A MMIC GaAs up-converter from 350 MHz to 1835 MHz realized both in a HBT diode-mixer topology and pHEMT resistive FET-mixer topology

Examensarbete

LITH-ITN-ED-EX--06/019--SE

A MMIC GaAs up-converter from

350 MHz to 1835 MHz realized both

in a HBT diode-mixer topology

and pHEMT resistive FET-mixer

topology

Anders Andersson

Joakim Östh

LITH-ITN-ED-EX--06/019--SE

A MMIC GaAs up-converter from

350 MHz to 1835 MHz realized both

in a HBT diode-mixer topology

and pHEMT resistive FET-mixer

topology

Examensarbete utfört i Elektronikdesign

vid Linköpings Tekniska Högskola, Campus

Norrköping

Anders Andersson

Joakim Östh

Handledare Per Gustafson

Handledare Martin Johansson

Examinator Adriana Serban Craciunescu

Rapporttyp Report category Examensarbete B-uppsats C-uppsats D-uppsats _ ________________ Språk Language Svenska/Swedish Engelska/English _ ________________ Titel Title Författare Author Sammanfattning Abstract ISBN _____________________________________________________ ISRN _________________________________________________________________

Serietitel och serienummer ISSN

Title of series, numbering ___________________________________

Nyckelord

Datum Date

URL för elektronisk version

Avdelning, Institution Division, Department

Institutionen för teknik och naturvetenskap Department of Science and Technology

2006-05-24

x

x

LITH-ITN-ED-EX--06/019--SE

A MMIC GaAs up-converter from 350 MHz to 1835 MHz realized both in a HBT diode-mixer topology and pHEMT resistive FET-mixer topology

Anders Andersson, Joakim Östh

Two mixers for up-conversion from an IF frequency of 350 MHz to a RF frequency of 1835 MHz have been designed and simulated to be used in Ericssons radio link system MINI-LINK. One mixer uses diodes in a balanced structure, and the other one use resistive FET-mixers, also in a balanced structure. Both implemented in a GaAs MMIC process; for the diode mixer TriQuint HBT2 and for the resistive FET-mixer TriQuint 0.25 um pHEMT. The mixers were designed to work with input LO-power of 0 dBm and an IF-power of -20 dBm. For the diode based mixer with an active LO balun the conversion gain is 5.7 dB, P-1dB 15 dBm and the LO-suppression -22 dB. For the resistive FET-mixer the conversion gain is 11 dB, IIP3 26 dBm, P-1dB 15 dBm and the LO-suppression -49 dB. The data given is based on simulations; no wafers have been processed at this time. The chip-area the final design will occupy is approximated to 1.8 mm^2 for the diode mixer and approximately 1.9 mm^2 for the resistive FET-mixer. For both of the mixer types an off-chip balun for the IF-frequency is the only external component needed.

Upphovsrätt

Detta dokument hålls tillgängligt på Internet – eller dess framtida ersättare –

under en längre tid från publiceringsdatum under förutsättning att inga

extra-ordinära omständigheter uppstår.

Tillgång till dokumentet innebär tillstånd för var och en att läsa, ladda ner,

skriva ut enstaka kopior för enskilt bruk och att använda det oförändrat för

ickekommersiell forskning och för undervisning. Överföring av upphovsrätten

vid en senare tidpunkt kan inte upphäva detta tillstånd. All annan användning av

dokumentet kräver upphovsmannens medgivande. För att garantera äktheten,

säkerheten och tillgängligheten finns det lösningar av teknisk och administrativ

art.

Upphovsmannens ideella rätt innefattar rätt att bli nämnd som upphovsman i

den omfattning som god sed kräver vid användning av dokumentet på ovan

beskrivna sätt samt skydd mot att dokumentet ändras eller presenteras i sådan

form eller i sådant sammanhang som är kränkande för upphovsmannens litterära

eller konstnärliga anseende eller egenart.

För ytterligare information om Linköping University Electronic Press se

förlagets hemsida

http://www.ep.liu.se/

Copyright

The publishers will keep this document online on the Internet - or its possible

replacement - for a considerable time from the date of publication barring

exceptional circumstances.

The online availability of the document implies a permanent permission for

anyone to read, to download, to print out single copies for your own use and to

use it unchanged for any non-commercial research and educational purpose.

Subsequent transfers of copyright cannot revoke this permission. All other uses

of the document are conditional on the consent of the copyright owner. The

publisher has taken technical and administrative measures to assure authenticity,

security and accessibility.

According to intellectual property law the author has the right to be

mentioned when his/her work is accessed as described above and to be protected

against infringement.

For additional information about the Linköping University Electronic Press

and its procedures for publication and for assurance of document integrity,

please refer to its WWW home page:

http://www.ep.liu.se/

A MMIC GaAs up-converter from 350 MHz to 1835

MHz realized both in a HBT diode-mixer topology

and pHEMT resistive FET-mixer topology

Master Thesis

by

Anders Andersson and Joakim Östh 2006

Department of Science and Technology Linköping Institute of Technology

Abstract

Two mixers for up-conversion from an IF frequency of 350 MHz to a RF frequency of 1835 MHz have been designed and simulated to be used in Ericsson’s radio link system MINI-LINK. One mixer uses diodes in a balanced structure, and the other one use resistive FET-mixers, also in a balanced structure. Both implemented in a GaAs MMIC process; for the diode mixer TriQuint HBT2 and for the resistive FET-mixer TriQuint 0.25 um pHEMT. The mixers were designed to work with input LO-power of 0 dBm and an IF-power of -20 dBm. For the diode based mixer with an active LO balun the conversion gain is 5.7 dB, P-1dB 15 dBm and the LO-suppression -22 dB. For the

resistive FET-mixer the conversion gain is 11 dB, IIP3 26 dBm, P-1dB 15 dBm and the LO-suppression -49 dB. The data given is based on simulations; no wafers have been processed at this time. The chip-area the final design

will occupy is approximated to 1.8

mm

2for the diode mixer and approximately

1.9

mm

2 for the resistive FET-mixer. For both of the mixer types an off-chip

balun for the IF-frequency is the only external component needed.

Contents 1 Introduction ...1 2 Theory ...2 2.1 Semiconductor devices ...3 2.1.1 HBT ...3 2.1.2 HEMT ...4 2.1.3 Schottky diode...4 2.2 Mixer basics ...5

2.2.1 General non-linear analysis ...5

2.2.2 Restricted analysis: the conversion matrix method ...11

2.2.3 Mixer parameters ...12

2.3 General non-linear phenomena ...14

2.3.1 Gain Compression ...15

2.3.2 Desensitization and Blocking ...15

2.3.3 Cross Modulation ...16

2.3.4 Intermodulation ...16

2.4 Resistive FET-mixer ...18

2.5 Balanced diode mixers ...22

2.5.1 Single-balanced diode mixer ...22

2.5.2 Double-balanced diode mixers...26

3 Requirements ...30 4 Design ...31 4.1 Diode mixer ...31 4.1.1 Mixer core ...31 4.1.2 Baluns ...35 4.1.3 RF power amplifier ...41 4.1.4 IF-balun ...42

4.1.5 The complete up-converter ...42

4.2 FET-mixer ...44

4.2.1 FET-mixer core ...44

4.2.2 Active balun...48

4.2.3 Passive balun...50

4.2.4 RF power amplifier ...50

4.2.5 The complete up-converter ...53

5 Results ...54 5.1 Diode mixer ...54 5.1.1 Active LO-balun...54 5.1.2 Passive LO-balun...57 5.1.3 LO amplifier...61 5.1.4 RF power amplifier ...62

5.1.5 The complete up-converter ...63

5.2 FET mixer...68

5.2.1 LO balun...68

5.2.2 RF amplifier...71

5.2.3 The complete up-converter ...73

6 Layout and chip area ...76

6.1 Diode mixer ...76

6.1.1 Active LO-balun...76

6.1.5 RF-amplifier...79

6.1.6 An estimation of the complete chip-area ...80

6.2 FET mixer...81

6.2.1 Layout of the mixer core...81

6.2.2 Layout of the active LO balun ...82

6.2.3 Layout of the output RF amplifier ...83

6.2.4 Layout of the whole up-converter ...83

7 Conclusion...84

8 Acknowledgements ...85

9 References...86

Appendix A ...90

Additional design: adaptive bias circuit ...90

Layout 91 Appendix B ...93 Workbenches ...93 Appendix C ...97 Maple calculation 1 ...97 Maple calculation 2 ...98 Maple calculation 3 ...99 Maple calculation 4 ...101 Maple calculation 5 ...102

1 Introduction

The purpose of this work is to investigate if it is possible to make an up-converter in MMIC technology that fulfills the performance requirements, and at the same time, is cheaper than the solution used by Ericsson today. All work has taken place at Ericsson AB in Mölndal using Agilent ADS for design and simulation.

In today’s solution an expensive filter is needed after the mixer to suppress the LO-signal, therefore it is highly desirable to suppress the LO-signal already in the mixer so that a cheaper filter can be used.

The most demanding requirements were

1 high linearity

2 good LO-suppression

It turned out that a resistive pHEMT FET-mixer and a HBT diode mixer are good candidates to requirement one, due to their inherent good linearity. To meet requirement two, a balanced topology is selected, that theoretical can suppress the LO to infinity.

The work was basically divided in four parts: literature study, design,

simulations and tuning, and finally the layout was made to estimate the chip area needed to manufacture the mixer.

The report is divided in sections in an attempt to make it easy for the reader to find the information he or she finds interesting. The reader that is familiar with basic mixer theory can with advantage skip the section dealing with basic mixer theory. The design and result chapters are also divided in different sections, one for each mixer type to make it easy to find the relevant information.

The reader mainly interested in the performance is directed to the results section, and especially to the summaries there.

2

Abbreviatoins and vocabulary

Agilent ADS – The company Agilents Avanced Design System, a computer aided electronics design program.

Balun – A baluns task is to convert an balanced signal to an unbalanced signal (and also conversely). A hybrid is often used as a balun.

DB – Duble balanced.

GB (GD) – Gain balance (Gain difference) HB – Harmonic balance.

HBT – Heterojunction bipolar transistor. HBT2 – A TriQuint specific HBT process. HEMT – High Electron Mobility Transistor IF – Intermediate Frequency

IM – InterModulation

IP3 (IIP3) – Interception Point Three, i.e third order interception point. IIP3 represents Input IP3.

LO – Local Oscillator.

LSSP – Large Signal S-Parameters, this refers to the ADS simulation controller.

MINI-LINK – Ericsson’s radio link system. P-1dB – Power of the 1-dB compression point. PB (PD) – Phase Balance (Phase differance) RF – Radio Frequency.

SB – Single Balanced.

SP – S-parameter (scattering parameter). TOI – Third Order Interception.

TriQuint – An American semiconductor company.

XDB – X DeciBell i.e this refers to the X:th order compression point simulaiton controller.

3 Theory

3.1 Semiconductor

devices

In this thesis three types of semiconductor devices have been used, namely bipolar transistors, FET transistors and diodes. Regarding the transistors it is the improved and modern Heterojunction Bipolar Transistor (HBT) and High

Electron Mobility Transistor (HEMT)1 that have been used. The diodes used

are Schottky diodes. Since these improved devices differ slightly from the traditional (BJT, FET and PN-diode), regarding performance and

manufacturing, each of them will be described briefly in the subsections below [1] [2].

3.1.1 HBT

In order to increase the speed of a regular BJT the doping of the base could be increased. This comes, unfortunately, with the drawback of decreased current gain. There are also physical limitations; the semiconductor cannot be doped to that great extent that is sometimes wanted. Therefore, an additional

material is added to the emitter and, thus, forming a heterojunction2. If the

additional material added is a material that easily releases electrons (for example Al or In) the consequence will be that more electrons are injected into the base from the emitter; and this without excessive doping. As a result this will create a much faster device than the regular BJT.

Today there are mainly two versions of the HBT. They are the so called

AlGaAs/GaAs HBT and InGaP/GaAs HBT. The latter, so called 2nd generation

HBT, is the one used in TriQuints HBT2 process, which also is the one used in this thesis work. This is said to perform better and be more reliable than the

1st generation.

The key features of the HBT are:

• High linearity

• High power gain

• Low cost

• Relatively high operating frequency

• Low noise

3.1.2 HEMT

The HEMT or, High Electron Mobility Transistor, is basically constructed as an ordinary FET besides that, similar to HBT, bandgap engineering technologies have been utilized to increase the performance and creating a channel with low losses. In the HEMT a 2D electron gas is responsible for the carrier transport, this electron gas have a very large electron density and a high mobility and this is the main reason for the special features of the HEMT, the interested reader is recommended to read chapter 7 in reference [4]. In this work a special version of the HEMT has been used, instead of using

additional materials with matching crystal lattices (for instance GaAlAs/GaAs) non-matching materials have been used (for instance InGaAs/GaAs). These HEMTs are called pseudomorphic HEMT or pHEMT. The reason that the pHEMT is used is that there was no HEMT device available in the design library used.

The key features of HEMT are:

• High linearity

• Very high cut-off frequency (at least 500 GHz have been reported) • Very low noise

3.1.3 Schottky diode

A commonly used diode in RF applications is the Schottky diode. Unlike the regular pn-junction diode the Schottky diode uses a metal-semiconductor junction. By using this configuration the diode becomes a majority carrier device, which means only electrons are injected and, by being a majority carrier device, there will be no time consuming electron-hole recombination. Due to this the Schottky diode is faster than a conventional pn-diode and therefore suitable for high speed switching RF-applications, such as mixers.

3.2 Mixer

basics

A mixers' prime function [2-9] is to translate one frequency to another. Mathematically this is done by multiplication of two signals at different

frequencies, for example between frequency

f

₁ and

f

₂. However the angular

frequency

ω

=

2 f

π

is used due to the trigonometric functions. How it works

mathematically is shown with the following identity

( ) ( )

1 2

(

1 2

)

(

1 2

)

1

cos

cos

cos

cos

2

t

t

t

t

t

t

ω

ω

=

⎡

_⎣

ω

−

ω

+

ω

+

ω

⎤

_⎦

(1.1)

Apparently, multiplication of two signals with different frequencies gives two new frequency components; one is the sum and the other the difference between the two frequencies. The wanted signal is selected in some suitable way, usually by filtering. The symbol for a mixer can be seen in Figure 1. For

the case of up-conversion3, the intermediate frequency (IF)

ω

_IF signal is

applied to the left. From the bottom the local oscillator (LO)

ω

_LO signal is

applied, and the output, the radio frequency (RF)

ω

_RF signal, is taken from

the right. The multiplication sign in the mixer symbol suggests it works by multiplication. Inside the mixer symbol some non-linear device can be found, for example diodes or transistors. Before moving on to investigating different mixer topologies, let us investigate more closely how the mixer works by analyzing it mathematically. Down-converter Up-converter LO LO RF IF RF IF

Figure 1 The mixer symbol for the cases of an up-converter and a down-converter.

3.2.1 General non-linear analysis

In the following, the variable

ω

_I is the input frequency, that is either

ω

_IF or

RF

ω

depending if the mixer is an up-converter or a down-converter.

A mixer is a non-linear device that is capable of frequency transformation due to the non-linear relationship between the input signals and the output signal. To describe it mathematically lets assume that

2 3 1 2 3 1 n i out i i

I

α

V

α

V

α

V

α

V

=

=

∑

=

+

+

+

_K

(1.2)

where V is the total input voltage,

α α α

₁

,

₂

,

₃

,

_K

are constants dependent of the non-linear device used. It is easy to see that the output signal can be written as above, because every function can be approximated by a power or Taylor series if needed.

Below follows a practical example to show how it is applicable on a simple FET device and then follows the more general case.

3.2.1.1 Practical example

Let us see how the drain current for a simplified FET model that uses the square law behavior in saturation can be written [5]. This is done to make the analysis a bit more concrete.

The LO-signal is applied to the source of the transistor and the input signal is

applied to the gate4 of the transistor, that is the RF-signal if it is a

down-converting mixer, and the IF-signal if it is an up-down-converting mixer. The circuit can be seen in Figure 2.

G D VtSine VLO VtSine VI V_DC gate_bias Vdc=VGS EE_MOS1 FET R Rload L DC_FEED3 C DC_Blck2 C DC_Blck L DC_FEED2 L DC_FEED V_DC drain_bias Vdc=drain_bias

Figure 2 A simple mixer circuit. The LO-signal is applied to the source and the input signal to the gate.

From Figure 2 it is evident that the gate to source voltage is

gs I LO GS

V

=

V

−

V

+

V

(1.3)

4

Often both the LO- and RF/IF-signal is applied to the gate, however using the source for the LO-signal is also valid because the potential between gate and source is what is important when the non-linear drain current is investigated.

where

V

_GS is the gate to source DC bias voltage. Under the assumption that the device is operating in the saturation region, the drain current can be expressed as 2 2 2 1 gs 2 gs gs D DSS DSS DSS DSS P P P V V V I I I I I V V V ⎛ ⎞ = _⎜ − _⎟ = − + ⎝ ⎠ (1.4)

Where V_P is the pinch-off voltage and

I

_DSS is the drain current for

V

_gs

=

0

A comparison with equation (1.2) and (1.4) shows that

P DSS V I 2 1 =−

α

, 2 2 P DSS V I =

α

and

V

=

V

_gs

in this case. If a more complicated function were used, for example the relationship for a diode or BJT transistor, then the exponential function needs to be approximated by a series expansion.

If

( )

cos I I I v =V

ω

t (1.5)

(

)

cos LO LO LO v =V

ω

t (1.6)

and equation (1.3) is inserted in equation (1.4) then the drain current I_D can

be found (here

ω

_I is the input signal, that is either

ω

_RF or

ω

_IF). The derivation

was done with help of the mathematical software MAPLE5; this can be seen in

appendix C, Maple calculation 1. From this calculation the up-converted as well as the down-converted frequencies can be found. The expression is

(

)

(

)

2

1

cos

cos

DSS I LO I LO I LO P

I

V V

t

t

t

t

V

⎡

⎣

ω

−

ω

+

ω

+

ω

⎤

⎦

(1.7)

The constant before the square bracket divided by the amplitude of the input

voltage

v

_I is defined as the conversion transconductance:

2 2 DSS I LO DSS LO C I P P I V V I V g V V V = = (1.8)

From the calculation (appendix C, Maple calculation 1) the following wanted components arises

The down-converted frequency:

(

)

2

1

cos

DSS I LO I LO P

I

V V

t

t

V

ω

−

ω

(1.9)

(

)

2

1

cos

DSS I LO I LO P

I

V V

t

t

V

ω

+

ω

(1.10)

And these undesired components arises DC component: 2 2 2

1

2

DSS P DSS P GS DSS GS P

I

V

I

V V

I

V

V

⎡

⎣

−

+

⎤

⎦

(1.11)

Fundamental (RF/IF) input frequency component:

( )

2

2

cos

DSS P I I P

I

V V

t

V

ω

−

(1.12)

Fundamental LO input frequency component:

(

)

2

2

cos

DSS P LO LO P

I

V V

t

V

ω

(1.13)

Second harmonic (RF/IF) input frequency component:

(

)

2 2

1

cos 2

2

V

_P

I

DSS

V

I

ω

I

t

(1.14)

Second harmonic LO input frequency component:

(

)

2 2

1

cos 2

2

V

_P

I

DSS

V

LO

ω

LO

t

(1.15)

Note that signal components that have multiples, greater than one, of either the LO-, RF- or IF-signal is called harmonics.

In order to minimize the influences of these unwanted components some actions are needed, for example by filtering the output signal. Filtering is usually needed anyway to select the desired output frequency. The above equations were derived for the specific case of a square law FET device, described by (1.4). That is every voltage component above order two in the output current is zero.

3.2.1.2 The general case

The general form of equation (1.9) to (1.15), for terms up to third-order will

now be considered. It is straight forward to analyze more terms if desired6.

The calculation that was made to get these results was done in MAPLE and can be seen in appendix C, Maple calculation 2.

Up-converted frequency component:

6

[

3

α

3V V VI LO bias+

α

2V VI LO

]

cos

(

ω

It+

ω

LOt

)

(1.16)

Down-converted frequency component:

[

3

α

3V V VI LO bias+

α

2V VI LO

]

cos

(

ω

It−

ω

LOt

)

(1.17)

DC term:

2 2 3 2 2

3 3 3 2 2 1

3

3

1

1

2

α

V V

LO bias

+

2

α

V V

I bias

+

α

V

bias

+

2

α

V

I

+

2

α

V

bias

+

α

V

bias (1.18)

Fundamental (RF/IF) input frequency component:

( )

2 3 2 3 3 3 2 1

3

3

3

2

cos

2

α

V V

LO I

4

α

V

I

α

V V

I bias

α

V V

I bias

α

V

I

ω

I

t

⎡

₊

₊

₊

₊

⎤

⎢

⎥

⎣

⎦

(1.19)

Fundamental LO input frequency component:

(

)

3 2 2 3 3 3 2 1

3

3

3

2

cos

4

α

V

LO

2

α

V V

LO I

α

V V

LO bias

α

V V

LO bias

α

V

LO

ω

LO

t

⎡

₊

₊

₊

₊

⎤

⎢

⎥

⎣

⎦

(1.20)

Second harmonic (RF/IF) input frequency component:

(

)

2 2 3 2

3

1

cos 2

2

α

V V

I bias

2

α

V

I

ω

I

t

⎡

₊

⎤

⎢

⎥

⎣

⎦

(1.21)

Second harmonic LO input frequency component:

(

)

2 2 3 2

3

1

cos 2

2

α

V V

LO bias

2

α

V

LO

ω

LO

t

⎡

₊

⎤

⎢

⎥

⎣

⎦

(1.22)

Third harmonic (RF/IF) input frequency component:

(

)

3 3

1

cos 3

4

α

V

I

ω

I

t

(1.23)

Third harmonic LO input frequency component:

(

)

3 3

1

cos 3

4

α

V

LO

ω

LO

t

(1.24)

Third-order intermodulation product:

(

)

(

)

2 3

3

cos 2

cos 2

4

α

V V

LO I

⎣

⎡

ω

LO

t

−

ω

I

t

+

ω

LO

t

+

ω

I

t

⎤

⎦

(1.25) and

It should be noted that no third-order or higher intermodulation products arose during the analysis of the simplified FET because only non-linearity up to the second-order was considered there.

The name third-order is because the sum of the multiples of the two

frequency components is three, two LO and one signal component in (1.25) for example, and 2+1 = 3 hence third-order. The third-order components will be discussed more closely in chapter 3.3.

The different frequency components are often illustrated in a graph in the frequency domain where the frequency and amplitude of the different components is shown. One example can be seen in Figure 3, this is in fact for the LO frequency of 1485 MHz and the IF frequency of 350 MHz used in

this work. The output is therefore the RF frequency7 of 1485+350=1835 MHz.

From Figure 3 the LO-signal at 1485 MHz is seen, as well as the harmonic 2LO at 2970 MHz. The third-order intermodulation product at

2LO-IF at 2620 MHz can be seen in this graph, as well as many other harmonics and intermodulation products. The vertical scale is in dBm.

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0.0 4.5 -150 -100 -50 -200 0 freq, GHz IF_ s p e c tr u m Output Spectrum

Figure 3 Output spectrum

7

3.2.2 Restricted analysis: the conversion matrix method

In the literature often the output current as a function of the RF/IF input signal is given in the following form

( )

( ) ( )

i t =g t v t (1.27)

where g t

( )

is the (trans) conductance,

( )

( )

( )

di t

g t

dv t

=

and is assumed to be a

function of the LO-signal only and v t

( )

is the applied RF/IF-signal. This way

to express the current i t

( )

is valid if the amplitude of the RF/IF-signal is much

lower than the amplitude of the LO-signal, then the LO is assumed to be solely responsible for mixing [8][9]

( )

(

)

1

cos

m n LO n

g t

g

n

ω

t

=

=

∑

(1.28)

( )

_Icos

( )

_I v t =V

ω

t (1.29)

It is therefore possible to divide the analysis of the mixer in two steps: 1 a non-linear analysis to determine the steady state time-varying

(trans)conductance g t

( )

, here only the LO and bias voltage is

considered to see how they affect g t

( )

2 a linear analysis to determine the small-signal performance.

The current i t

( )

is then simply

( )

( ) ( )

(

)

( )

1

cos

cos

m n LO I I n

i t

g t v t

g

n

ω

t

V

ω

t

=

⎛

⎞

=

_{= ⎜}

_⎟

⎝

∑

⎠

(1.30)

Separating the products that arises when the multiplication in (1.30) is made gives

( )

0 Icos I g V

ω

t (1.31)

(

)

(

)

1

_cos

_cos

2

I LO I LO I

g

V

_⎣

⎡

ω

t

−

ω

t

+

ω

t

+

ω

t

⎤

_⎦

(1.32)

(

)

(

)

2

cos 2

cos 2

2

I LO I LO I

g

V

_⎣

⎡

ω

t

−

ω

t

+

ω

t

+

ω

t

⎤

_⎦

(1.33)

(

)

(

)

3

cos 3

cos 3

2

I LO I LO I

g

V

_⎣

⎡

ω

t

−

ω

t

+

ω

t

+

ω

t

⎤

_⎦

(1.34)

( )

From this, it is clear that the output components is of the form

n

ω

_LO

−

ω

_I and

LO I

n

ω

+

ω

for

n

=

1, 2,3,

_L

. Harmonics of the LO-signal exists but no

harmonics of the input RF/IF-signal and no DC component. But from the general analysis, as well as from the example above for the simplified FET mixer, harmonics of both the LO- and RF/IF-signal as well as a DC

component appeared in the output!

A question is now arising, how valid is the restricted analysis? The simple answer is that it is valid while the LO-signal is much greater than the RF/IF-signal. To see this, a somewhat cumbersome mathematical calculation was made to compare the complete non-linear analysis with this restricted

analysis. In this comparison, many terms are missing in the i t

( )

=g t v t

( ) ( )

approximation. This is expected because only the LO-signal is treated as a non-linear function, not the applied RF/IF-signal.

The comparison is made by comparing the result of the complete non-linear analysis in “equ2” with the restricted analysis in “equ1” shown in the MAPLE

calculation in appendix C, Maple calculation 3. Clearly every term that has

V

_I

is the same in both “equ1” and “equ2” but higher order terms as

V V

_I2

,

_I3

,

_K

are missing, as well as the DC components. Now, if the quotient between the LO and input signal is much greater than one, these components will have a negligible effect on the output signal and the general analysis can be

simplified to the restricted analysis by discarding these higher order terms of

2 3

,

,

I I

V V

_L

and the DC term.

The complete non-linear analysis must be performed if the LO- and RF/IF-signal is of the same magnitude or if intermodulation products are of interest, which they usually are.

3.2.3 Mixer parameters

Now when it has been seen, that mixing is due to the non-linear relationship between input and output, let us define some important parameters.

3.2.3.1 Conversion gain

The conversion gain8 is defined as the amplitude of the output RF (IF) signal

divided by the input IF (RF) signal amplitude.

3 2 3 2 3 3 I LO bias I LO LO bias LO I V V V V V conversion gain V V V V

α

+

α

_α

_α

= = + (1.35)

The conversion gain is often expressed in dB

8

From the equations for conversion gain or conversion loss, apparently the LO-signal amplitude is an important factor. Other important parameters such as the third-order interception point, IP3 and the 1-dB compression point

1dB

P

₋ will be defined later. These parameters are also dependent of the

α α α

₁

,

₂

,

₃

,

_K

in the non-linear transfer characteristics.

(

)

20 log

dB

conversion gain = conversion gain (1.36) For mixers with no gain, often conversion loss is used.

1

conversion loss

conversion gain

=

(1.37)

dB dB

conversion loss

= −

conversion gain

(1.38)

3.2.3.2 Isolation

Isolation (IS) between ports is an important parameter and is defined by the ratio of power available from the source to the power dissipated in the load at the same frequency.

3.2.3.3 Suppression

The suppression is defined as the power difference between two signals at the same port. For example in this case the LO-suppression relative to the RF-signal is defined as the LO-power at the output minus the RF-power at the output.

3.2.3.4 Impedances

The input and output impedance is defined as

, IN OUT

V at excitation frequency

Z

I at excitation frequency

=

(1.39)

3.3

General non-linear phenomena

There are many important phenomena in a non-linear system like a mixer. This section deals with the basic theory behind it and why it is important [10].

Assume that the output y t

( )

of a non-linear device is a function of the input

signal x t

( )

and can be written as

( )

( )

( )

( )

2

( )

3 1 2 3 1 n n n

y t

α

x t

α

x t

α

x t

α

x t

∞ =

=

∑

=

+

+

+

_K

(1.40)

where

α α α

₁

,

₂

,

₃

,

_L

are constants. For simplicity, only the components to

order three are considered here, but it is easy to extend the order if required. From (1.40) many interesting phenomena of a non-linear system can be derived. If

( )

cos

( )

x t = A

ω

t (1.41) then (1.40) becomes

( )

( )

2 2

( )

3 3

( )

1 cos 2 cos 3 cos

y t =

α

A

ω

t +

α

A

ω

t +

α

A

ω

t (1.42)

Simplifying and collecting terms, y t

( )

can be written as

( )

( )

(

)

( )

2 3 2 1 3 3 2 3 2

3

cos

2

4

cos 2

cos 3

2

4

A

y t

A

A

t

A

A

t

t

α

_α

_α

_ω

α

α

_ω

_ω

⎛

⎞

=

+

_⎜

+

_⎟

+

⎝

⎠

+

(1.43)

From this, the term with the input frequency

ω

, is the fundamental term:

( )

3 1 3

3

cos

4

A

A

t

α

α

ω

⎛

₊

⎞

⎜

⎟

⎝

⎠

(1.44)

The nth harmonic term is

(

)

cos

where

K

is a constant and

n

=

1, 2,3,

_K

In (1.43) the second-order (n=2) harmonic is

(

)

2 2

cos 2

2

A

t

α

_ω

and the third-order(n=3) harmonic is

( )

3 3

cos 3

4

A

t

α

_ω

.There is also a DC component in the output,

2 2

2

A

α

, although

the input signal, x t

( )

,had no DC term.

3.3.1 Gain Compression

The small-signal gain of a circuit is usually obtained with the assumption that

the harmonics are negligible. Assume that

α

₁ is much greater than all the

other factors, so the higher order terms can be neglected. The small-signal

gain is then

( )

( )

1

y t

x t

=

α

In reality, as the input level increase, the small-signal gain starts to decrease

if

α

₃

<

0

, since ₁

3

₃ 3

4

A

A

α

+

α

is a decreasing function of

A

. This effect is

quantified by the 1-dB compression point,

P

_−1dB defined as the input signal

level that causes a small-signal gain drop by 1 dB. Mathematically:

(

)

2 1 3 1

3

20 log |

|

20 log |

|

1

4

A

dB

α

α

α

⎛

₊

⎞

₌

₋

⎜

⎟

⎝

⎠

(1.46) 1 1 3 0.145 | | dB A

α

α

− = (1.47)

3.3.2 Desensitization and Blocking

When a weak desired signal, A₁cos

( )

ω

₁t and a strong interferer A₂cos

( )

ω

₂t

is applied to a circuit, the strong signal reduces the gain of the circuit and the weak desired signal experiences a vanishingly small gain; this is called

desensitization. For a sufficient large

A

₂ the gain becomes zero and the

weaker signal is blocked. To see this assumex t

( )

= A₁cos

( )

ω

₁t +A₂cos

( )

ω

₂t ,

then the output is (from (1.40))

( )

3 2

( )

1 1 3 1 3 1 2 1

3

3

cos

4

2

y t

=

_⎜

⎛

α

A

+

α

A

+

α

A A

_⎟

⎞

ω

t

+

⎝

⎠

K

(1.48) if

A

₁

<<

A

₂ then

( )

2

( )

1 3 2 1 1

3

cos

2

y t

=

⎛

_⎜

α

+

α

A

⎞

_⎟

A

ω

t

+

⎝

⎠

K

(1.49)

The gain of the desired signal A₁cos

( )

ω

₁t is therefore ₁

3

_{3 2}2

2

A

α

+

α

. If

α

₃

<

0

the gain is decreasing with

A

₂, this is called desensitization. For some value

of

A

₂, the gain becomes zero and the signal is blocked.

3.3.3 Cross modulation

When a variation in the amplitude of a strong interferer affect the amplitude of the weak and wanted signal, is called cross modulation. It is easy to see this if

( )

y t from (1.49) is considered. Now, if the amplitude

A

₂ is changing, the

amplitude of the wanted signal

ω

₁ also is changing. This phenomenon is most

important when many signals are processed at the same time.

3.3.4 Intermodulation

When two (or more signals) with different frequencies are applied to a non-linear system, frequencies that are not harmonics of the input frequencies arise. This is called intermodulation, IM.

This arises from the mixing of the two signals when their sum is raised to a power greater than unity. To see how, assume

( )

1cos

( )

1 2cos

( )

2

x t = A

ω

t +A

ω

t , if this is inserted in (1.40) the resulting

intermodulation products are

(

)

(

)

1 2: 2A A1 2cos 1t 2t 2A A1 2cos 1t 2t

ω ω ω α

= ±

ω

+

ω

+

α

ω

−

ω

(1.50)

(

)

(

)

2 2 1 2 3 1 2 1 2 3 1 2 1 2

3

3

2

:

cos 2

cos 2

4

A A

t

t

4

A A

t

t

ω

=

ω ω

±

α

ω

+

ω

+

α

ω

−

ω

(1.51)

(

)

(

)

2 2 2 1 3 2 1 2 1 3 2 1 2 1

3

3

2

:

cos 2

cos 2

4

A A

t

t

4

A A

t

t

ω

=

ω ω

±

α

ω

+

ω

+

α

ω

−

ω

(1.52)

and these fundamental products

( )

3 2 1 1 3 1 3 1 2 1

3

3

cos

4

2

A

A

A A

t

α

α

α

ω

⎛

₊

₊

⎞

⎜

⎟

⎝

⎠

(1.53)

( )

3 2 1 2 3 2 3 2 1 2

3

3

cos

4

2

A

A

A A

t

α

α

α

ω

⎛

₊

₊

⎞

⎜

⎟

⎝

⎠

(1.54)

The third-order IM products

2

ω ω

₂

−

₁and

2

ω ω

₁

−

₂are the most interesting

ones. The reason is that the difference between

2

ω ω

₂

−

₁and

2

ω ω

₁

−

₂ is in

the vicinity of

ω

₁ and

ω

₂, if the difference between

ω

₁ and

ω

₂ is small. This

small frequency difference makes it almost impossible to filter these unwanted frequency components.

To measure the IM distortion, a tone test can be made. In a typical

two-tone test;

A

₁

=

A

₂

=

A

. The ratio of the amplitude of the third-order output

product to

α

₁

A

, defines the IM distortion. The unit used here is dBc, “c” means

“with the respect to the carrier”.

The third-order IM is so important so that a performance metric has been defined for this, called the third-order interception point. This is measured with

a two-tone test where

A

₁

=

A

₂

=

A

is chosen so that higher order non-linearity

terms are negligible and the gain is relatively constant and equal to

α

₁. The

fundamental product increases in proportion to

A

, whereas the third-order IM

product, increases as

A

3. The third-order interception point is defined to be

the interception of these two lines as the name suggest. The horizontal coordinate of this point is called the input interception point, IIP3 and this is the voltage input amplitude. The vertical coordinate of this point is called the

output interception point, OIP3 and is the corresponding voltage output

amplitude. It is common to express these metrics in the unit dBm, in that case the voltage quantities is converted to power.

If

A

₁

=

A

₂

=

A

then a simple expression for IP3 can be derived under the

assumption that ₁

9

₃ 2

4

A

α

>>

α

, the result is 1 3 3 4 | | 3 IIP A

α

α

= (1.55)

In practice, the IM and fundamental is measured for small values of

A

then

3.4 Resistive

FET-mixer

In a resistive FET-mixer’s the resistance between the drain and source is modulated by the LO-signal, applied to the gate of the FET. With no applied bias at the drain the slope of the I-V curves can be changed by the applied

gs

V

voltage, the LO-signal, and therefore the conductance can change very

much.

Ideally, the LO switch the conductance between an on-state, when the device is near forward turn-on, and an off-state when the device is in pinch-off. The

I-V curves for small

V

_ds and different

V

_gs for the Cold-FET9 model (used in this

work) can be seen in Figure 4. From this figure it is clear that the I-V

relationship is almost linear for low

V

_ds voltages. The threshold voltage for this

FET is -0.95 V. The right graph in Figure 4 is a close up of the left one, from

this, the drain to source resistance for

V

gs

=

0

and

V

gs

= −

0.8

was estimated

as

R

dV

dI

=

. For

V

gs

=

0

the value is about 15 ohm and for

V

gs

= −

0.8

the

value is about 300 ohm. From the graph it can be seen that the resistance is

approaching infinity when

V

_gs is approaching the threshold voltage. The

conductance shows an inverse relationship with

V

_gs voltage.

-0.4 -0.2 0.0 0.2 0.4 -0.6 0.6 -40 -30 -20 -10 0 10 20 30 -50 40 VGS=-2.000 VGS=-1.800 VGS=-1.600 VGS=-1.400 VGS=-1.200 VGS=-1.000 VGS=-0.800 VGS=-0.600 VGS=-0.400 VGS=-0.200 VGS=0.000 VDS D C .ID S .i , m A

Device I-V Curves

-0.1 0.0 0.1 -0.2 0.2 -12 -9 -6 -3 0 3 6 9 12 -15 15 VGS=-2.000 VGS=-1.800 VGS=-1.600 VGS=-1.400 VGS=-1.200 VGS=-1.000 VGS=-0.800 VGS=-0.600 VGS=-0.400 VGS=-0.200 VGS=0.000 VDS D C .ID S .i , m A

Device I-V Curves

Figure 4 I-V curves for a FET operating as a resistive FET-mixer.

For very small values of

V

_ds the drain current can be modeled by Shockley

theory [13], however here an approximate analysis [14] is made to predict the conversion gain or conversion loss.

Let us assume that the conductance can be described by the following equation

(

)

if 0 if g P g P ds g P K V V V V G V V ⎧ _{− >} ⎪ = ⎨ ≤ ⎪⎩ (1.56)

where

K

is the slope of the channel conductance,

V

_gis the gate voltage and

P

V

the pinch-off voltage.

9

To see if this is reasonable a S-parameter simulation was done to estimate

ds

G

as a function of

V

_g. Here the input resistance, looking into the drain, was

simulated. So influences from the drain resistance, for example, are also included here. From the simulation results shown in Figure 5 the assumption

that

G

_ds is a linear function of

V

_g seems pretty good at low

V

_g, for higher

value of

V

_g it can be seen that the curve deviate from being a linear curve.

But according to the Shockley theory this is expected.

-0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 -1.0 0.0 0.02 0.04 0.06 0.00 0.08 Gate_bias 1/ real( Z (1 ,1 )) Gd(Vg)

Figure 5 Conductance as a function of gate bias.

An equivalent circuit, as seen from the LO-port, can be made. The LO-signal is applied to the gate of the transistor. Remember also that the drain and source is assumed to be DC short circuit. The equivalent circuit for the LO is

shown in Figure 6. The impedance ZDLO is assumed to be short-circuit for

the LO-signal at the drain terminal, ZGLO is assumed to be the generator

impedance for the LO and short circuit for all other frequencies. The left hand circuit in the figure can be simplified to the one shown to the right in the figure,

under the assumption that

C

_gs and

C

_gd have approximately the same value

and

R

_sand

R

_d also have approximately the same value. This results in that

no current will flow through

G

_d because the potential is the same on the left

and right side, therefore it can be removed. From this equivalent circuit it is also possible to estimate the input impedance for the LO-signal.

G S D D S G C Cgs1 C Cgd1 R Rg1 R ZGLO1 VtSine LOsrc1 R ZDLO1 R Rd1 R Rs1 R Rs R Rd R ZDLO VtSine LOsrc R ZGLO R Rg C Cgd C Cgs R Gd

D S S D G R Rd1 R ZDRF1 VtSine RFsrc1 R Rs1 R Gd1 R Gd C Cgd C Cgs R Rg R Rs R ZGRF VtSine RFsrc R ZDRF R Rd

Figure 7 Equivalent circuit for the input signal, looking in to the drain.

An equivalent circuit for the input RF-signal, looking in to the drain, can also be developed; this is shown in Figure 7. The simplification of the left circuit to the right one in Figure 7 is not too obvious. But here the fact that the

reactance of

C

_gs and

C

_gd is much greater than

R

_s and

R

_d is used.

From the right circuit in Figure 7, the small-signal drain current can be

derived. Assume that RFsrc1=VRFcos

(

ω

RFt

)

then the small-signal current

can be derive as

(

)

(

)

(

)

(

)

(

)

[

]

, , ,

cos

1

cos

1

RF RF d d s d g LO d g LO RF RF d g LO d s

V

t

I

ZDRF

R

R

G V

G V

V

t

G V

R

R

ZDRF

ω

ω

=

=

+

+

+

+

+

+

(1.57) if we let

(

)

₍

₎

_[

(

,

)

_]

,

1

d g LO LO d g LO d s

G V

f

t

G V

R

R

ZDRF

ω

=

+

+

+

(1.58) then (1.57) becomes

(

)

cos

(

)

d LO RF RF I = f

ω

t V

ω

t (1.59)

If the transistor is biased near pinch-off,

G V

_d

(

_{g LO,}

)

can be expressed as

(

)

,

(

)

,

cos

if

0.5

0.5

0 if 0.5

1.5

g LO LO LO d g LO LO

KV

t

t

G V

t

ω

φ

π ω

φ

π

π ω

φ

π

⎧

+

−

<

+ <

⎪

= ⎨

<

+ <

⎪⎩

(1.60) Hence (1.58) becomes

(

)

,

(

)

₍

₎

, cos 1 [ ] cos g LO LO LO d s g LO LO KV t f t R R ZDRF KV t

ω

φ

ω

ω

φ

+ = + + + + (1.61)

From (1.61) it is apparent that more LO power, and therefore a higher

,

g LO

V

voltage, increases conversion gain or decreases conversion loss.

The drain current can now be predicted by the derived equations. To get the

conversion gain the function f

(

ω

_LOt

)

needs to be described by a Fourier

transform, only the first term

g

₁ is needed. It can be calculated numerically

from the following integral

( ) ( )

2 1 2

1

cos

g

f x

x dx

π π

π

−

=

∫

(1.62)

where f x

( )

is the same function as in (1.61)

The down-converted or up-converted drain current (

I

_d) is then

(

)

1 , cos 2 RF d IF LO RF g V i =

ω

t−

ω

t (1.63)

(

)

1 , cos 2 RF d RF LO IF g V i =

ω

t+

ω

t (1.64)

Now the conversion gain or loss can be calculated. The conversion gain is

simply 1

2

g

3.5

Balanced diode mixers

In today’s mixers the requirements on linearity, port-to-port isolation, noise, IM-suppression and so on are high, and a single ended mixer just is not enough. The solution to this is to use a balanced mixer topology which, inherently, performs well on these aspects. The drawback with this method is that it requires more LO-power since it uses two or more diodes. It can also be hard, and even impossible, to add bias to the diode which degrades the conversion gain.

The operation of single-balanced (SB) and double-balanced (DB) mixers will be described respectively in the following two sections. [2] [3] [9] [36] [37]

3.5.1 Single-balanced diode mixer

Figure 8 shows one example of a single-balanced mixer using two

antiparallel diodes, a hybrid and a band-pass filter. The hybrid can either be a 180°-hybrid or a 90°-hybrid, but the latter one not commonly used in up-converters since its disability to produce positive mixing frequencies

(

ω

_LO

+

ω

_RF), which will be shown later. In Figure 8, the 180°-hybrid is used

which, ideally, phase shifts the LO-signal 180° on the

Δ

-port and leave the

phase of the LO untouched at the

Σ

-port. The IF-signal will be in phase at

both of these ports10.

V_LO* V_IF V_LO V_IF C I_RF2 I_RF1 B A BPF_Butterworth BPF1 Diode D2 Diode D1 Port RF Num=3 Hybrid180 HYB2 IN ISO Port LO Num=2 Port IF Num=1

Figure 8 Single-balanced diode mixer.

Under the first half period of the LO-signal the unshifted part of the signal (V_LO) will make D1 conduct (if diodes are treated as switches) and the shifted part of the LO (V_LO*) will make D2 conduct. That is, in other words, both diodes are short circuit to the IF. Conversely, both diodes will be open circuits to IF during the second half period of the LO, this causes mixing in the same “switch-like” manner as in a single diode mixer. Since IF is in phase over the diodes and the diodes conduct simultaneously the RF-current, created by the conductance switching, will simply be summed in node C.

10

Since the RF-signal is inserted in phase, a balun can be used for the LO, and the RF is simply connected to the diodes directly.

Since the IF and LO are connected to mutually isolated ports of the hybrid (if such is used) their isolation depends solely on the performance of the hybrid. Ideally the single-balanced mixer in Figure 8 should have very good LO to RF isolation. This is because when the two LO-signals (one phase shifted 180°)

enters node C11 they both cancels each other, ending up with no LO at the

output. However, this is only ideally, in reality there is no such thing as a perfect 180° phase shift and there does not exist perfectly matched diodes either.

Considering the diode as a non-linear device the current produced in the diode can, again, be approximated using power series. Thus

2 3

1 2 3

Diode

I

=

α

V

+

α

V

+

α

V

+K

(1.65)

describes the total current in the diode, where

V

is the total voltage over the

diode and

α α α

₁

,

₂

,

₃ are constants. Independent of whatever phase shift that

may occur in the hybrid one of the diodes is reversed with respect to the other; this will cause the voltage of one of the diodes to be opposite of the other. Currents and voltages over the two diodes are shown in a simplified picture of the mixer in Figure 9. Here the voltage over D1 will change sign, and hence the two currents I1 and I2 are

2 3 1 1 1 2 1 3 1 2 3 2 1 2 2 2 3 2 I V V V I V V V

α

α

α

α

α

α

= − + − + = + + + K K (1.66)

respectively. The total current at the output is

2 1 RF

I

=

I

−

I

(1.67) I_RF=I2-I1 C B A I1 I2 - V2 + - V1 + Port RF Port P1 Port P2 Diode D2 Diode D1

In the general case, the voltages appearing at node A and B, and also over the diodes are

1 1, 1, 2 2, 2, IF LO IF LO V V V and V V V = + = + (1.68) where 1, 1, 1, 1, 2, 2, 2, 2, cos( ), cos( ), cos( ) and cos( ) IF IF IF IF LO LO LO LO IF IF IF IF LO LO LO LO V A t V A t V A t V A t

ω

ϕ

ω

ϕ

ω

ϕ

ω

ϕ

= + = + = + = + (1.69)

where A_IF and

A

_LO is the amplitude of the IF- and LO-signals,

ω

_IF and

ω

_LOis

their frequencies and the

ϕ

’s is the phase shift that occurs in the hybrid.

If the hybrid in Figure 8 is used there will be a 180-degree phase shift from

the LO to the

Δ

-port and thus the voltages, V1 and V2, over the diodes will

be 1 2

cos(

)

cos(

)

cos(

)

cos(

)

IF IF LO LO IF IF LO LO

V

A

t

A

t

and

V

A

t

A

t

ω

ω

ω

ω

=

+

=

−

(1.70)

If V1 and V2 are inserted in the expression for the current at the output, and

after some trigonometry12, the following expression is found

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

3 2 1 3 2 3 2 3 2 3 3 1 1 cos 3 2 3 cos 2 cos 2 2 2 cos cos 3 3 2 cos 2 RF IF IF IF LO IF LO IF LO IF LO IF LO IF LO IF IF LO IF IF I I I A t A A t t t t A A t t t t A A A A t

α

ω

α

ω

ω

ω

ω

α

ω

ω

ω

ω

α

α

α

ω

= − = + − + + − − − + + ⎛ _{+ +} ⎞ ⎜ ⎟ ⎝ ⎠ (1.71) 12

Some useful trigonometric identities are cos(2 ) 1

2 2

( cos( ))

2

xt

As seen in the expression (1.71) above and also according to Maas [3] the current at the output contains no spurious responses where m and n both are

even. m and n are the coefficients appearing in

±

m

ω

_IF

±

n

ω

_LO. Also the

spurious’ arising from the cases when m is even and n is odd vanishes.

Mixing harmonics occurs only at order k, where

k

=

|

m n

+

|

. Wanted mixing,

when m and n are positive and 1, occurs, but also down-conversion when

1

m= − andn=1.

If the LO and IF input signals are interchanged the phase shift occurs at the IF instead. Thus 1 2

cos(

)

cos(

)

cos(

)

cos(

)

IF IF LO LO IF IF LO LO

V

A

t

A

t

and

V

A

t

A

t

ω

ω

ω

ω

=

+

= −

+

(1.72)

And the current appearing at the output now becomes

(

)

(

)

3 2 2 1 1 3 3 2 3 2

3

2

3

cos(

t)+

2

3

cos(

2

)

cos(

2

)

2

1

cos(3

)

2

2

cos(

)

cos(

)

RF LO LO LO IF LO LO IF LO IF LO IF LO LO IF LO IF LO IF LO

I

I

I

A

A

A A

cA A

t

t

t

t

cA

t

A A

t

t

t

t

α

α

α

ω

ω

ω

ω

ω

ω

α

ω

ω

ω

ω

⎛

⎞

=

− =

_⎜

+

+

_⎟

⎝

⎠

−

+

+

+

−

−

+

+

+

(1.73)

A quick comparison between the results (1.71) and (1.73) shows that the only difference is that the elimination of the spurious’ that arose from the case when m was even and n odd are reversed, that is spurious’ are eliminated when m is odd and n even.

A more interesting result is when a 90-degree hybrid is used. In the same manner the voltages becomes

1 2

cos(

)

cos(

)

cos(

)

cos(

)

IF IF LO LO IF IF LO LO

V

A

t

A

t

V

A

t

A

t

ω

ω

ω

ω

= −

+

=

−

(1.74)

The expression for the total output current becomes very long and is therefore presented in appendix C, Maple calculation 4. The main issue is however that there exist no frequency components where m and n are one and positive, only the down-conversion case. Hence a 90-degree hybrid could not be used in an up-converter. The result also shows that there is no “m even, n odd/m odd, n even” suppression in this case.

However, there is a solution to circumvent the unfortunate lack of up-conversion responses, and that is to connect the diodes parallel. This configuration, on the other hand, will only result in up-conversion, the down-conversion components vanish in this case.

To conclude this discussion the choice of hybrid preferably falls on a 180-degree if up-conversion is wanted. Considering what kind of spurious signals wanted (or unwanted) at the output the interconnection of the input signals to the hybrid becomes important. It might however be wise, in this application, to choose the one that causes 180-degree shift since this will eliminate the LO at the output due to the virtual ground at node C.

3.5.2 Double-balanced diode mixers

A double-balanced mixer [2] [3] [9] [36] [37] is actually two single-balanced diode mixers connected together, so whatever good features that comes with SB also comes with DB. To begin with, a DB uses a separate balun for the IF and RF-signal (the RF is tapped at the centre-tap of the IF balun) which gives it good IF to RF-isolation. This is due to the fact that the RF-port can be treated as a virtual ground when connected to the centre-tap of the balun, and therefore, the balance of this balun is important if high IF-suppression is wanted. Secondly, the LO to RF- and IF-isolation is caused in the same way as in SB. A basic DB mixer can be seen in Figure 10, here the input baluns are realized using ideal transformers.

Figure 10 A double-balanced mixer using ideal transformers as input baluns. Again, due to symmetry, the nodes A and A’ is seen as virtual grounds to the LO and, conversely, the nodes B and B’ is seen as virtual grounds to the IF.

LO IF B B' A' A Port RF_out XFERTAP XFer2 Port IF1 Port IF2 XFERTAP XFer1 Port LO2 Diode D1 Diode D4 Diode D3 Diode D2 Port LO1