1

Mixer with Diodes

2

Index1. Down-converter and Up-converter

2. Characteristic parameters of mixers

3. Diodes

• Schottky diodes

• Quadratic analysis

4. Directional couplers

• Rat Race

5. Single diode mixer

6. Balanced mixer

7. Polarization circuits

Down-converter and Up-converter

Mixers are non-linear 3-port devices that are usedto translate a signal from one frequency to another. In particular, mixers can operate a translationtowards low frequencies (down-converter) or high frequencies (up-converter)

I F

L O

(down-converter)

R F

(up-converter)

I F R F

L O

3

When the mixer operates as a down-converter the RF signal (continuous or modulated wave) is sent to the RF input of the mixer, whereas the signal of a localoscillator is applied at the LO input, and the intermediate frequency signal (IF) is generated at the output

fIF = fLO - fRF

When operating as an up-converter, the two mixer inputs are intermediate frequency and local oscillatorsignals, whereas the radio frequency signal is generatedat the mixer output

fRF = fLO - fIF

A mixer can operate in heterodyne mode with 2 different frequencies at the two inputs (i.e. in transmitter and receiver apparatus)

Or it can operate in homodyne mode with two signals at the same frequency at the two inputs (i.e. phase comparator apparatus)

4

Characteristic parametersof mixers

In the following, some parameters will be defined in order to characterize a mixer

In particular, these parameters will be referred to the «down-converter» operating mode

Conversion lossConversion loss is defined for passive mixers and it is the ratio, expressed in dB, between the RF input power and the IF output power:

It is due to mismatches, losses in the resistive elements, and losses connected to the generation of signals atfrequencies other than fRF and fIF

Typical values of conversion loss for passive mixers varybetween 6 and 12 dB

IF

RF10C

P

Plog10)dB(L

5

Conversion GainConversion gain is defined for acive mixers and it is the ratio, expressed in dB, between the IF output powerand the RF input power:

Typical values of conversion gain for active mixers varybetween 5 and 30 dB

𝐺𝑐 = 10 𝑙𝑜𝑔10𝑃𝐼𝐹

𝑃𝑅𝐹

Isolation

Isolation gives a measure in dB of the ratio between the input power at a port and the output power at anotherport (at the same frequency)

Particularly critical in the mixer is the OL-RF isolationsince local oscillator level is always higher than that of the RF signal

Isolation values of at least 20 ÷ 30 dB must be achieved

J

I10

P

Plog10)dB(I

6

Noise factor

Noise factor is defined as the ratio between the following two ratios: the input available signal power over the available noise power (RF), and the output availablesignal power over the available noise power (IF):

This parameter, is always greater than 1

IO

OI

O

O

I

I

NS

NS

N

S

N

S

F

SensitivitySensitivity is the minimum power of the signal that can be converted and that gives rise to an appreciable output signal. The sensitivity (S) of the mixer is defined for S0/N0 = 1 and therefore we have:

S = F NI

where :

NI = <Vi2>/4R = 4KTB /4R = KTB

S = F KTB

SdBm = NFdB -174 dBm/Hz + 10 log10 B (where: T=290K)

7

1 dB compression point

In a mixer the intermediate frequency output poweris directly proportional to the radiofrequency input power. This is true up to a fixed level of the input power. Above this level the output grows more slowlyand is defined "compressed"

The 1 dB compression point (P1dB) is defined as the power input level whereby the output is 1 dB belowthe level predicted by the linear extrapolation of the curve

Dynamic range

For low input power the output is masked by the noise. For powers higher than the sensitivity S the output input link is linear (in power) until the 1 dB compressionpoint (P1dB) is reached. The dynamic range is defined asthe range of input values between S and P1dB

dynamic range

1 dB

S P1dB PIN(RF)

POUT(IF)

8

Intermodulation products

The nonlinear behavior of the mixer also producesintermodulation distortion

If two frequency signals f1 and f2 are applied at the input of the mixer such that f1 - f2 < fIF, furtherfrequencies are generated (f0 - 2f1 + f2 and f0 - 2f2 + f1) and they are placed near the IF frequency

The power of these signals increases as the cube of RF input power

These signals are called third order intermodulationproducts because the sum of the multiplicativecoefficients of f1 and f2 is equal to 3

The intermodulation distortion is specified by givingthe third order intercept point (P3 in the figure below) that is input RF power level at which the line thatextrapolates the power trend of the intermodulation products intersects that which extrapolates the curve of the IF signal power

prodotti di intermodulazione del terzo ordine

POUT (IF)

PIN (RF) P3

Third order intermodulation products

Third order intercept point

1 dB/dB

3 dB/dB

9

Diodes

Schottky Diodes

Tempo V

Iout(t)

Tem

po

I

Vp cos 1t

1eI)V(I V

S

10

Schottky diodes are realised with metal semiconductorjunctions

Schottky diodes have an inverse saturation current higherthan the one of a conventional PN diode and, consequently, can have a zero thershold voltage (V = 0)

Schottky diodes operate with an unipolar currentconsequently their diffusion capacity (CJ), that is relatedto the variation of minority carriers, is negligible

CJ RJ

RS

ID

VD Cj 0

I(V,t) = I0 + i[v(t)]V(t) = V0 + v(t)

Quadratic analysis

2

0

0VV

2

2

0

0VV

0 )VV(V

)V(I

2

1)VV(

V

)V(I)V(I)V(I

Voltage and current of the diode can be expressed as the sum of a static component and of an alternating component:

Under small signals approximation we can consider a Taylor series expansion of the current, around the polarization point, stopped at the second order

11

J

2

J R2

)t(v

R

)t(v)t(i

The inverse of the dynamic resistance (RJ) is the tangentin V0 to the I-V curve of the diode. RJ is a function of the bias current and typically assumes values of the order of k in the absence of bias or for polarizations close to zero, and values of a few tens of ohms for direct biasesgreater than 0.2 V

0S0V

S

0VV

IIeIV

)V(I

J0VV

2

2

RV

)V(I

0VVJ V

)V(I

R

1

The coefficients of the expansion are given by:

Directional Couplers

1

2 3

4

b3

a3

b4

a4

b2

a2

b1

a1

The directional coupler presents a main line (1-4) and a secondary line (2-3)

The purpose of the coupler is to transfer a knownfraction of power from the main line to the secondary line with directional behavior

12

00SS

00SS

SS00

SS00

S

4241

3231

2423

1413

The power entering in port 1 must exit to port 3 and 4 without coupling with the port 2, whereas the powerentering in 2 must exit in 3 and 4 without coupling with port 1

Similar properties must apply for the port 3 and 4. Finally, each port must be matched when the others are closed on matched loads.

In conclusion, the scattering matrix of an ideal DC will be

31

IN

P

Plog10C

21

31

P

Plog10D

21

IN

P

Plog10I

2

31S

1log10C

0

2

1*INININ

Z2

1Re

2

1P

aIV

2

1

2

31

0

2

3

0

31 SZ2

1

Z

1

2

1P ab

2

21

2

31

S

Slog10D

2

21S

1log10I

The main parameters used to characterize a directional coupler are the coupling C, the directivity D and the insolation I which are defined as follows:

with these definitions C, D and I are positive quantities

2

1

2

21

0

2

2

0

21 SZ2

1

Z

1

2

1P ab

13

Rat Race - 180° Hybrid

1

3 2

4

Piano di simmetria

+

- V

Y0 Y0 Y0

Y0 Y0

Y0 Y0

Y0

2

0Y

𝑍𝐶 = 𝑍0 2 = 70.7 𝑌𝐶 =𝑌0

2

𝐶 = 3 𝑑𝐵 = 10𝑙𝑜𝑔101

𝑆31 2

𝑆312 = 10

−310 =

1

2

𝑆31 =1

2

For the Rat Race:

and it results:

simmetry plane

𝑉1 =𝑉02

𝑉2 =𝑉0

2𝑒−𝑗𝛽4

𝜆

4 +𝑉0

2𝑒−𝑗𝛽2

𝜆

4 =𝑉0

2𝑒−𝑗

2𝜋

𝜆4𝜆

4 +𝑉0

2𝑒−𝑗𝛽2

𝜆

4 =

𝑉0

2(cos 2𝜋 – j sen 2𝜋) +

𝑉0

2(cos 𝜋 – j sen 𝜋) = 0

𝑉3 =𝑉0

2𝑒−𝑗

2𝜋

𝜆

𝜆

4 +𝑉0

2𝑒−𝑗

2𝜋5

𝜆

𝜆

4 =𝑉0

2𝑒−𝑗

𝜋

2 + 𝑉0

2𝑒−𝑗

𝜋

2 = -j𝑉0

𝑉4 =𝑉0

2𝑒−𝑗

2𝜋

𝜆

3𝜆

4 +𝑉0

2𝑒−𝑗

2𝜋

𝜆

3𝜆

4 =𝑉0

2𝑒−𝑗

3𝜋

2 + 𝑉0

2𝑒−𝑗

3𝜋

2 = j𝑉0

𝑆31 =𝑏3

𝑎1=

𝑉3

𝑉1=

−𝑗

2𝑆41 =

𝑏4

𝑎1=

𝑉4

𝑉1=

𝑗

2

Putting:

we have:

and:

Taking into account the term we achieve: 1

2

14

0011

0011

1100

1100

2

jS

-j

j

1-1

The whole scattering matrix of the Rat race is:

The S matrix can be easilyconstructed observing thateach /4 line gives rise to a phase delay of /2 (-j multiplication)and observing that each column ofthe matrix is achieved by feedingthe corresponding port

𝑏3 = 𝑆31𝑎1 + 𝑆32𝑎2 = −𝑗𝑎1

2+ −𝑗𝑎2

2=

−𝑗

2(𝑎1 + 𝑎2)

𝑏4 = 𝑆41𝑎1 + 𝑆42𝑎2 = 𝑗𝑎1

2+ −𝑗𝑎2

2=

−𝑗

2(𝑎2 - 𝑎1)

2

1 3 + sum

4 - difference

If ports 1 and 2 are fed together we obtain:

Signals are added at port 3 (sum) and subtractedat port 4 (difference)

15

Single diode mixer

Due to the presence of the capacitor (C) the diode is in parallel for the sum signal but in series for the IF signal which crosses the filter and arrives at the output

Since the equivalent circuit of the small signal diode isnot a simple 50 resistor, a matching network is insertedto match the "+" output of the DC to the diode

In this way, for an ideal DCD, the RF and LO inputs are also matched and there is infinite isolation between the two

Hibrid 180°

RF

LO

Matching

networkLow pass

filterIF

+

-50

Note that with this scheme, good RF-LO insulation isobtained, but the operating band is small since the 180°DCs are frequency selective

Moreover, high conversion losses occurs because the 180° DCD operates the division of the input powerbetween two outputs and therefore half of the RF signalstrength is lost

The low pass filter is used to prevent the harmonicsgenerated by the diode from propagating towards the IF output

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2

JJ

J

2

J

)t(v2

G)t(vGR2

)t(v

R

)t(v)t(i

v(t) = V0sin(0t) + VRsin(Rt)

2RR00J

RR00J tsenVtsenV2

GtsenVtsenVGti

In the quadratic approximation the link between voltageand current across the Schottky diode is of the type:

If we set: vLO = V0sin (0t) and vRF = VRsin (Rt) we have:

which inserted in the above equation gives:

tcosVV2

GtcosVV

2

Gt2cos

4

VG

4

VGt2cos

4

VG

4

VGtsenVGtsenVGti

R0R0J

R0R0J

R

2RJ

2RJ

0

20J

20J

RRJ00J

Exploiting the various terms we achieve

If 0 > R in the current expression the pulsationsare present in the order:

0 - R, R, 0, 2R, 0 + R, 20

where 0 - R = I is the intermediate frequency

17

tsenVtsenVtsenVtv IIRR00

2IIRR00J

IIRR00J

tsenVtsenVtsenV2

G

tsenVtsenVtsenVGti

The previous analysis assumed an open output. If the output is closed on a 50 Ohm load the intermediate frequency signal (0 - R = I) "recirculates" in the diode. The actual voltage applied to the diode will therefore be:

By inserting v(t) in the quadratic diode approximation weachieve:

The other frequencies (20, 2R, etc.) do not recirculate as they are blocked by the band pass filter. Also note that in the above equation the phase displacementbetween the various terms has been neglected

tcosVVtcosVV

tcosVVtcosVV

tcosVVtcosVV

t2cos2

V

2

Vt2cos

2

V

2

Vt2cos

2

V

2

V

2

G

tsenVGtsenVGtsenVGti

RIRIRIRI

I0I0I0I0

R0R0R0R0

I

2I

2I

R

2R

2R

0

20

20J

IIJRRJ00J

(1)

18

and therefore, with 0 > R the following pulsations are present in the current:

0 - R = I (intermediate frequency)2I

R - I (image frequency)R

0

0 + I (image frequency)2R

0 + R (sum frequency)20

The terms R - I and 0 + I are called image frequencies since they are respectively the mirrors of 0 with respect to R and R with respect to 0

In conclusion, with a sinusoidal signal 0 applied to the LO input and a sinusoidal signal R applied to the RF input, atthe output of the diode there is the intermediate frequency but also a series of frequencies that can interfere with the desired signal (I)

In particular, if image frequencies recirculate in the diode, they can mix with OL or RF and can give rise to terms at I

with arbitrary phase which can reduce the level of the output signal (i.e. IM - R = R - I - R = -I )

Note that tones at the image frequencies can create interferences even when they occur at the RF input together with the desired frequency. In fact, in this case, there is the direct conversion of these signals to the IF frequency and therefore their superposition with the useful signal. To solve this problem, an input filter can be inserted

19

Finally, note that the analysis carried out stems from having considered a quadratic model for the diode

If we consider a series development of the currentextended to orders greater than 2 ° (analysis for large signals), further tones are found at the frequenciesn 0 + m R (called spurius signals)

Balanced mixer

AD 180° filtro passa bandaaa

RF

LO

IF

v1

v2

i2

i1

+

This type of mixer uses a 180 ° DC for conversionfollowed by a pair of diodes.In the figure matching networks and polarizationnetworks have been omitted

20

tsenVtsenVtV RR00

tsenVtsenVtV RR00

22J

2J2 tv2

GtvGti

22J

2J2 tv2

GtvGti

At the "+" output of the DC we have:

At the "-" output of the DC we have:

in this expression the further factor of reduction of the tension due to the DC has been neglected.Note that diode 1 is in the same situation describedabove. Diode 2 is reversed, so we have:

2/1

tcosVVtcosVV

tcosVVtcosVV

tcosVVtcosVV

t2cos2

V

2

Vt2cos

2

V

2

Vt2cos

2

V

2

V

2

G

tsenVGtsenVGtsenVGti

RIRIRIRI

I0I0I0I0

R0R0R0R0

I

2I

2I

R

2R

2R

0

20

20J

IIJRRJ00J2

The study for diode 2 can be conducted by observingthat the actual voltage applied to this diode is:

v2 (t) = - V0sin (0 t) + VRsin (Rt) + VIsin (I t)

By inserting this equation in the diode quadraticequation we have:

(2)

21

At the output of the device the currents (1) and (2) addup and therefore the opposite terms are eliminated, in particular the image frequency at the pulsation:

R - I = R - (0 - R) = 2 R - 0

is eliminated, while, the pulsation term:

0 + I = 0 + (0 - R) = 2 0 - R

is not eliminated

For the intermediate frequency currents have the samesign and therefore the two outputs add up

A further advantage of this type of mixer is that itallows the elimination of any noise superimposed on the amplitude of the local oscillator

This noise, seen as an additive signal n (t) gives rise to an output to the "+" and "-" ports in phase oppositionand therefore the resulting intermediate frequencycurrents are in phase opposition and are canceled atthe output

Therefore, any additive noise on the amplitude of the local oscillator is eliminated but this effect is notextended to the oscillator phase noise

22

Diode Polarization

Diode in Parallel to the transmission line:

To polarise the diode we use a resistor and a voltagegenerator

R

B V

B

In this circuit, however, there are two types of problems

First of all, due to the electrical continuity of the circuit, all the other devices present on the line are polarized

Secondly, the radio-frequency signal sees a short circuitrepresented by the generator

Therefore, we have to decouple the continuous operationof the circuit from the high frequency operation. Decoupling is commonly obtained by introducingcapacitors to the right and left sides of the diode and an LC circuit (low pass) between the diode and the generator

23

Everything is schematized as in the figure. For the DC current the capacitor CB behaves like an open circuit while the inductance ‘L’ is a short-circuit and therefore the diode is correctly polarized

At high frequencies, the behavior of the reactiveelements is reversed and therefore the group L-C blocks the path of the radio-frequency signaltowards the generator

Z0 = 50

RB VB

CB

L

C

Z0 = 50

CB

Diode in series to the transmission line:

On the basis of considerations similar to thosemade previously, we have as a possible polarizationscheme the one shown in the figure

RB

CB L

LA

C VB

CB

Z0 = 50 Z0 = 50

24

At frequencies greather than 1 GHz, inductor parasiticare not negligible. So we can realize a polarizationnetwork using planar technology as reported in the figure

The open line in A behaves in B, after the /4 section, as a short for the RF. This short is in parallel to the polarization resistance of the diode, so the overall behavior is still that of a short

Moving a further section /4 the short in B is seenby the main line as an open and therefore does notaffect the signal along the line itself

Note that the open line is generally made with a wider microstrip in order to ensure good powersupply isolation over a wide band

25

This can be explained observing that a wider strip has a higher characteristic admittance and therefore the short-circuit type behavior that occurs in B for l = /4 ismaintained over a wide band

To complete the diode polarization, an additional short (C in the figure) is used followed by a /4 line. the circuitallows the closure of the DC curernt but is seen by the main line as an open circuit

In order to avoid the spep discontinuity in B,the typical realization of the polarizatio networkis that showed in the figure