ChapterVII MIXERSandDETECTORS - RF toolboxrftoolbox.dtu.dk/book/Ch7.pdf · VII-1 Mixer Basics 3 characteristic in Fig.1(b). With both input signals set to zero, the transistor DC

Class Notes, 31415 RF-Communication Circuits

Chapter VII

MIXERS and DETECTORS

Jens Vidkjær

NB235

ii

Contents

VII Mixers and Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1VII-1 Mixer Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

A Prototype FET Mixer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Example VII-1-1 Square-Law FET mixer . . . . . . . . . . . . . . 7Example VII-1-2 Square-Law FET mixer - continued . . . . . 9

VII-2 Differential Stage Mixers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15VII-3 Gilbert-Cell Mixers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

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VII Mixers and Detectors

The fundamental operation of a mixer is to multiply two signals. One of them is oftena modulated signal and the other a stable local oscillator output, and the purpose of the pro-cess is to shift the carrier frequency in the modulated signal while keeping the informationcarrying envelope - real or complex - intact. If one of the carriers is at zero frequency, DC,this means that we are either shifting from or to baseband signals. In that case the circuits mayalso be called modulators and demodulators respectively. Demodulator circuits are sometimesincluded in a group of circuits known as detectors, although this term also comprises caseswhere baseband information is extracted from a modulated signal without guidance from alocal oscillator. Finally, circuits that shifts from one carriers frequency to another are some-times also called frequency converters.

Mixing processes were considered from a system point of view in sec.I-5. It was seenthat even an ideal multiplication process had undesirable effects that must be dealt with arounda multiplying circuit. There was removal of either unwanted sum or unwanted differencefrequency signal components by post filtering. Protection against image responses by pre-filtering was another problem. In this chapter we shall present more details about circuitimplementations and consider limitations on ideal performances due to noise and non idealcomponents.

This chapter is preliminary. You may find additional information on the subject inref’s [1],[2], or [3].

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2

VII-1 Mixer Basics

Multiplying voltage or current waveforms is a nonlinear process. A simple exampleis the resultant response if two signals are applied to a nonlinear device. We have already seensome consequences in the discussion of limitations in almost linear circuits in section VI-4.There, considerations were made from the point of view, that the mixing process deteriorateslinear performance by introducing distortion and intermodulation signal components atfrequencies, which may differ from the input signal frequencies. Dealing with mixers,however, we shall now organize the circuits to provide the best possible translation from onefrequency range to another.

A Prototype FET Mixer

In the preceding section IV-4 we saw how signal distortion was introduced when

Fig.1 FET transistor that is biased and driven for mixer applications. Figure (b) shows asimplified transistor transfer characteristic.

device characteristics departed from linearity. In mixers it is the same nonlinear devicecharacteristics that are used to give the desired frequency translations. To fix ideas, Fig.1demonstrates in principle how a FET may be employed as a mixer when two input signals areapplied to the gate. We assume that the transistor is biased, driven, and loaded in a way thatprevents saturation effects, so its operation may be described solely by its nonlinear transfer

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3VII-1 Mixer Basics

characteristic in Fig.1(b). With both input signals set to zero, the transistor DC bias conditionsare fixed by

where the bias current to the drain flows through the RF-choke Lchk. VGG is the gate bias

(1)

voltage from the battery. When signals are applied, we suppose that the corresponding currentcomponents are related to the input signal through a Taylor series expansion of the transfercharacteristic around the gate bias voltage VGG. Voltage vg denotes the deviation voltageaccording to

Following the notation that was introduced in section IV-4, the drain current, when the two

(2)

time varying signals Vrf and Vlo are applied, is expressed

Inserting the two signal components, where

(3)

it is seen, that the second order term in the Taylor expansion provides a drain current

(4)

(5)

component, which is proportional to the product of the two input signals ,Vrf and Vlo. If welet the RF signal be represented by a single tone so both applied signals are sinusoidal,

the generation of frequency components in the drain current follows basically the scheme that

(6)

formerly gave the distortion components in Eq.IV-136 for Taylor expansion components upto third order. In present terms this equations reads,

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4 Mixers, Modulators, Demodulators, and Detectors

This result required repeated use of the following basic relations

(7)

Had we proceeded to even higher order of terms, recursive use of the last identities would

(8)

reveal that each order of expansion terms introduces components where, in double sidedrepresentation based on the identity,

the frequencies become all possible combinations that may be written by

(9)

(10)

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5VII-1 Mixer Basics

The resultant frequency from this process may be simpler than the expression above suggestsat a first glance. For instance, the left term of the third order components in Eq.(7)(c1)develops as

Here the resultant frequency keeps no indication of the Taylor expansion order in the non-

(11)

linear mixing characteristic, N=3, that caused this particular frequency component. Note,however, that the amplitudes hold information about the order. In the example above, the twocontributions are both of third order in voltages, VRF3

3 and VRFVLO2 respectively.

A common way of expressing the resultant frequency components from a mixingprocess in literature is1

According to the discussion above, there is no simple one to one relationship between the

(12)

integer set n,m and Taylor expansion order(s) for the mixing nonlinearity. The only point thatcan be states that a mixing frequency given by Eq.(12) cannot originate from an expansionterm of order less than |n|+|m|.

Returning to basic mixing properties, the product term from Eq.(4) provides both thesum and the difference frequency drain current components in Eq.(7)(b2). Which one to usedepends on the application at hand. In a receiver the typical mixer function is to shift thefrequency region of an incoming RF-signal down to a lower intermediate frequency, IF, range.Had we used the diagram in Fig.1 directly we would get output voltage components at allcombination frequencies in the Taylor expansion of the current. A simple way to sort out thedesired component is indicated by Fig.2, where the load of the FET is made by a parallelcircuit that is tuned to IF at the difference frequency

If the Q-factor for the tuned circuit is high enough, the IF current component provides the

(13)

dominant contribution to the output voltage while all other current components are practicallyshort circuited. In this case, the output voltage is given by the IF component Id,if in the draincurrent,

(14)

Consult ref’s [1], [2], or [3], for more elaboration on this subject.

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Like the case of distortion in almost linear circuits, residual signal components at all but the

Fig.2 Simple FET transistor mixer, where the parallel circuit L, C is tuned to the IFfrequency.

desired IF frequencies are called spurious signals.

Sometimes, the ratio of the current amplitude at the desired output frequency, herethe IF, over the input RF voltage amplitude is called the conversion transconductance2. Theratio of the desired frequency output voltage amplitude over the input RF voltage amplitudeis correspondingly called the conversion voltage gain or just the conversion gain if it is clearfrom the context that we are dealing with voltages, not powers. In the present case we have

Introducing conversion quantities, we may now express

(15)

(16)

(17)

2 ) The Danish term is "blandingsstejlhed".

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7VII-1 Mixer Basics

so Gcnv and Acnv are used the same way in amplitude calculations as the traditional transcon-ductance and voltage gain factors are used in conventional circuit characterization. It must bekept in mind, however, that conversion transconductances and conversion gains relate currentand voltage components of different frequencies and, furthermore, these quantities dependsnot only on simple device and DC biasing properties, but also on the local oscillator signal,VLO. In order not to distort the information in a modulated signal envelope when the carrierfrequency is changed by a mixer, the conversion transconductances and gains are supposedto be independent of the RF signal level, VRF. Like transconductance and gain, concepts likecompression, intermodulation, or intercept points are inherited from the almost linear amplifierterminology to characterize distortions introduced in mixers, now with the complication, thatthe input and output frequency ranges of significance are no longer coincident.

Example VII-1-1 Square-Law FET mixer

In ideal form the transfer characteristic of a n-channel FET that is biased for normaloperation is a part of a parabola,[4]. It ranges from zero current at the so-called pinch-offor threshold voltage VP - a negative voltage - and the saturation current IDSS at zero gate-source voltage as expressed by

First and second order derivatives of the drain currents are

(18)

and all higher order derivatives are zero, so the Taylor expansion in Eq.(3) for a given gate

(19)

biasing, VGG, is given through

As seen, the second order b-term is independent of the bias voltage VGG, i.e. constant across

(20)

the whole gate voltage range where the transistor conducts current. In this range the b termdetermines the conversion transconductance, which becomes

The maximum conversion transconductance limit in the expression is approached if the RF

(21)

signal is small compared to the local oscillator signal, which is set to the maximum amplitude,

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where the transistor is conducting all times. That is VLO= ½|VP|, gate bias VGG= ½VP, andassuming VRF << VLO as sketched in Fig.3.

Example VII-1-1 end

Fig.3 Local oscillator driving of a square-law FET mixer for max. continuous modeconversion transconductance. The RF signal, VRF, is much smaller than VLO.

The assumption of a small RF signal compared to the local oscillator signal - whichgave the maximum conversion gain in the example above, applies commonly to the initialmixers in radio receivers. The assumption provide the background for a method of calculatingconversion gains in mixer circuits by the so-called time-varying transconductance approach3, which is an alternative to the Taylor expansions we have considered so far. Here the trans-conductance, which is calculated like the "a" term in the Taylor series expansion, is taken asa function of both the DC bias and the large local oscillator time dependent signal. When thesmall RF signal is applied, the resultant mixing components in the drain current are calculatedlike conventional small signal current components by multiplying the RF gate voltage by atransconductance. However, the latter is now time dependent, so we may express the mixingcomponents in the drain current by

To find a particular mixing component with a given time dependent transconductance

(22)

waveshape, it must be expanded in a Fourier series,

3 ) The approach is also called "Large Signal - Small Signal" analysis.

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9VII-1 Mixer Basics

Here, it is assumed that the transconductance waveshape Gd,mix(t) is symmetrical with respect

(23)

to t=0. If this is not the case we must elaborate the Fourier expansion correspondingly. To finda particular mixing components with a given RF component of frequency ωRF, each Fouriercoefficient gives rise to two mixing products, one sum and one difference frequency betweenthe RF and one of the local oscillator components, usually the fundamental local oscillatorfrequency component at ωLO corresponding to m=1 in

Conversion transconductance is still defined like Eq.(15) by the ratio of the desired frequency

(24)

component in the drain current over the RF signal amplitude, so we have to select one ofabove component by filtering. In time-varying transconductance terminology, the conversiontransconductance becomes

where gmix,m is the Fourier coefficient of appropriate order in the expansion of the trans-

(25)

conductance waveshape.

Example VII-1-2 Square-Law FET mixer - continued

There are several practical drawbacks in the square-law mixer circuit that wasconsidered in the preceding example. One of them is that gate bias and local oscillator voltagedepends on the pinch voltage of the FET. This parameter is commonly subject to largespreadings, say 100% or more. Precise tracking would therefore require adjustment capabilitiesin the circuit and tuning in production. We may reduce the direct dependencies of the pinchvoltage with a higher local oscillator amplitude than we had before. In consequence, the FETmust be non conducting - i.e. cut off - in part of the oscillator period. Assume that the RFsignal is small compared to the local oscillator amplitude, so it totally dominates the draincurrent waveshape and the switching instants of the FET. The drain current pulse train isillustrated in Fig.4, and the corresponding time varying transconductance is shown in Fig.5.It is given though

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Since the current characteristic Id(Vgs) is square-law when the transistor conducts current, the

Fig.4 Current pulses in square-law FET mixer with large sinusoidal LO drive. The RFsignal is small, VRF VLO.

(26)

corresponding transconductance is a linear function of Vgs. Therefore - using a sinusoidal localoscillator - the transconductance waveshape becomes a train of sine-tips. We have alreadyconsidered the Fourier expansion of this waveshape in conjunction with the power amplifierdiscussion in chapter 5. The transistor opening angle θ, i.e. the portion of a period where thetransistor conducts, may still be used as the controlling parameter. Thereby the conversiontransconductance is expressed

(27)

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11VII-1 Mixer Basics

The ratio ym/yp represents the Fourier coefficient and it is taken from Table 5-1 using the

Fig.5 Time varying transconductance in square-law FET mixer with large, sinusoidallocal oscillator drive.

appropriate local oscillator mixing harmonic component number m. The peak value in thetransconductance pulse train is gm(Vgs,max). If Vgs is driven up to 0 V, the peak value is

A few extracts from Table 5-1 in the case of fundamental frequency local oscillator mixing,

(28)

Table I Fourier coefficient of sinetips. Extracts from table 5-1.

θ y1 /yp y2 /yp y3 /yp

0 0.0000 0.0000 0.0000

90 0.3102 0.2562 0.1811

180 0.5000 0.2122 0.0000

270 0.5326 0.0439 -0.0311

360 0.5000 0.0000 0.0000

m=1, are summarized in Table I. It is seen here that in the limit case of full conduction,θ=360°, we get a conversion conductance in agreement with the maximum limit in the previ-

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ous result from Eq.(21). Between full conduction and half-time conduction at θ=180°, the

Fig.6 Time-varying transconductance in square-wave driven FET mixer.

conversion transconductance exceeds this value by a small amount. If the conduction anglegoes below θ=180°, conversion transconductance reduces correspondingly. However, thesmaller the conduction angle, the smaller is the significance of the actual pinch voltageparameter size, |VP|, since the local oscillator amplitude VLO must rise correspondingly. Witha maximum gate to source voltage of zero volts, which implies that the gate bias is VGG=-VLO , the relationships between the two are expressed through

Instead of driving by a sinusoidal local oscillator, the transconductance could besquare-waved using a square-waved local oscillator signal. In the simple case sketched inFig.6, where the conduction angle is 180°, the conversion transconductance becomes

Here the 4/π factor is the Fourier coefficient of a square-wave that is normalized to the

(30)

interval [+1,-1], so the amplitude to be used is ½gm(Vgs,max). Square-wave local oscillatorsignals are feasible in many cases, since they resemble outputs from limiter circuits that maybe used to minimize the effects of local oscillator amplitude noise and fluctuations in sensitiveequipment like radio receivers.

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13VII-1 Mixer Basics

Driving the mixer with a local oscillator signal that is so large that the transistorbecomes non-conducting, like it was done in this example, has the consequence, that mixingtakes place around harmonic components of the local oscillator signal. We have assumed thatany undesired or spurious frequency component is removed by subsequent filtering, but clearlythe more spurious components generated, the more efforts must be given to the filteringproblem. When the FET mixer was used in continuous mode in the foregoing example, nohigher order mixing took place, so the price paid by making the mixer less sensitive toparameter variations by enlarging the local oscillator drive could be that the filtering require-ments are tightened.

Example VII-1-1 end

The FET mixer prototype we have considered in this section was introduced as avehicle to exemplify how the required multiplying function may be realized through anonlinear characteristic. Besides this very basic property there are several more concerns foremploying mixers in RF-circuits which we shall deal with below. If we should use theprototype FET mixer as an outset for a practical application, we should probably rearrange thecircuit as sketched in Fig.7, where the RF and LO signals are separated so each of them getsa ground terminal. ZRF and ZLO represent the generator impedance of the RF and LO sourcesrespectively and they are applied through coupling capacitors Ccpg and Ccps. Biasing of thetransistor is made through RG and RS1,RS2. We ascribe no DC voltage across RG as the idealFET requires no gate bias current. The gate source biasing is establish by the DC currentthrough the transistor and RS1+RS2. It is assumed that the FET has a negative thresholdvoltage VP.

Fig.7 Practical realization of a FET mixer including biasing. A negative threshold vol-tage is assumed.

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VII-2 Differential Stage Mixers

The differential amplifier structure that was considered in Chapter 5 may also serve

Fig.8 Differential stage mixer coupling. Ccpli and Ccple are coupling and decouplingcapacitors. R1 to R3 are bias resistors. Filtering is made outside this circuit.

as a mixing circuit. One way of doing this is sketched in Fig.8. The local oscillator signal isapplied to the differential input terminals while the common tail current, which formerly wasa constant bias current, now is overlaid by the RF signal through a conventional commonemitter stage around transistor Q3. Assuming small signal conditions for the RF signal, the tailcurrent is expressed through

Here, appropriate bias resistor settings establish the DC current level, I0 . Inserting into the

(31)

large signal expression for the differential amplifier, Eq.5-138, the differential output currentmay now be written

(32)

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To see the mixing properties, we assume initially that both the RF and the local oscillatorsignals are sinusoidal,

and, furthermore, that the local oscillator amplitude is small compared to Vt, so if suffice to

(33)

use the first, linear term in the Taylor series expansion

Now the differential current may be written

(34)

(35)

No mixing is associated with the first term since it contains only the local oscillator frequency

(36)

component. The second term holds the mixing product

By subsequent filtering, the desired intermediate frequency component, either

(37)

must be selected. It is represented by the differential current

(38)

The corresponding conversion parameters now take the forms

(39)

(40)

(41)

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17VII-1 Differential Stage Mixers

It is often desirable to use a local oscillator signal that is larger than the sizeassumed by the approximation in Eq.(35). With growing local oscillator signal we shall, likethe limiter development in Section 5-3, reach a limit where the two differential transistors Q1and Q2 are operated as antagonistic switches. In that case we may approximate the result ofthe hyperbolic tangent function by a square-wave and approximate,

Now the differential current becomes

(42)

Again, the first term contributes nothing to the mixing function. It holds only fundamental and

(43)

harmonic frequency components of the local oscillator signal. The last term, however, holdmixing products around the local oscillator frequency and higher harmonic components.Isolating one component from mixing with local oscillator fundamental frequency gives

The corresponding conversion parameters, which are independent of the local oscillator signal

(44)

level, now become

If the output of a mixing circuit includes unmixed fundamental and higher harmonic

(45)

(46)

frequency components of both the RF and the LO signals before the IF filter is applied, themixer is called unbalanced. If one of these family of components are suppressed beforefiltering, the mixer is called single balanced, if both families are absent, the mixer is calleddouble balanced. In this terminology, the differential stage mixer above is single balanced.Besides selecting the proper IF components, the subsequent IF filter has to suppress the localoscillator components down to levels where they do no harm. An alternative is to employ twocross-coupled differential stages to achieve double balancing by the so called Gilbert Cellmixer, which is discussed next.

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VII-3 Gilbert-Cell Mixers

Cross coupling two differential stage mixers that are current biased through a

Fig.9 Gilbert-Cell. The input terminals must be properly biased in addition to differentialinputs VLO and VRF. All current expressions assume transistor current gains αfequal to one.

common DC tail current gives the structure in Fig.9. It is called a Gilbert Cell after itsinventor. Compared with the single differential stage mixer, also the RF signal is now appliedto a differential input port. As seen in the figure, there are three differentially operatedtransistor pairs in this configuration, and to investigate the mixing function, we shall makerepeated use of the differential stage results from Section 5-3. It is convenient, therefore, tointroduce the input signal normalization that was used formerly, i.e.

The common tail current is here kept at a constant DC value, I0. The differential current in

(47)

the bottom transistor pair, which is driven by the RF signal, becomes

(48)

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where αf is the common base current gain of the transistors. Commonly it has a value slightlybelow one, say 0.98. Including the effect of the bottom differential current, ΔI0, the differentialand tail currents for the two LO signal operated transistor pairs are expressed

Finally, the two differential terms above subtracts to the final output differential current ΔIout,

(49)

(50)

We may use the Taylor series expansions and assumptions from Eqs.(34),(35) to approximate

(51)

Compared with the similar expressions from the single differential stage mixer like Eq.(36),

(52)

it is seen that only a product term remains. There is no separate LO signal terms, so theGilbert Cell has clearly doubly balanced mixer function. For the same reason, a Gilbert Celldriven by small input signals is sometimes called a pure four-quadrant multiplier.With sinusoidal input signals

the frequency component of desired intermediate frequency IF, where

(53)

has the amplitude

(54)

(55)

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Thereby, the conversion parameters for a Gilbert Cell Mixer operated by small signals at boththe RF and the LO ports become,

(56)

(57)

With a large signal input to the local oscillator port, i.e. xlo>6 or VLO>150 mV, theupper differential pairs are operated like switches as sketched in Fig.10. The direction of the

Fig.10 Gilbert-Cell circuit used as a switching mixer. The RF differential current isreverted when the local oscillator voltage VLO shifts between high positive (a) andnegative (b) levels.

RF-signal controlled differential current, ΔI0, is changed according to the sign and in turns thefrequency of the LO signal. In the limit, where we assume ideal instant switching, the LOcontrolled hyperbolic tangent factor in Eq.(51) should be replaced by the Fourier expansionof a square-wave like Eq.(42). With this replacement we get

Maintaining the assumption of a small RF signal, and mixing around the fundamental LO

(58)

frequency component, the output differential current IF terms have amplitudes of

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The mixing parameters in switched operation now become

(59)

(60)

(61)

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Problems

P.VII-1

Fig.11 shows the principle for a mixer with a square-law FET. Both the RF and the

Fig.11

LO signals are sinusoidal at frequency fRF = 450MHz and fLO = 520 MHz respec-tively. The load circuit has quality factor QIF = 100, and it is tuned to fIF = 70 MHz.The load resistor is RL=1kΩ. Capacitor Cdcp is a decoupling capacitor. It is assumedthat the RF signal is much smaller then the local oscillator amplitude.

Find the bias resistor RE and the local oscillator amplitude VLO that give mean draincurrent equal to 10 mA and forces the transistor peak current to IDSS.

Sketch the time varying transconductance - minimum and maximum values - and findthe conversion transconductance. What is the IF output voltage if an RF signal of2mV is applied to the RF input port?

Fig.12

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In practice, the RF signal is applied through a transformer as shown in Fig.12. Theinput circuit is tuned to fRF and has quality factor QRF = 20 including the effect ofthe generator resistance Rg.

What is the image frequency of the mixer. Estimate the IF output voltage if an imagefrequency input signal of 2mV amplitude is applied to the RF input port?

P.VII-2

Fig.13 shows a differential stage mixer. The bias current without any RF signal is set

Fig.13

by resistor R0, which is chosen to give I0=5mA. The local oscillator frequency is 120MHz and the amplitude VLO is set to provide a differential amplitude of ΔIc,LO = 2.5mA at the local oscillator frequency if no RF signal is applied. The RF signal is offrequency 100 MHz is applied through coupling capacitor C0. The load circuit L, Cis tuned to a intermediate frequency of 20 MHz. Including the load resistanceRL=1kΩ, the load circuit has Q factor equal to 100.

What is the conversion transconductance of the stage and the corresponding localoscillator voltage amplitude, VLO ? Find the IF output voltage VIF if the RF signalis an unmodulated carrier of 5 mV.Note, it is not assumed in this problem that VLO is << Vt.

An amplitude modulated RF signal,

where m is the modulation index and the baseband frequency fBB << fRF, is applied.

(62)

The resultant intermediate frequency output is written

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where the frequency dependency of the resultant modulation index, mIF, is caused by

(63)

the frequency characteristic of the tuning circuit.

At which baseband frequency is the resultant modulation index mIF reduced 3dBcompared to the input index m ?

The envelope of the IF output is distorted due to the nonlinear transfer characteristicof the mixer. Assume a low baseband frequency and show that the 2nd and 3rdharmonic contribution to the total harmonic distortion of the IF envelope for smallRF signals, VRF << Vt = 25mV are

The exponential function series expansion may be useful to answer the last question,

(64)

(65)

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References and Further Reading

[1] R.S.Carson, Radio Communication Concepts: Analog, Wiley 1990.

[2] S.A.Maas, Microwave Mixers, 2nd ed., Artech House 1993.

[3] S.A.Maas, Nonlinear Microwave Circuits, Artech House 1988.

[4] A.S.Sedra, K.C.Smith, Microelectronic Circuits, 3rd ed., Saunders, 1991.

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Index

Conversion Gain . . . . . . . . . . . . . . . 6Conversion Transconductance . . . . . . 6Conversion Voltage Gain . . . . . . . . . 6Demodulators . . . . . . . . . . . . . . . . . 1Detectors . . . . . . . . . . . . . . . . . . . . 1Double Balanced Mixer . . . . . . . . . . 17Frequency Converters . . . . . . . . . . . . 1Gilbert-Cell . . . . . . . . . . . . . . . . . . . 18Large Signal - Small Signal Anal-

ysis . . . . . . . . . . . . . . . . 8Mixer

balanced . . . . . . . . . . . . . . . . . 17Conversion Gain . . . . . . . . . . . 6double balanced . . . . . . . . . . . . 17unbalanced . . . . . . . . . . . . . . . 17

Mixers . . . . . . . . . . . . . . . . . . . . . . 1FET basics . . . . . . . . . . . . . . . 2Gilbert-Cell . . . . . . . . . . . . . . . 18

Modulators . . . . . . . . . . . . . . . . . . . 1Single Balanced Mixer . . . . . . . . . . . 17Spurious Signals . . . . . . . . . . . . . . . 5Switching Conversion Gain

differential stage mixer . . . . . . . 17Gilbert cell mixer . . . . . . . . . . 21

Time-Varying Transconductance . . . . 8

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ChapterVII MIXERSandDETECTORS - RF toolboxrftoolbox.dtu.dk/book/Ch7.pdf · VII-1 Mixer Basics 3 characteristic in Fig.1(b). With both input signals set to zero, the transistor DC