltages V1
and V2 as:
μ>μ0
 Combine Faraday’s law and
definition of self and mutual
inductance to describe the
coupled inductors as a two-port
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Fundamentals of Integrated Transformers:
from Principles to Applications
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Dot Convention
dot convention
swapped port
polarity
 Current flows into the dot of one coil  the induced voltage has positive
polarity at the dotted terminal of the other coil  M is positive
 If we swap the polarity of one electrical port (e.g. V’2, I’2)  180° phase shift
 M is negative
 Self inductances L1, L2 are always positive, but mutual inductance can be M≷0
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© 2020 IEEE
International Solid-State Circuits Conference
Fundamentals of Integrated Transformers:
from Principles to Applications
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Flux Leakage
 In general, some of the flux density B generated by one coil does not link to
the other coil  flux leakage
μ>μ0
 Only the smaller coil surface (between S1 and S2) crossing the flux of B1 and
B2 is relevant for the mutual inductance
 Hence M2 ≤ L1L2  Define magnetic coupling k
 |k| ≤ 1
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© 2020 IEEE
International Solid-State Circuits Conference
Fundamentals of Integrated Transformers:
from Principles to Applications
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Equivalent Circuit for Coupled Inductors
leakage
inductance
 Can describe coupled inductors with an equivalent circuit:
magnetizing
inductance
ideal
transformer
 Magnetizing inductance  models the need to establish magnetic flux
 Leakage inductance  models flux leakage
 Ideal transformer with ratio 1:nk  V’2/V1 = -I’1/I2 = nk


Turn ratio:
Impedance transformation  V’2/I2 = (-V1/I’1)n2k2
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International Solid-State Circuits Conference
Fundamentals of Integrated Transformers:
from Principles to Applications
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Magnetic Transformer
 The coupled inductors make a magnetic transformer, that approximates an
ideal transformer if |k|1 and L1∞
|k|1
L1∞
 Materials with high permeability μ, coils with many turns and large coil
section (to have large inductance), and tightly wound coils (to have |k|1)
help implementing a good transformer  Transformers are inherently bulky
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© 2020 IEEE
International Solid-State Circuits Conference
Fundamentals of Integrated Transformers:
from Principles to Applications
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Inductor Series Resistance
 Coil wire has resistivity ρ≠0

Inductor is lossy
μ>μ0
 Assuming a wire with diameter d
the inductor series resistance is:
Wire section
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© 2020 IEEE
International Solid-State Circuits Conference
Total coil length
Fundamentals of Integrated Transformers:
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Inductor Quality Factor
 Assume to connect the inductor to a
lossless capacitor  LC resonator with
resonance frequency
 The resonator quality factor is defined as:
μ>μ0
Stored energy
Dissipated power
 Only the coil has losses, that we assume only due to the wire resistance R

The inductor quality factor is:
Andrea Bevilacqua
© 2020 IEEE
International Solid-State Circuits Conference
Fundamentals of Integrated Transformers:
from Principles to Applications
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Integrated Inductors
center tap
 Integrated inductors are implemented as a planar version of the solenoid
geometry  spiral shape
 Can have several shapes: circular, octagonal, square, etc…
 Can be symmetric and have a center tap
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© 2020 IEEE
International Solid-State Circuits Conference
Fundamentals of Integrated Transformers:
from Principles to Applications
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Estimation of Integrated Inductance
 Computation of the inductance is complex. Can approximately say [Mohan99]:
Square
Octagonal
K1
2.34
2.25
K2
2.75
3.55
Assuming octagonal coil, w=10μm, s=5μm:
L [nH]
N
dout [μm]
0.1
1
70
1
2
170
10
7
300
Fill factor
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International Solid-State Circuits Conference
Fundamentals of Integrated Transformers:
from Principles to Applications
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Limitations of Integrated Inductors
 Magnetic materials are typically not available (μ=μ0)  larger device size for
a given inductance
 Planar device  Non-uniform H field  L prop. to coil diameter, not area
 Spiral shape  Limited coupling between turns (inner turns have smaller
area)  Inductance is not proportional to N2  larger size

Inner turns contribute less additional inductance and more loss  hollow spiral
layout (typically limit fill factor ρd<0.5)
 Inductance is proportional to Np with p<2, but series resistance still
approximately increases with N  Q is proportional to Lb with b≈0.2, 0.3
 The circle has the best area/perimeter ratio  highest Q

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
Circular inductor are not always allowed by design rules
Regular polygon can approximate a circle  Octagon is usually all right
Square inductor has higher Q than rectangular inductor
 Bottom line: try use hollow (i.e. few turns) octagonal inductors
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Fundamentals of Integrated Transformers:
from Principles to Applications
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Inductor Self Resonance
 Parasitic capacitance between inductor traces and substrate and from trace to
trace  inductor turns into a capacitor beyond self resonance frequency ωsrf
 Approximately:

 Want operation frequency to be a fraction of ωsrf  there is a maximum
value of inductance that can be used for a given operation frequency
 progressively more difficult to use large inductances at higher frequencies
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Fundamentals of Integrated Transformers:
from Principles to Applications
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High Frequency Losses
Simulation
at 300 MHz
 At higher frequency the current is not
uniformly distributed in the coil trace
 The current tends to flow on a thin region
near the trace surface (skin effect)


The thickness of this region is related to the
skin depth
Trace resistance increases as δ decreases

Simulation
at 60 GHz
 Current tends to flow along the least
inductance path  crowds at inner coil side
 Substrate coupling increase losses
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Capacitive coupling
Eddy currents
More relevant for larger inductors
Current density
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Fundamentals of Integrated Transformers:
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Integrated Transformers
 Stacked layout
 k=0.6-0.8
 Larger capacitive
parasitics
 Lower coil made of
thinner metal
 Coplanar
interwound layout
 k=0.5-0.7
 Can use thicker
metal for both coils
Andrea Bevilacqua
Fundamentals of Integrated Transformers:
from Principles to Applications
© 2020 IEEE
International Solid-State Circuits Conference
 Concentric
coplanar layout
 k=0.3-0.5
 Minimum capacitance
between primary and
secondary
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Transformer Design Considerations
 The requirements on desired magnetic coupling influence the transformer
layout to be selected

Larger k trades off with larger capacitive parasitics between primary and
secondary coils
 Typical achievable k ranges from 0.2 to 0.8  leakage inductance is never
negligible
 To have n»1 or n«1 we need one inductor much larger than the other, but:


Larger inductors have a lower self resonance frequency and more substrate losses
Mutual coupling depends on the coil with the smaller area  limited magnetic
coupling if coils have very different sizes
 Typical achievable turn ratio for integrated transformers: 0.5<n<2
 For a given technology, same achievable Q as for inductors, except if:


One coil is implemented in a thinner metal (stacked layout)
Layout requires the use of lower (and much thinner) metals for the underpasses
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Design Example (1)
dout ≈ 280μm
w ≈ 9μm
[Vallese09]
 Operation frequency: 3—5 GHz
 Coplanar interwound layout
 Relatively large turn ratio
Andrea Bevilacqua
© 2020 IEEE
International Solid-State Circuits Conference
L1 [nH]
L2 [nH]
k
Q1
Q2
1.8
6.4
0.7
8
10
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Design Example (2)
dout ≈ 100μm
w ≈ 4μm
[Padovan16]
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Operation frequency: 15—40 GHz
Hybrid interwound/stacked layout
Moderate magnetic coupling
Compact size
Andrea Bevilacqua
© 2020 IEEE
International Solid-State Circuits Conference
L1 [pH]
L2 [pH]
k
Q1
Q2
680
580
0.65
21
20
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from Principles to Applications
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Physical Principles Summary
 In integrated processes high permeability materials are typically not available
 large area for inductors and transformers
 Self and mutual inductance depend on device geometry

Mutual inductance depends on the area of the smaller coil
 integrated transformers have limited magnetic coupling
 Self resonance frequency limits the maximum inductance for a given
frequency of operation
 Quality factor increases (although weakly) with inductance for small
inductors; use of large inductances is limited by substrate losses and ωsrf
 Skin effect, current crowding and substrate coupling limit Q at higher
frequencies
 Cannot effectively couple inductors with very different inductance values
 limitation in achievable transformer turn ratios
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Fundamentals of Integrated Transformers:
from Principles to Applications
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Interim Q&A Session
Please ask questions that you feel is
essential to follow the rest of this tutorial
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Fundamentals of Integrated Transformers:
from Principles to Applications
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Outline
 Introduction
 Physical principles


Self and mutual inductance
Integrated transformers
 Interim Q&A session
 Transformers in amplifiers



Baluns
Couplers and combiners
Matching networks
 Transformers in oscillators



Doubly-tuned resonators
Varactor coupling
Multiple resonance tanks
Andrea Bevilacqua
© 2020 IEEE
International Solid-State Circuits Conference
Fundamentals of Integrated Transformers:
from Principles to Applications
31 of 78
Balun
 Can use transformer for single-ended to differential signal conversion
balanced port
if L2=L3
and k12=k13
 If L2=L3 and k12=k13  No common-mode ac signal at balanced port
 Layout symmetry is essential
 Can use center tap to provide bias at balanced port  Galvanic isolation
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© 2020 IEEE
International Solid-State Circuits Conference
Fundamentals of Integrated Transformers:
from Principles to Applications
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Balun Non-Idealities
 Magnetizing and leakage inductance are
unwanted but unavoidable
ideally
actually
 How do we cope with it?
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International Solid-State Circuits Conference
Fundamentals of Integrated Transformers:
from Principles to Applications
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Embedding Balun Parasitics in a Ladder
 Can conveniently embed magnetizing and leakage inductance in amplifier
input/output networks

Example: LNA with inductive degeneration  Use transformer parasitics as part of
a ladder network
explicit cap +
pad parasitics
2-section
ladder
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© 2020 IEEE
International Solid-State Circuits Conference
Fundamentals of Integrated Transformers:
from Principles to Applications
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Coupled Input and Degeneration Inductors
 Coupling Lg and Ls in an inductively degenerated amplifier allows to reduce
the inductor size
additional
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International Solid-State Circuits Conference
Fundamentals of Integrated Transformers:
from Principles to Applications
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Layout of Coupled Lg and Ls
 Coupling Lg and Ls leads to a particularly
compact and convenient layout, especially
in a differential design
 Can easily route signals to transistors
[Padovan16]
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© 2020 IEEE
International Solid-State Circuits Conference
Fundamentals of Integrated Transformers:
from Principles to Applications
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DM/CM Inductance
 Take two identical symmetrically-wound coils
 bifilar transformer with unitary turn ratio
 Different behavior in differential mode or
common mode
 Differential mode: I2=I1 For each branch:
I1
 Common mode: I2=-I1 For each branch:
 Can also change coil winding to get k<0
I2
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© 2020 IEEE
International Solid-State Circuits Conference
Fundamentals of Integrated Transformers:
from Principles to Applications
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Differential Artificial T-Line
 Can use symmetric bifilar transformer to implement
a differential artificial t-line with compact layout

Example: lumped-element Wilkinson divider
[Caruso14]
section of differential artificial t-line
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International Solid-State Circuits Conference
Fundamentals of Integrated Transformers:
from Principles to Applications
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Quadrature Hybrid
 Lumped element implementation of quadrature hybrid directional coupler
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Useful for I/Q generation
Power combining (balanced amplifier, Doherty amplifier)
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International Solid-State Circuits Conference
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from Principles to Applications
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Design of Quadrature Hybrid
 Design equations:
 Operating frequency:
 Layout example of a
differential implementation
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© 2020 IEEE
International Solid-State Circuits Conference
[Park15]
Fundamentals of Integrated Transformers:
from Principles to Applications
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Series Power Combiner
 Combine the power of NS amplifiers
by series connecting the secondary
windings of identical transformers
 Load voltage is shared among
amplifiers: VL=NSV2  decreased
voltage stress
 Intrinsic load impedance
transformation: IL=-I2, VL=NSV2
If k=1  m=n
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© 2020 IEEE
International Solid-State Circuits Conference
Fundamentals of Integrated Transformers:
from Principles to Applications
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Parallel Power Combiner
 Combine the power of NP amplifiers
by parallel connecting the secondary
windings of identical transformers
 Intrinsic load impedance
transformation: IL=- NPI2, VL=V2
 Can also do series/parallel combiner
 As for the balun, we need to embed
transformer parasitics into the
combiner network (e.g. as a ladder)
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International Solid-State Circuits Conference
Fundamentals of Integrated Transformers:
from Principles to Applications
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Transformer-Based Matching Networks
interstage network
output network
input network
 Transformers are broadly used in input, interstage and output networks
 Inherent impedance transformation
 It is key to embed parasitics in network design  ladder or doubly-tuned
networks
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© 2020 IEEE
International Solid-State Circuits Conference
Fundamentals of Integrated Transformers:
from Principles to Applications
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Doubly-Tuned Matching Network (1)
explicit or
parasitic caps
 When shunt capacitances C1 and C2 are not negligible  doubly-tuned
matching network


Two magnetically-coupled LC tanks
Fourth-order resonator
 We derive equivalent 2nd order circuits in order to more easily evaluate:


The transformed load impedance at resonance
The equivalent loss resistance  network efficiency
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© 2020 IEEE
International Solid-State Circuits Conference
Fundamentals of Integrated Transformers:
from Principles to Applications
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Doubly-Tuned Matching Network (2)
 The loading effect of RL on the
reactive network is quantified by:
 If QS»1  two parallel resonances

Transformer behavior
 If QS<1  degenerate case


explicit or
parasitic caps
Single resonance
Impedance inverter behavior
 Network behavior depends on ratio
ξ of LC products of coupled tanks
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© 2020 IEEE
International Solid-State Circuits Conference
Fundamentals of Integrated Transformers:
from Principles to Applications
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Parallel Resonances (High-QS Case)
 The two parallel resonance frequencies are:
 Pole splitting occurs as the magnetic coupling k is increased:
ξ«1
ξ=1
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© 2020 IEEE
International Solid-State Circuits Conference
Fundamentals of Integrated Transformers:
from Principles to Applications
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Equivalent Circuit in the High-QS Case
 In the neighborhood of the parallel resonances the doubly-tuned matching
network can be approximated as a second-order circuit
ωL
ωH
 Load impedance gets transformed by factor Av21
 R’=RL/(Av21)2
 For ξ = 1  |Av21|=n regardless the value of k
Andrea Bevilacqua
© 2020 IEEE
International Solid-State Circuits Conference
Fundamentals of Integrated Transformers:
from Principles to Applications
R’(ωH) is
proportional to ξ2
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Loss Resistance at ωL
 Req models the losses of the matching network  can use to assess network
efficiency (smaller Req  lower efficiency)
 At ωL a higher magnetic coupling results in a larger Req,L
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Fundamentals of Integrated Transformers:
from Principles to Applications
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Loss Resistance at ωH
 Similarly at the higher frequency resonance we have:
 Req,H is approximately proportional to ξ2
 A larger magnetic coupling results in a smaller parallel loss resistance Req,H
 higher k results in lower efficiency
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Fundamentals of Integrated Transformers:
from Principles to Applications
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Low-QS Case
impedance
inverter
 If QS is low  C2 is negligible as it is shunted by a low RL
 Circuit degenerates and shows a single parallel resonance at
 In the neighborhood of resonance the circuit behaves as an
impedance inverter  Zin(ωS)=(Z0)2/(RL+Req,S)
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from Principles to Applications
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Design of the Matching Network
matching network
efficiency
(power gain)
High-QS Case
Low-QS Case
 More degrees of freedom but more complexity than second-order tank
 Design constraints: desired input impedance and operating frequency ω0
 Desired design goal: minimization of the power loss of the doubly-tuned
transformer matching network  want to maximize network efficiency
 High-QS case: for a given operating frequency it is more convenient to work
at the lower parallel resonance  ωL=ω0
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© 2020 IEEE
International Solid-State Circuits Conference
Fundamentals of Integrated Transformers:
from Principles to Applications
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Impedance Step-Down





Typical situation in the design of a (CMOS) power amplifier
Efficiency η is maximized if Req,L is maximized
Req,L is maximized if transformer inductances and k are maximized
Req,L is maximized if ξ is about unity (L2C2≈L1C1)
Impedance transformation ratio achieved by using turn ratio n
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from Principles to Applications
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Step-Down Design Example
Q1=Q2=20





pad capacitance
η
87%
k
0.85
L1
75pH
L2
150pH
C1
200fF
C2
100fF
30GHz 18dBm differential power amplifier with 1V supply  R’=25Ω
Assume given pad capacitance C2=100fF
Select ξ=1, maximize inductance and k
Set ωL=ω2/(1+k)1/2=ω0
Set n=1.4 to transform RL=50Ω into desired R’
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Fundamentals of Integrated Transformers:
from Principles to Applications
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Impedance Step-Up
 Often the case in inter-stage matching networks
 Large inductors and small RL  low QS case  R’≈(Z0)2/RL
 Should minimize Req,S  minimize n
 There is an optimum value of the magnetic coupling k that minimizes Req,S
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© 2020 IEEE
International Solid-State Circuits Conference
Fundamentals of Integrated Transformers:
from Principles to Applications
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Step-Up Design Example
Q1=Q2=20





η
56%
k
0.5
L1
220pH
L2
110pH
C1
166fF
C2
100fF
30GHz inter-stage network for a cascode amplifier: RL=10Ω, R’=200Ω
Set n=0.7 and select k=0.5 which is optimum if Q1=Q2
Desired impedance transformation ratio yields Z0 and thus sets L2
Adjust C1 to set ωS=ω1/(1-k2)1/2=ω0
Value of C2 is not relevant as long as it is not too large
Andrea Bevilacqua
© 2020 IEEE
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Fundamentals of Integrated Transformers:
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Broadband Matching Network
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Can use doubly-tuned transformer network to achieve broadband operation
To achieve R’(ωL)=R’(ωH)=RL/n2  ξ=1
Minimize ripple between ωL and ωΗ  |k|QS=1
The choice of k sets the bandwidth
Req,L > Req,Η but Req,Η proportional to ξ2  can equalize losses increasing ξ
Increasing ξ over unity increases bandwidth  can compensate decreasing k
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Broadband Matching Design Example
Q1=Q2=20
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k
0.85
L1
405pH
L2
810pH
C1
133fF
C2
125fF
amplifier gain
matching network
efficiency
(power gain)
Wideband 30GHz power amplifier
Maximize bandwidth  k=0.85
Set n=1.4 to transform RL=50Ω into R’=25Ω
First set ξ=1 and |k|QS=1  obtain L1, L2, C1, and C2
Then increase ξ to 1.9 by decreasing C1 to 133fF to equalize response
Passband is from 8 to 53GHz
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Transformers in Amplifiers Summary
 Transformers are key in amplifier design as they provide Galvanic isolation,
single-ended to differential conversion, power/signal combining and
impedance transformation
 Coupling inductors also enables area savings and different circuit behavior in
differential and common mode operation
 Transformers can be conveniently used in lumped-element implementations
of distributed components
 In integrated implementations magnetizing and leakage inductances cannot
be made negligible  it is imperative to embed them into the design:
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Ladder networks
Doubly-tuned networks
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Outline
 Introduction
 Physical principles
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Self and mutual inductance
Integrated transformers
 Interim Q&A session
 Transformers in amplifiers
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Baluns
Couplers and combiners
Matching networks
 Transformers in oscillators
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Doubly-tuned resonators
Varactor coupling
Multiple resonance tanks
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Oscillators with Transformer Resonator
 Use doubly-tuned network as
resonator  two possible
oscillation modes at ωL and ωH
 Connect resonator to negative
resistance  1-port oscillator
 Close feedback around
resonator  2-port oscillator
 Topologies that combine both
approaches are possible
1-port oscillator
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2-port oscillator
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Start-up of 1-Port Oscillator
Q1/Q2=1
region with
oscillations
at ωL
boundary depends
on Q1/Q2
region with
oscillations
at ωH
 Oscillations start if GmReq>1
 Req,H is proportional to ξ2 while Req,L is weakly dependent on ξ=(L2C2)/(L1C1)
 oscillations start at ωH if ξ is large enough, otherwise start at ωL
 Can select oscillation mode choosing the port where -Gm is connected
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Start-up of 2-Port Oscillator
Root locus
Gm<0
ω0=ωH
Gm>0
ω0=ωL
 Oscillations start if Av21GmReq>1
 Sign of Av21 is positive at ωL and negative at ωH  choosing the sign of Gm
(the connections of the transconductor) we select the oscillation mode
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Tank Equivalent Reactive Components
 Leq,L,H and Ceq,L,H have non-trivial expressions, but:
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Notice that Ceq,L,H is larger than both C1 and C2
For ξ=1  Leq,L,H=L1(1±|k|)/2 and Ceq,L,H=2C1
 The resonator quality factor is QL,H=Req,L,H/(ωL,HLeq,L,H)=ωL,HCeq,L,HReq,L,H
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Quality Factor of Transformer Resonator
Q1/Q2=1
Q1/Q2=1
region with
oscillations at ωH
in 1-port oscillator
ΨL
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ΨH
Resonator quality factor can be written as: QL,H=Q1(ωL,H) ΨL,H(ξ,k,Q1/Q2)
ΨL,H is symmetric in ξ only if Q1/Q2=1  if Q1/Q2=1 choose ξ=1 for best Q
If operating at single frequency ω0  Operate at ωL=ω0 for higher Q
If want to use both modes  Can get QL≈QH as Q1 increases with ωL,H
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Comparison with LC tank
parallel inductors
tank at ω0
transformer
resonator at ωL=ω0
Same Q
series inductors
tank at ω0
 Mutual inductance increases Q of transformer
resonator at ωL
 The same happens if primary and secondary
are wired as a single inductor
 No intrinsic Q improvement due to transformer
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Phase Noise
 Neglecting flicker noise, phase noise is:
transistor excess
noise factor
 Transformer resonator helps achieving a
smaller Req for a given Q
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For a target amplitude of oscillation V1
phase noise improves but current
consumption increases
1-port osc.
2-port osc.
 Can leverage Av21 to decrease excess
noise factor F in 2-port oscillator
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Varactor Coupling
varactor
bipolar
 Transformer provides low-loss ac varactor
coupling, particularly at mm-waves
CMOS
 Can use all tuning curve with single-supply
 Can use turn ratio to decrease swing on the
varactor (especially useful for pn-varactors)
 If C1«C2  ξ»1 and ωL≈ω2  tuning as in LC tank
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Using Multiple Resonances
 Using both modes of oscillation at ωL and ωH expands the tuning range
[Bevilacqua06], [Li12]
 Setting ωH to 2nd harmonic resonance decreases 1/f noise upconversion
[Shahmohammadi15], [Murphy15]
 Simultaneous resonances at fundamental and 3rd harmonic increases the
slope of the voltage waveform  class-F oscillator [Babaie13]
 Tuning ωH to 3rd harmonic enables 3rd harmonic extraction  implicit
frequency multiplication [Zong16], [Hu18]
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Example of Dual-Mode Oscillator
Phase noise at 3MHz offset
 Dual mode resonator with simple mode-selection circuitry
[Bhat19]
 40% tuning range from 25-to-38GHz
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Example of 2nd Harmonic Resonance
 4GHz inverse class-F oscillator [Lim18]
 Limits 1/f noise upconversion into phase noise
 Improves the power efficiency (1.2mW power
consumption)
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Example of Harmonic Generation
Phase noise at 1MHz offset
10GHz oscillator
30GHz buffer
 10GHz VCO with 3rd harmonic
extraction [Hu18]
 14% tuning range
 Low 1/f3 flicker corner
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Papers to See at ISSCC 2020
 Session 17 “Frequency Synthesizers & VCOs”
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17.4 “A 18.6-to-40.1GHz 201.7dBc/Hz FoMT Multi-Core Oscillator Using E-M
Mixed-Coupling Resonance Boosting”
17.9 “A 9mW 54.9-to-63.5GHz Current-Reuse LO Generator with a 186.7dBc/Hz
FoM by Unifying a 20GHz 3rd-Harmonic-Rich Current-Output VCO…”
 Session 24 “RF & mm-Wave Power Amplifiers”
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24.2 “A Reconfigurable Series/Parallel Quadrature-Coupler-Based Doherty PA in
CMOS SOI with VSWR Resilient Linearity and Back-Off PAE for 5G MIMO Arrays”
24.6 “An Instantaneously Broadband Ultra-Compact Highly Linear PA with
Compensated Distributed-Balun Output Network…”
24.7 “A 15 dBm 12.8%-PAE Compact D-Band Power Amplifier with Two-Way
Power Combining in 16nm FinFET CMOS”
 Session 29 “Emerging RF & THz Techniques”
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29.3 “Non-Magnetic 0.18μm SOI Circulator with Multi-Watt Power Handling Based
on Switched-Capacitor Clock Boosting”
Andrea Bevilacqua
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Fundamentals of Integrated Transformers:
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Summary
 Microelectronic planar processes limit the obtainable magnetic coupling k and
turn ratio n of integrated transformers
 Magnetizing and leakage inductance will not be negligible, so try and embed
them in your design
 Magnetically coupling inductors allows to decrease coil footprints and help
implement lumped element versions of distributed passive circuits
 Transformers are key to amplifiers’ design, especially at mm-waves, because
they provide low loss ac coupling, yield impedance transformation, and
enable power combining
 Transformers do not increase the resonators’ Q per se, but they are useful to
make oscillators with low phase noise and wide tuning range
Andrea Bevilacqua
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International Solid-State Circuits Conference
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Andrea Bevilacqua
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Fundamentals of Integrated Transformers:
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Andrea Bevilacqua
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20.[Padovan15] F. Padovan, M. Tiebout, K. L. R. Mertens, A. Bevilacqua and A. Neviani, "Design of
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Andrea Bevilacqua
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27.[Zong16] Z. Zong, M. Babaie and R. B. Staszewski, "A 60 GHz Frequency Generator Based on a
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29.[Bhat19] A. Bhat and N. Krishnapura, "26.3 A 25-to-38GHz, 195dB FoMT LC QVCO in 65nm LP
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30.[Lim18] C. Lim, J. Yin, P. Mak, H. Ramiah and R. P. Martins, "An inverse-class-F CMOS VCO with
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achieving 196.2dBc/Hz FOM," in IEEE ISSCC, 2018, pp. 374-376.
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