IEEE JOURNAL OF SOLID-STATE CIRCUITS, VOL. 48, NO. 9, SEPTEMBER 2013
`
`2067
`
`A Multiband RF Antenna Duplexer on CMOS:
`Design and Performance
`
`Mohyee Mikhemar, Member, IEEE, Hooman Darabi, Senior Member, IEEE, and Asad A. Abidi, Fellow, IEEE
`
`Abstract—An RF duplexer has been fabricated on a CMOS IC
`for use in 3G/4G cellular transceivers. The passive circuit sustains
`large voltage swings in the transmit path, and isolates the receive
`path from the transmitter by more than 45 dB across a bandwidth
`of 200 MHz in 3G/4G bands I, II, III, IV, and IX. A low noise am-
`plifier embedded into the duplexer demonstrates a cascade noise
`figure of 5 dB with more than 27 dB of gain. The duplexer inserts
`2.5 dB of loss between power amplifier and antenna.
`Index Terms—Antenna tuning unit, autotransformer, balancing
`network, bridge network, CMOS, diplexer, duplexer, electrical
`balance, FDD, full-duplex, hybrid transformer, isolation, noise
`matching, reciprocal circuit, transmitter leakage, 3G.
`
`I. INTRODUCTION
`
`M ULTIBAND operation is today a de facto requirement
`
`for all commercial cellular handsets. A typical 2G/3G
`cellular transceiver as shown in Fig. 1 covers seven frequency
`bands to encompass four 2G bands and three 3G bands. Al-
`though the RF transceiver is integrated on a single CMOS chip,
`it needs four external SAW filters and twelve matching compo-
`nents. While there is progress in miniaturization, such as inte-
`grating the two power amplifiers (PA)s for 2G into one module,
`the numerous off-chip filters and duplexers continue to hand-
`icap this multiband approach in cost and area. With the advent
`of 4G, this approach will likely become impractical.
`A multiband transceiver is needed along the lines of Fig. 2,
`which may be thought of as an antenna-ready radio with every-
`thing integrated on a single CMOS chip. It consists of the ex-
`isting transceiver integrated with a multi-mode multi-band PA,
`all necessary filters, and duplexers [1]. A multi-mode, multi-
`band power amplifier is in the research phase [2], [3]; a multi-
`band RF filter for 2G operation was demonstrated in [4], [5];
`and a multi-band duplexer is now described here.
`
`Fig. 1. A simplified block diagram of the RF board in a 2G/3G cellular phone.
`
`Fig. 2. True multi-band multi-mode radio transceiver.
`
`II. SYNTHESIS OF THE MULTIBAND DUPLEXER
`The duplexer is a network with three ports to which are con-
`nected, respectively, the antenna, the transmitter output, and the
`receiver input as shown in Fig. 3(a) [6]. For concurrent full-du-
`plex operation, the network should ideally isolate the transmitter
`
`Manuscript received November 11, 2012; revised April 05, 2013; accepted
`April 30, 2013. Date of publication June 20, 2013; date of current version Au-
`gust 21, 2013. This paper was approved by Associate Editor Jan Craninckx.
`M. Mikhemar was with the Electrical Engineering Department, University of
`California, Los Angeles, CA 90095 USA, and is now with Broadcom Corpora-
`tion, Irvine, CA 92617 USA (e-mail: mohyee@broadcom.com).
`H. Darabi is with Broadcom Corporation, Irvine, CA 92617 USA.
`A. A. Abidi is with the Electrical Engineering Department, University of Cal-
`ifornia, Los Angeles, CA 90095 USA.
`Color versions of one or more of the figures in this paper are available online
`at http://ieeexplore.ieee.org.
`Digital Object Identifier 10.1109/JSSC.2013.2264626
`
`(TX) output from the receiver (RX) input; convey the available
`output power from the PA to the antenna (ANT); and transfer
`the voltage induced on the antenna to the receiver input with al-
`most no attenuation. The theory of duplexers has been studied
`extensively, and it is known that the gyrator makes an ideal du-
`plexer with constant driving point resistance at all ports [7].
`In 3G wireless systems, the duplexer should isolate the RX
`from the PA by 50 dB or more to prevent saturation of the re-
`ceiver or damage to the LNA input. Furthermore, it must be able
`to withstand TX voltages as large as 15 V. On both these counts,
`a passive realization seems to be the most promising. An ac-
`tive feedforward cancellation is described in [8], where an LMS
`adaptive filter produces an out-of-phase replica of the TX wave-
`form that is subtracted from the LNA output, and while this ar-
`rangement resembles the isolating function of the duplexer, it is
`not a substitute.
`
`0018-9200 © 2013 IEEE
`
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`IEEE JOURNAL OF SOLID-STATE CIRCUITS, VOL. 48, NO. 9, SEPTEMBER 2013
`
`Fig. 3. Duplexer filtering requirements for 3G/4G FDD operation.
`
`How are the three-port duplexers constructed that are used in
`today’s full-duplex mobile telephones? SAW duplexers [9], [10]
`operate in well-defined non-overlapping narrow bands of fre-
`quency for TX and RX, while providing an approximately con-
`stant resistance at the TX port. They consist of RX and TX SAW
`bandpass filters with a steep inter-band transition, so that the RX
`filter presents a small input reactance across the TX sub-band.
`An integrated
`line1 then transforms this to a large reactance
`at the TX filter output where the antenna is also connected. Thus,
`across the TX band, the SAW filter connected to the PA output
`is terminated essentially only by the antenna impedance.
`
`III. HYBRID TRANSFORMER
`The hybrid transformer’s roots stretch back to the earliest
`years of telephony [11]–[13]. In the pre-electronic telephone
`handset it served to isolate the microphone from the earpiece,
`enabling signals on a pair of wires at each transducer to travel
`bidirectionally on a two-wire loop to the central office, while
`suppressing crosstalk from microphone to the headset.
`Since the hybrid transformer is a four-port, its analysis can
`become complicated when tackled without a guiding intuition
`such as Friedheim’s heuristic approach [14], which helps in ex-
`tracting the circuit’s essential properties more straightforwardly.
`A hybrid transformer is, at its heart, a bridge circuit with cer-
`tain useful null, or conjugacy, properties [12], [15]. We start with
`an analysis of a symmetrical hybrid circuit. The autotransformer
`hybrid consists of a coil with a terminal at the center tap as well
`as at the two ends Fig. 4(a,b) [12], [16]. Ports are defined at the
`coil’s three terminals with respect to a separate common node,
`or ground, and a fourth (floating) port is defined by the terminals
`at the coil’s two ends. The fourth port could equally well be de-
`fined by the terminals of a second, closely coupled, coil which
`links the magnetic flux of the first coil; this is the classic hy-
`brid transformer Fig. 4(c) [12]. A discrete version of the hybrid
`transformer has recently been demonstrated as an RF duplexer
`in the 800 MHz band [17]. In this paper we describe the first
`realization of an autotransformer hybrid constructed on CMOS
`for RF duplexing in the 2 GHz band.
`
`A. Ideal Hybrid Autotransformer
`In keeping with the circuit’s eventual use, we label its ports
`ANT, TX, BAL, and RX (Fig. 5). Assume that equal value re-
`sistors
`are attached between the ANT and BAL ports and
`common, a resistor
`is attached across the RX port, and
`some other
`across the TX port. Stimuli may be introduced
`
`1The physical length of the line is set by the very short acoustic wavelength.
`
`Fig. 4. Duplexer coil configurations a) autotransformer with single-ended RX
`b) autotransformer with differential RX c) transformer with differential RX and
`common-mode rejection.
`
`Fig. 5. Hybrid autotransformer model for a) TX mode b) RX mode.
`
`as one or more independent voltage sources in series with the re-
`sistor branches, or independent current sources in parallel with
`them. When subject to multiple stimuli, the response of this
`linear circuit is the superposition of individual responses.
`Suppose a series voltage
`is inserted in series with the re-
`sistor connected to the TX port (Fig. 5(a)). In the spirit of [14],
`we analyze as follows. The symmetry of the circuit suggests
`that equal currents flow in the resistors connected to the ports
`TX and BAL. But since the currents flow from the center tap
`to the two ends of the coil, they will create equal but opposite
`magnetic fluxes that cancel to induce zero voltage across the RX
`port. Since the voltage across the coil is zero, an equal voltage
`appears across the equal valued resistors in the ANT and BAL
`branches, verifying the hypothesis that they are carrying equal
`currents. The circuit must have this as its unique solution, and
`we have established that the RX port is isolated from the stim-
`ulus applied to the TX resistor; in other words, that the TX and
`RX ports are conjugate. This term is a useful reminder that con-
`jugacy is a reciprocal property; we will put it to use in later anal-
`ysis. When we view the resistors at the ANT and BAL ports as
`elements in two arms of a bridge, where the two coupled halves
`of the coil comprise the other two elements, we recognize that
`isolation arises from the null at the balance of the bridge. In this
`ideal case the bridge balance depends only on equal impedances
`being present at the ANT and BAL ports, and therefore TX will
`remain isolated from RX at all frequencies.
`Viewed in a different way, we can say that the hybrid au-
`totransformer isolates TX from RX by presenting the transmit
`waveform in common-mode at the two RX terminals, with re-
`spect to the common terminal which we will call ground. And
`as we will now show, it conveys the antenna voltage to these
`terminals in differential mode, as is desired.
`
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`MIKHEMAR et al.: A MULTIBAND RF ANTENNA DUPLEXER ON CMOS: DESIGN AND PERFORMANCE
`
`2069
`
`(Fig. 5(b)).
`Consider a stimulus voltage in series with
`Since a bridge circuit is unchanged when its circuit diagram is
`rotated by 90 , it is reasonable to expect that this port too will
`have its own conjugate at the BAL port opposite. This is indeed
`the case, a property called biconjugacy [7], [12]. This means
`that the voltage at the ANT terminal will drop entirely across the
`coil, leaving zero volts at BAL2. Since the coil is center-tapped,
`KVL in the BAL -TX loop implies that half the voltage must
`drop across the resistor
`.
`Owing to the configuration of resistors and the constraints
`imposed by an ideal autotransformer, biconjugacy requires that
`the termination resistances must have specific ratios. When this
`symmetrical hybrid coil is wound on a core of infinite perme-
`ability, the currents entering the dots in the two halves of the
`coil must cancel, whatever the coil voltage. This means that if
`a current
`flows through
`it can only complete the loop
`if a current
`flows through
`and returns through either
`the ANT or the BAL terminals. Similarly, when driven from
`the ANT port, the voltage across
`is 2
`the voltage across
`. It follows that to fulfill these conditions simultaneously,
`the two resistors must be related as
`
`(1)
`carries no current under ANT stimulus, it might
`Since
`seem that it can take any value; but only a particular value will
`isolate RX in transmit mode. Indeed several considerations de-
`termine the choice of balance resistor
`:
`1) It must balance the bridge so that the TX and RX ports are
`conjugate.
`2) It must enable extraction of all the available power3 from
`the voltage source attached to TX.
`3) It must consume some small fraction of this power, al-
`lowing the rest of it to flow into the ANT port.
`In a symmetrical bridge where the hybrid coil is tapped at its
`center, TX-to-RX isolation requires that
`. This,
`as we have shown, leads to the constraint that
`.
`If this constraint were not met by the inherent values of
`(which is usually 50
`) and
`(which is the PA’s output re-
`sistance), an ideally lossless impedance transformer would be
`employed to scale
`until the constraint is met. In the end,
`though, only half the available power from the transmitter would
`arrive at the ANT port, while the other half would be lost to the
`BAL port. In most cases of interest here, this sacrifice in TX
`power is not acceptable.
`Sartori [12] has shown that by shifting the tap off-center,
`power may be diverted away from the BAL to the ANT port.
`Our analysis of the asymmetrical circuit derives figures-of-merit
`that are better suited to the RF application, using arguments that
`we believe are more fundamental and conceptual.
`Since in general
`, the power transmission from
`TX to ANT should be characterized by transducer power gain
`. This is defined as [18], [19]
`
`(2)
`
`where available power from a source is defined in footnote 3.
`When the transmitter is connected through a lossless matching
`network to the antenna,
`: since no active ele-
`ments are involved, this is the highest possible power gain. Now
`when the transmitter couples to the antenna through the hybrid
`transformer, then owing to loss in
`it must be that
`.
`When the hybrid is symmetrical and the resistors are related as
`described above,
`.
`We will now see how asymmetry in the hybrid coil can raise
`towards (but never equal to) the maximum value of 1. Let
`the number of turns in the coil at the ANT end be
`, and
`at
`the BAL end. To isolate RX from TX, bridge balance requires
`that
`
`(3)
`
`At balance, the voltage will be equal at all three terminals, ANT,
`TX, and BAL, which means that the three resistors to ground
`appear all in parallel. Then to extract the available power from
`TX it is required that
`
`(4)
`
`is intrinsic to the
`signifies conductance. Since
`where
`antenna, which through (3) also determines
`, the TX PA’s
`source resistance should be scaled through a lossless impedance
`transformer to satisfy the equality in (4). From now on,
`will refer to this transformed resistance. With simple analysis,
`this implies a transducer gain of
`
`(5)
`
`As a check, in the symmetric hybrid where
`.
`this expression gives the correct value of
`To apply it to the asymmetric hybrid, we will insert into (5) the
`balance condition from (3) and the impedance match condition
`from (4). Then with straightforward manipulations we see that
`for maximum transducer gain the various conductances must be
`related as
`
`and this maximum gain is
`
`(6)
`
`(7)
`
`Since TX and ANT ports are individually matched4, then (7)
`also specifies the available gain
`between these two
`ports5. To reach the maximum available gain of 1 requires that
`. This stands to reason since in the limit of infinite
`turns ratio, the condition imposed by (3) will force
`which means that no power will be lost to the now unloaded
`BAL port.
`
`4Not necessarily to each other.
`is the driving point impedance into the
`ANT port, and
`.
`at the TX port, but
`5Available gain [20] is the ratio of the available power at the output port to
`the available power from the source driving the input port.
`
`2An ideal transformer can support a voltage across a winding, even when no
`current flows through it.
`3Available power from
`quantity.
`
`, where
`
`is an RMS
`
`is
`
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`
`For good TX efficiency, the circuit designer must select some
`reasonable turns ratio in the asymmetric hybrid. What will then
`be the consequence on RX operation? The appropriate figure-of-
`merit to judge this is noise factor at the RX port with the input
`at the ANT port. The larger the noise factor above 1, the less
`sensitive is the receiver.
`A clear understanding of Friis’ definition of noise factor [20]
`simplifies calculation. In this loaded hybrid coil, as in any re-
`ciprocal circuit, the noise factor between two ports is merely
`the inverse of the available gain from the input to output port.
`We wish to know the available gain from ANT to RX. Available
`gain is a property of the Thevenin equivalent of the input source,
`and of the Thevenin equivalent at the output port with the input
`source attached: it does not depend on the load attached to the
`output port [19].
`If
`is disconnected from the RX port leaving two open
`terminals (Fig. 5(b)), then for conjugacy to hold to the \bal
`port the open circuit voltage between these terminals must be
`(the other half drops across
`). The impedance
`between the terminals of the RX port when
`must
`now be found to complete the Thevenin equivalent. But when
`RX is made conjugate to TX, to balance the bridge a voltage at
`TX produces zero voltage across RX, even when the terminals
`at the RX port are open. It then follows from reciprocity [21,
`Ch.16, Fig.4.4] that a current source attached to the RX termi-
`nals will induce zero current in the TX terminal. Since the cur-
`rent at the TX terminal must be the sum of the currents flowing
`into the tap from the two parts of the coil, these currents must
`each be zero. Therefore, the impedance across the RX terminals
`is
`in series with
`, which is the only available path
`for current to flow between the terminals.
`We now have the necessary information to calculate avail-
`able gain. The power available from the antenna source
`(RMS) is
`
`(8)
`
`and from the immediately preceding calculations, the power
`available at the RX port is
`
`(9)
`
`where we make use of the relation (3) that must be satisfied
`for conjugacy. Thus the available gain from ANT to RX is given
`by the ratio
`
`and its inverse gives the noise factor from ANT to RX,
`
`(10)
`
`(11)
`
`The design tradeoff is now clearly before us. For efficient
`transmission, the available gain from TX to ANT should ap-
`proach its maximum of 1, which (7) tells us is a consequence
`
`of a large turns ratio
`. But (11) shows that the noise
`factor at the RX port rises with this ratio, leading to a worse re-
`ceiver. The two dependencies are contrary. Why? Because for
`low loss in transmission the balancing resistor must be chosen
`very large (and the turns ratio adjusted according to (3)), but
`then its large noise voltage appears at RX with unity gain, over-
`whelming the antenna’s noise voltage that arrives there accom-
`panying the signal. Noise from the balancing resistor thus de-
`grades the signal-to-noise ratio at RX.
`
`IV. ANALYSIS OF ON-CHIP DUPLEXER
`An asymmetric autotransformer hybrid coil is to be realized
`on a CMOS chip and evaluated as a wideband duplexer. This
`coil is a planar inductor tapped at some point in its middle. It is
`a simpler structure than a classic hybrid transformer, which re-
`quires a second closely coupled planar coil that will add to the
`total parasitic loss through its winding resistance. Electromag-
`netic field simulations are used to design and optimize the in-
`ductor geometry for lowest loss and a sufficiently high self-res-
`onance, and to extract from this geometry an equivalent circuit
`for simulations.
`An on-chip realization will depart in significant ways from
`the properties of the ideal hybrid coil duplexer that we have
`analyzed in the previous section.
`1) Since there is no magnetic core in the coil, its magnetizing
`or self-inductance will be determined by the designer to
`some value on the order of nanohenries; it will certainly
`not be infinite.
`2) The ports will be terminated no longer by pure resistances,
`but by a complex and variable antenna impedance [22] and
`interconnection parasitics, and in the case of the RX port
`in our prototype circuit, by a pure capacitance. This will
`destroy biconjugacy.
`3) While the ideal duplexer maintains isolation between TX
`and RX through bridge balance at all frequencies, in prac-
`tice due to quite different impedances in the bridge arms
`(one containing the off-chip antenna, the other an on-chip
`balancing network), bridge balance and the ensuing null
`will typically hold only in a narrow band around one fre-
`quency.
`4) Transmission from ANT to RX, on the other hand, is a
`straightforward broadband transfer function independent
`of a null, that rolls off at high frequencies due to parasitic
`capacitances.
`While in the ideal duplexer simple, well-defined expressions
`specify the available gain from transmitter to antenna and the
`associated RX noise factor (and as we will recall, neither of
`which is 1 in spite of the ideal transformer), resistor losses in
`the windings of the on-chip coil and other losses will degrade
`both these quantities by amounts that can only be predicted with
`accurate field simulations that model the various parasitics.
`
`A. Equivalent Circuit of On-Chip Duplexer
`A coil with a center tap (Fig. 6(a)) is described completely,
`except for parasitic resistance and capacitance, by a T-network
`of three uncoupled inductors (Fig. 6(b)). Two inductors are of
`value
`and
`, and the third is negative,
`
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`MIKHEMAR et al.: A MULTIBAND RF ANTENNA DUPLEXER ON CMOS: DESIGN AND PERFORMANCE
`
`2071
`
`shifting [23], but because of limitations of space the steps cannot
`be included here. It concludes with
`
`(12)
`
`where
`is the total capacitance at the ANT port. This is the
`bandpass transfer function of an
`resonant circuit, with a
`peak value of 1 at its resonant frequency
`.
`The coil inductance should be chosen so that this resonant fre-
`quency is centered roughly on the band of interest, 2 GHz in
`our case. At a few GHz,
`which means that
`the quality factor of the resonant circuit is roughly 1 and the
`transfer function presents a very broad peak with no need for
`accurate tuning. This is a good approximation to the ideal auto-
`transformer, whose transfer function is 1 at all frequencies.
`This hybrid coil is not biconjugate, because its RX port is
`terminated by a capacitor, not by the resistor specified in (1).
`Therefore, the driving point impedance at the ANT port is a fre-
`quency-dependent complex quantity. We show in Section V-B
`that it is in fact a constant resistor over the frequencies of in-
`terest with a very small reactive part. Section VI presents sim-
`ulations and measurements of the reflection coefficient at this
`driving point.
`
`C. TX-to-RX Isolation
`The duplexer relies entirely on the null arising from bridge
`balance to isolate the RX port from the power amplifier that
`drives the TX port. Consequently, small amounts of imbalance
`can cause an unacceptable feedthrough. Therefore, great care
`must go into balancing the two arms of the duplexer over a
`frequency range that spans the transmit as well as the receive
`bands, since the phase noise spectral tail of the large transmitted
`signal falls in the receive band and raises the noise floor.
`Unbalance in the duplexer bridge arms arises mainly from
`unpredictable antenna impedance and interconnect cables. Para-
`sitics of connection to and from the chip will play a smaller role.
`The duplexer is a single tapped planar coil, but since the tap is
`off-center, the inductance of the two coil segments,
`and
`,
`must be found accurately from electromagnetic simulations or
`from experimental characterization of the coil as a two-port net-
`work. Parasitic capacitance to substrate will produce frequency
`dependence in
`and
`.
`Let
`signify the
`Norton equivalent admittance connected to the ANT port, and
`the admittance at the
`balance port. In our circuit the balance network is a digitally
`selected array of resistors in shunt with a capacitor array, and
`when inductance to the ground plane is included even
`will acquire a small frequency dependence. Isolation from TX
`to RX requires the following two conditions to hold:
`1) At a given transmit carrier
`frequency
`there
`,
`exists a digital selection for
`force
`that will
`, where
`is the attenua-
`tion necessary to provide the desired isolation.
`2) If at the same time the receiver is tuned to a carrier fre-
`quency
`, then
`there also.
`
`Fig. 6. Hybrid autotransformer model with a) practical autotransformer model
`b) autotransformer equivalent circuit.
`
`in the three expressions arise from
`. The signs of
`of value
`the relative sense of the windings as shown by the dots.
`and
`is the self-inductance between one of the end terminals of
`the coil and the center tap, when the terminal at the other end is
`left floating.
`At the RX port, our on-chip duplexer drives a low-noise am-
`plifier whose input is purely capacitive. The loading effect is
`modelled by two equal capacitors connected from the ANT and
`the BAL terminals to ground. The antenna attaches to the chip
`through a bonding pad, whose capacitance
`appears in par-
`allel. The loss resistance and capacitance to substrate of the
`on-chip coil are found to have only a small effect, and are ig-
`nored.
`the an-
`is good enough to predict
`This simple circuit
`tenna-to-RX transmission function, whose analysis now
`follows. But the TX to RX isolation analyzed in the subsequent
`section depends on bridge balance, which is sensitive to small
`variations in the two bridge arms. Then the equivalent circuit
`must be supplemented, as we show in Section V-C, with stray
`inductances and other parasitic elements.
`
`B. Antenna-to-RX Transmission
`
`Fig. 6(b) shows the equivalent circuit that gives the transfer
`function from the antenna, which is modelled nominally as a 50
`voltage source, to the RX port whose open circuit voltage ap-
`pears at the gates of two FETs. Since this circuit must also iso-
`late the RX port from the transmitter, bridge balance requires
`that the admittances of the branches on the ANT and BAL ports
`are in the same ratio as
`, which we call the
`skew factor of the asymmetric hybrid autotransformer. In prac-
`tice this constraint is satisfied at the BAL port by choosing
`. When the antenna and interconnect are mod-
`elled as a shunt
`network at the duplexer’s ANT port, a bi-
`nary capacitor array in parallel with
`to obtain balance by
`adjusting
`,
`When the antenna appears inductive as electrically small an-
`tennas will, the inductance is tuned with a second capacitor array
`attached to the ANT port; otherwise an inductor array would be
`needed at the BAL port, which is impractical. The loading effect
`of LNA input capacitance is dealt with in a later section. The
`analysis that follows uses theorems of reciprocity and source
`
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`
`The first condition amounts to determining the balance
`network’s ability to reach a null, given spreads in the antenna
`impedance; and the second condition specifies that the null
`must attenuate over a wide enough bandwidth to encompass
`both transmit and receive frequencies. We will deal with these
`requirements one at a time.
`1) Isolation in TX Band: In the manner of (3), the balancing
`conductance must follow the relation
`
`(13)
`
`To account for variations in the antenna conductance,
`must be made adjustable through an array of trimming resistors.
`This specifies a certain range and resolution. Range must be
`wide enough to encompass the full spread of
`, while res-
`olution limits how small the attenuation can be. In an array of
`resistors, range and resolution together will dictate the number
`of bits.
`
`arises from antenna reactance, cables, connectors,
`board traces, the chip package and the LNA, and must be
`compensated in the balance network by a digitally trimmable
`capacitance (Fig. 6(b)) obeying the inverse relation,
`
`frequency. A key aspect of design is to predict this bandwidth.
`Parasitic reactances such as bondwire inductance that appear in
`the balance network will introduce a frequency dependency in
`the real and imaginary parts of the balancing admittance, i.e.
`. These various considera-
`tions determine the isolation bandwidth that is finally achieved.
`The following analysis shows a method to estimate this band-
`width. Ignoring some large terms that change slowly with fre-
`quency, the TX to RX transfer function is given to within an
`order of magnitude by
`
`(15)
`
`Suppose that under digital control,
`close to a wireless
`has enough range that at a frequency
`channel of interest, it can make this transfer function zero7. The
`question is: as frequency is now swept but the control words are
`frozen, how rapidly does the transfer function depart from zero?
`Or, more precisely, given some
`, over what frequency in-
`terval surrounding
`is the magnitude of the right-hand side of
`(15) less than ?
`This interval will always be small compared to
`. So using
`a series expansion around
`and retaining only the first term,
`
`(14)
`
`(16)
`
`If this were a complete description, then with sufficient reso-
`lution in the shunt
`network the duplexer could be balanced
`to isolate TX from RX over all frequencies. But the model is in-
`adequate for two reasons. First, reflections from nearby objects
`will cause the antenna to appear reactive as well as (radiation)
`resistive, and what is worse, its reactance will change as the mo-
`bile wireless device moves in its environment [22]. If this is to
`be balanced by a shunt
`network, then, in general, the vari-
`able span of
`and
`must be widened to cover likely
`eventualities.
`Today’s transceivers employ an antenna tuning unit (ATU)
`that estimates and tracks the changing impedance by measuring
`the VSWR in the antenna feed cable. The ATU has been a sub-
`ject of research for many years [24], [25], and has matured to the
`point today [26] that it is used in mass-produced handsets to ad-
`just antenna VSWR to 2:1 or less at all times, thus lowering the
`average PA current drain. We will show later how the balance
`network as realized on our prototype is good enough to give the
`desired isolation in a mobile handset, provided a state-of-the-art
`commercial ATU is present on the antenna side of the duplexer6.
`2) Isolation in RX Band: The bridge will balance exactly at
`only one frequency because the antenna-side admittance com-
`prises, in general, two frequency-dependent terms
`. However, it can continue to provide useful isolation
`over a bandwidth of tens of MHz or more surrounding the null
`
`6This is not a weakness of this RF duplexer, but is inherent in any method of
`isolation that relies on balancing a bridge. The most widespread use of the hybrid
`transformer to date was in pre-electronic telephone sets, where a temperature-
`dependent resistor (varistor) in the balance network adapts to the typically large
`spreads in resistance of incoming loops [13]
`
`The same linear approximation will apply to the imaginary parts
`of admittance,
`and
`. Then, in the vicinity of
`,
`(15)simplifies to
`
`(17)
`
`The isolation to RX worsens linearly with frequency offset, at
`a rate set by the frequency derivative of the ratio of the real parts
`and the ratio of the imaginary parts of the admittances at the
`ANT and BAL ports. If these ratios are constant with frequency,
`that is, if the network at the BAL port is an admittance-scaled
`replica by
`of the network on the ANT port, the null will hold
`for all frequencies. Since it is almost impossible to make these
`two networks identical, both derivatives of ratios in (17) must
`be kept small by careful design if we are seeking a large attenu-
`ation across a useful bandwidth8. We will use this expression to
`predict the isolation bandwidth in our experimental prototype,
`and show that it is close to measurement.
`
`7Without well-thought out networks attached to the BAL port and a capacitive
`trim network at the ANT port, it may be impossible in practice to reach the null
`condition when parasitic elements are included.
`8For a correct understanding of this null, it should be noted that the -domain
`transfer function from
`to
`contains a pair of complex conjugate poles
`that are found by replacing
`in (12) by the complex frequency , as well as
`a pair of complex conjugate zeros. With sinusoidal stimulus, if the zeros lie on
`the
`-axis of the -plane a true null will be observed. However, if the zeros lie
`less than a distance away from the axis, while a null is not obtained the desired
`attenuation is seen over some non-zero bandwidth.
`
`Farmwald and RPX Exhibit 1070, pg. 6
`Farmwald and RPX v. ParkerVision
`IPR2014-00948
`
`

`

`MIKHEMAR et al.: A MULTIBAND RF ANTENNA DUPLEXER ON CMOS: DESIGN AND PERFORMANCE
`
`2073
`
`coil’s trace resistance, likely caused by skin effect at these fre-
`quencies. In a practical transceiver operating within a cellular
`infrastructure, a commonly used tradeoff is to strive to improve
`TX efficiency–since it limits battery life–at the expense of RX
`sensitivity. Accordingly we position the tap on the hybrid coil
`away from the center towards the ANT port so that in the equiv-
`alent circuit,
`,
`, and
`.
`This is more meaningful than to specify the location of the tap as
`a ratio of two non-integer numbers of turns. Then
`and
`, so the skew factor
`.
`Neglecting losses in the coil, (18) predicts that
`and
`. Full electro-
`magnetic simulations that include metal and substrate losses and
`imperfect coupling between the coil segments show that these
`two quantities will be slightly worse, 2.5 dB and 4.6 dB, respec-
`tively. Substrate loss is kept low by constructing the coil in the
`uppermost layers of metal [27].
`
`B. LNA Design
`The antenna drives the duplexer at the ANT port through
`an ATU to correct f