Download Fig. 12 Measured characteristics of a 2.4 GHz bandpass filter (FDK

(e) FET Mixers : have conversion gain (not loss)
Pozar (RF Ch 7)
* R. A. Pucel, D. Masse, and R. Bera, "Performance of GaAs MESFET Mixers at X Band," IEEE
Tram.Microwave Theory and Techniques, vol. MTT-24, pp. 351-360, June 1976.
Single-Ended FET Mixer
 There are several FET parameters that offer nonlinearities used for mixing
 The strongest is the transconductance, gm, when the FET is operated in a
common source configuration with a negative gate bias (Vgs )
 When used as an amplifier, the gate bias voltage is near zero, or positive, so the
transconductance is near its maximum value, and operates as a linear device.
 When the gate bias Vgs is near the pinch-off region:
 where the transconductance approaches zero
 a small variation of gate voltage causing a large change in transconductance
leading to a nonlinear response.
7-2
 Thus the LO voltage can be applied to the gate of the FET to pump the transconductance
to switch the FET between high & low transconductance states
=> provide mixing as the switching model
(see the Diode Large-Signal Model for Mixer)
 RF chokes are used to bias the gate at a negative voltage near pinch-off and to provide
a positive bias for the drain of the FET.
 A bypass capacitor at the drain provides a return path for the LO signal, and a
LPF provides the final IF output signal
* Based on the standard unilateral equivalent circuit for a FET.
vRF (t )  VRF cos RF t

vLo (t )  VLO cos LO t
 Zg = Rg + jXg : Thevenin source impedance for the RF input port
 ZL = RL + jXL : Thevenin source impedance at the IF output port.
 LO port has a real generator impedance of ZO
=> since we are not concerned with maximum power transfer for the LO signal.
 The same as for the large-signal analysis of the diode mixer, the LO pumped
FET transconductance is espressed as a Fourier series of harmonics of LO
signal:

 g n cos n LO
g (t )  g 0  2
n 1




2008
not having an explicit formula for the transconductance
must rely on measurements for values of g n ,
in the switching model, the desired down-conversion is due to (n = 1)
only need g1 coefficient & the typical measured value in the range of 10 mS
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H.-R. Chuang, EE NCKU
7-3
 Conversion gain of the FET mixer can be found as (?)
2
P
Gc  IF avail 
PRF avail
VDIF RL / Z L
VRF
2
2
/ 4 Rg
2
4 Rg RL VDIF
*(see ch3, p.3-26, conjugate matching formula)

2 V
RF
Z
L
 VDIF : IF drain voltage
 Zg & ZL chosen for maximum power transfer at the RF and IF ports.
 The RF signal across the gate-to-source capacitance is given as:
VcRF 
&
VRF
VRF
(*)

jRF C gs [( Ri  Z g )  (1 / jRF C gs )] 1  jRF C gs ( Ri  Z g )
vcRF (t )  VcRF cos RF t
From

g (t )  g 0  2


 g n cos(nLOt )
n 1

g m (t ) vcRF (t )  [ g 0  2
 g n cos n(LOt )]VcRF cos(RF t )
n 1
 g 0VcRF cos(RF t )  2 g1VcRF cos( RF t ) cos( LOt )  
 The down-converted IF signal can be extracted from the second term by using
the usual trigonometric identity:
g m (t )[vcRF (t ) |  IF ]  g1 VcRF cos(IF t ) (?)
 Then the IF component of the drain voltage (in phasor form) is (by using (*) )
 R Z 
VDIF  ( g1 VcRF )Rd // Z L    g1 VcRF  d L 
 Rd  Z L 

 Rd Z L 
VRF


  g1
 1  j RF C gs ( Ri  Z g )  Rd  Z L 


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H.-R. Chuang, EE NCKU
7-4
 The conversion gain GC (before conjugate matching) is then
from
4 Rg RL VDIF
Gc 
2 V
RF
Z
L
2
&

 Rd Z L 
VRF


VDIF   g1
 1  j RF C gs ( Ri  Z g )  Rd  Z L 


we have
VRF

( Rd Z L )

g


1
4 Rg RL
 1 jRF C gs ( Ri  Z g )  Rd  Z L
Gc |not

2
VRF
ZL
matched
2
2
 2 g1Rd 
Rg
RL


2
2
 RF C gs  [( R  R )  ( X  1 ) ] [( R  R ) 2  X 2 ]
i
g
g  C
d
L
L


RF gs

By conjugately matching the RF & IF ports:
( Rg  Ri , X g  1/ RF Cgs , RL  Rd , X L  0 )
2

 2 g1Rd 
Ri
Rd
2 g12 Rd

Gc  

 RF C gs  [( 2 R ) 2  (0) 2 ] [( 2 R ) 2  (0) 2 ] 4 C 2 R
i
d
RF gs i


* Practical mixer circuit generally use matching circuits to transform the FET
impedance to 50 ohm for the RF, LO & IF ports.
2008
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H.-R. Chuang, EE NCKU
7-5
 Diode Large-Signal Model for Mixer (Pozar RF P233)
I (V )  I s (eV  1)
v(t )  VRF  VLo  Vr cos r t  Vo cos o t
For a diode small-signal approximation
I Vo  v(t )   I o  vGd  v 2 Gd 2   ...................
Gd
2
G
 2d
 v 2 Gd 2  

Gd
4
Vr cosr  Vo cosot 2
V 2 cos2 rt  2VrVo cosrt cosot  Vo2 cos2 ot 
r


V 2  V 2  V 2 cos 2 t  V 2 cos 2 t  2V V cos   t  2V V cos   t 
r
o
r
o
o
r o 
r 
o
r o
r
o

 r



de siredIF signal
- Usually LO power (typical 5~10 dBm) will violate the small signal approximation
of the diode
- Large signal model is needed for fully nonlinear analysis
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H.-R. Chuang, EE NCKU
7-6
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7-7
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H.-R. Chuang, EE NCKU
7-8
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7-9
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7-10
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H.-R. Chuang, EE NCKU
7-12
EXAMPLE 7.2 MIXER CONVERSION GAIN
A single-ended FET mixer is to be designed
A single-ended FET mixer is to be designed for a wireless local area network receiver operating at
2.4 GHz. The parameters of the FET are: Rd = 300 ?, R, = 100, Cgs =0.3 pF, and g\ = 10 mS.
Calculate the maximum possible conversion gain.
Solution
This is a straightforward application of the formula for conversion gain given
in (7.54):
Note that this value does not include losses due to the necessary impedance matching networks.
FIGURE 7.12 A dual-gate FET mixer.
g (t )  g0  2 g1 cos LO t  2 g 2 cos 2LO t
 I RF   g 0 g1 g 2  VRF  I RF Rg 

 I    g g g   I R
0
1   IF IF

 IF   1

 I IM   g 2 g1 g 0   I IM Rg


g1VRF
I SC   I IF R  0 
IF
1  g 0 Rg  g 2 Rg
VOC  VIF I 0 
IF
g1VRF
2 g12 Rg
 g0 g 2 Rg  g0 (1  g0 Rg )
2 g12 Rg
I SC
GIF 
 g0 
VOC
1  g 0 Rg  g 2 Rg
PIF  avail 
I SC
2
4GIF2
V
PRF  avail  RF
4 Rg

(1  g0 Rg  g 2 Rg ) g0 (1  g0 Rg  g 2 Rg )  2 g12 Rg
P
Lc  RF  avail 
PIF  avail
g12 Rg
Lc 

2( x  a)( x  a  b)
bx
xopt  a ( a  b)
Lc  min 


2 a  a ( a  b) a  b  a ( a  b)
b a ( a  b)
g n  I s I n (VLO )  I s
2008
eVLO
2VLO
  2a 
a ( a  b)
b
2  2 1 
1 b / a
1 1 b / a

H.-R. Chuang, EE NCKU
7-13
b
2 g12

1
a g 0 ( g 0  g 2)
Te 
nT
( Lc  1)
2
g (t ) 

1
2
n

sin
cos n LO t
2 n 1 n
2

1

2
n
i (t )  g (t )vRF (t )   cos RF t  
sin
cos nLO t cos RF t 
n
2
n 1
 2





1
2
n
cos(RF  RF )t  cos(RF  nLO )t 
VRF cos RF t  
sin
2
n
2


n 1
Mixer
Type
Diode
Conversion
Noise
1dB
3rd Order
Gain
Figure Compression
Intercept
-5 dB
FET
5-7dB
6 dB
-6 to –1 dBm
7-8dB
5 dBm
5 to 6 dBm
20 dBm

g (t )  g 0  2  g n cos n LO
n 1
2
VDIF RL
P
Gc  IF  avail 
PRF  avail
VcRF 
ZL
2
VRF
4 Rg
2
4 Rg RL VDIF

2 V
RF
Z
L
VRF


j
jRF C gs  Ri  Z g 

RF C gs 


2


VRF
1  jRF C gs ( Ri  Z g )
g m (t )vcRF (t )  g 0VcRF cos RF t  2 g1VcRF cos RF t cos LO t  
g m (t )vcRF (t ) |RF  g1VcRF cos IF t
 R Z 
 Rd Z L 
 g1VRF


VDIF   g1VcRF  d L  
R

Z
1

j

C
(
R

Z
)
R

Z
L
RF gs i
g  d
L
 d
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H.-R. Chuang, EE NCKU
7-14
2
 2 g1Rd 
Rg
RL

Gc |not

 RF C gs  
2  R  R 2  X 2
matched 



d
L
L
1
 R  R 2 X 
 
i
g
g


RF C gs  




Gc 



g12 Rd
2
42RF C gs
Ri
Using the small-signal approximation of (7.20) gives the total diode current as
The first term in (7.24) is the DC bias current, which will be blocked from the IF by the DC blocking
capacitors. The second term is a replication of the RF and LO signals, which will be filtered out
by the low-pass IF filter. This leaves the third term can be rewritten using trigonometric identities
as
This result is seen to contain several new signal components, only one of which produces the desired
IF difference product. The two DC terms again will be blocked by the blocking capacitors, and
the 2&>RF, 2&>Lo^ and &)RF + (ULO terms will be blocked by the low-pass filter. This leaves
the IF output current as
where (UIF = O)RF ?a LO is the IF frequency. The spectrum of the down- converting single-ended
mixer is thus identical to that of the idealized mixer shown in Figure 7.1b.
Large-Signal Model
While the small-signal analysis of a mixer demonstrates the key process of frequency conversion,
it is not accurate enough to provide a realistic result for conversion loss. This is primarily
because the power supplied to the mixer LO port is usually large enough to violate the smallsignal approximation. Here we consider a fully nonlinear analysis of a resistive diode mixer [3][4], with the goal of deriving an expression for the conversion loss defined in (7.10). The term
"resistive" in this context means that reactances associated with the diode junction and package
are ignored, to simplify the analysis. Our results should be useful in understanding the nonlinear
operation and losses of the diode mixer, but for actual design purposes modem computer-aided
design (CAD) software is preferred [5].Such software can model the diode nonlinearity, as well
as the effects of diode reactances and impedance matching networks.
We again assume a diode I-V characteristic as given by (7.18), with a relatively low-level RF
input voltage given by (7.22), and a much larger LO pump signal given by (7.23). A DC bias
current may also be present, but will not directly enter into our analysis. As we have seen from
the small-signal mixer analysis, these two AC input signals generate a multitude of harmonics
and other frequency products:
&)RF
RF input signal (low power)
&IIF = &IRF ?'"LO IF output signal (low power)
(DIM = (ULO ?t^iF image signal (low power)
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H.-R. Chuang, EE NCKU
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(ULO
LO input signal (high power)
?>LO
harmonics of LO (high power)
racfLO ?'MIF
harmonic sidebands of LO (low power)
is leaves the IF output current as
In a typical mixer, harmonics of the LO and the harmonic sidebands are terminated reactively, and
therefore do not lead to much power loss. This leaves three signal frequencies of most
importance: &IRF, a>w, and n)iw To evaluate conversion loss, we will find the available power
of the RF input signal, the power of the IF output signal, and the power lost in the image signal.
The image signal is important because it is relatively close in frequency to the RF signal, and
thus sees essentially the same load. We will see that approximately half the input power gets
converted to the image frequency. Note that the image term at frequency &>IM = WLO - "RF =
^LO - <^RF was not explicitly shown in the small-signal expansion of (7.24), since this product
is generated by the u3 term of (7.20).
Under the assumption that the RF input voltage is small, we can write the AC diode current as a
Taylor series expansion about the LO voltage as
This Taylor series is similar in form to (7.20), as used for the small-signal analysis, but with the
important difference that the expansion point here is about the LO voltage, where as(7.20) was
expanded about the DC bias point. The first term in (7.26) is due only to the LO input, and does
not enter into the calculation of conversion loss. The second term is a function of the RF and LO
input voltages, and will provide a good approximation for the three products at frequencies ww,
w^, and &>IM, with a large LO pump signal. The coefficient of the second term has dimensions
of conductance, so we can use (7.18) to write the differential conductance as
Then for small input voltages v(t) we can write the resulting diode current as
We see from (7.27) that g(t) is a real number (consistent with our description as a resistive mixer)
and is a periodic function of the LO frequency. Thus, g(t) can be expressed as a Fourier cosine
series in terms of harmonics of w^o'with Fourier coefficients given by
where In(x) is the modified Bessel function of order n, defined in Appendix B. Now let the AC diode
current consist of three components at the frequencies &)RF, (DIF, and
where /pp, /IF, and /[M are the amplitudes of the RF, IF, and image signals to be determined, If the
RF voltage of (7.22) is applied to the diode through a source resistance Rg, and the IF and image
ports are terminated in load resistances /?ip and Rg, respectively, then the voltage
FIGURE 7.4 Equivalent circuit for the large-signal model of the resistive diode mixer.across the
diode can be written as
(7.32)
The equivalent circuit consists of a three-port network, with one port for each of the frequency
components at &)RF, a>ip, and a>m[, as shown in Figure 7.4. We assume the terminations for
the RF and image ports are identical, because aiRp is very close to u>wi, while the termination
for the IF port may be different.
2008
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H.-R. Chuang, EE NCKU
7-16
Using the first three terms of the Fourier series of (7.29) for the diode differential conductance
gives
(7.33)
Multiplying the voltage of (7.32) by the conductance in (7.33), and matching like frequency terms
with the current of (7.31) gives a system of three equations for the unknown port currents:
(7.34)
where VRF is the source voltage, and the gn 's are defined in (7.30). Note that multiplication of
(7.32) by (7.33) creates several frequencies in addition to oipp, (UIF, and (UIM, but we assume
these frequencies to be reactively terminated so that they do not lead to significant power
dissipation.
The easiest way to find the available power from the IF port is to first find the Norton equivalent
source for the IF port. As shown in Figure 7.5, this consists of a current source equal to /sc, the
short-circuit current at the IF port, and Gip, the conductance seen looking into the IF port. This
conductance can be found as GIF = /sc/ Voc, where Voc is the open-circuit voltage of the IF port.
The short-circuit IF port current can be found by setting R^p == 0 in (7.34) and solving for /ip.
After some straightforward algebra, we obtain
FIGURE7．5
Norton equivalent circuit for the IF port of the large-signal model of the resistive diode mixer.
The open-circuit IF port voltage is found by setting /ip = 0, and solving (7.34) for Vy.
Then the Norton conductance of the IF port is
The available output power at the IF port is
and the available input power from the RF source is
So from (7.10) the conversion loss is (not in dB)
Note that the conversion loss does not depend on the IF port termination, /?ip, because of the use of
available powers. It does depend on Rg, the RF and image port terminations, so it is possible to
minimize the conversion loss by properly selecting Rg. If we let x = I//?., a = go + g2?and b =
2g^/go, then (7.39) can be rewritten as
(7.40)
Differentiating with respect to x and setting the result to zero gives the optimum value of x as
for which the minimum value of conversion loss is
(7.41)
We can evaluate this result by approximating values for go, gi, and g^. For an LO input power of
10 mW, VLO is about 0.707 V rms, and a = 1/28 mV, so aV^n, the argument of the modified
Bessel functions for gn given in (7.30), is approximately 25. Thus the modified Bessel functions
can be approximated asymptotically using the large-argument formula given in Appendix B, and
the gn's simplified as
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H.-R. Chuang, EE NCKU
7-17
and the minimum conversion loss of (7.42) reduces to L(. = 2, or 3 dB. This means that half the
RF input power is converted to IF power, and half is converted to power at the image frequency.
In principle this result could be improved by terminating the image port with a reactive load, but
it is usually difficult in practice to separate the image termination from the RF termination. Also,
this result is highly idealized in that it assumes no power loss at higher harmonic frequencies, and
it ignores diode reactances.
This same model can be used to derive the SSB noise temperature for the resistive mixer as
(7.45)
where n is the diode ideality factor and T is the physical temperature of the diode [3].
Switching Model
The large-signal model suggests that the diode mixer can be viewed as a switch. As the LO
voltage cycles between positive and negative values of cosecant, the diode becomes conducting
or nonconducting, respectively. Thus, the diode conductance (the ratio of diode current to diode
voltage) switches between large values and zero at the same rate as the LO voltage. Figure 7.6
shows a typical diode conductance waveform, where T = ITCJW^O is the period of the LO
waveform.
The conductance waveform of Figure 7.6 can be calculated directly from the diode V-I
characteristic of (7.18), or from the Fourier series representation of (7.29). But since a
conductance greater than a few Siemens is essentially a short circuit, we can approximate the
diode conductance as the square wave shown in Figure 7.7. This square wave has a Fourier
transform given by
(7.46)
which is similar in form to the Fourier series of (7.29).
An equivalent circuit of the diode mixer then consists of the RF input voltage applied across a
load resistor in series with an ideal switch, as shown in Figure 7.8. The time-varying
FIGURE 7.6 Conductance waveform of a mixer diode pumped with a large-signal
FIGU運7．7
LO voltage waveform and idealized square-wave diode conductance waveform for
the switching model of a diode mixer.
switch conductance is given by (7.46). The diode current can be found by multiplying the RF input
voltage of (7.22) by the conductance of (7.46):
Filtering all but the lowest-frequency component for the n = 1 term of the summation gives the
desired IF output as
The switching model is useful for mixers of any type, including the FET mixer discussed in the
following section. Note that the switching model of a mixer can be considered as a linear, but
time-varying, circuit.
FIGURE 7.8 Equivalent circuit for the switching model of the diode mixer.
7.3 - FET MIXERS
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H.-R. Chuang, EE NCKU
7-18
Mixers can also be implemented by using the nonlinear properties of transistors. FETs, in
particular, offer low noise characteristics and easy integration with other circuitry such as
switches and low-noise amplifiers. Transistor mixers can provide conversion gain, but their noise
figure is generally not as good as can be obtained with diode mixers. PET mixers also offer
higher dynamic range. The following table compares the characteristics of typical diode and FET
mixers.
Because a FET mixer has conversion gain, but usually worse noise figure, the proper comparison
with a diode mixer should include the cascade effect of adjacent stages.
In this section we will analyze the single-ended FET mixer, and derive an expression for its
conversion gain. We will also discuss a few other popular FET mixer configurations.
Single-Ended FET Mixer
There are several FET parameters that offer nonlinearities that can be used for mixing but the
strongest is the transconductance, ^, when the FET is operated in a common source configuration
with a negative gate bias. Figure 7.9 shows the variation of transconductance with gate bias for a
typical FET. When used as an amplifier, the gate bias voltage is near zero, or positive, so the
transconductance is near its maximum value, and the transistor operates as a linear device. When
the gate bias is near the pinch-off region, where the transconductance approaches zero, a small
variation of gate voltage can cause a large change in transconductance, leading to a nonlinear
response. Thus the LO voltage can be applied to the gate of the FET to pump the
transconductance to switch the FET between high and low transconductance states, and provide
mixing in much the same manner as the switching model discussed in the previous section.
The circuit for a single-ended FET mixer is shown in Figure 7.10. A diplexing coupler is used to
combine the RF and LO signals at the gate of the FET. An impedance matching net-work is also
usually required between the inputs and the FET, which typically presents a verylow input
impedance. RF chokes are used to bias the gate at a negative voltage near pinch-off and to
provide a positive bias for the drain of the FET. A bypass capacitor at the drain pro-vides a return
path for the LO signal, and a low-pass filter provides the final IF output signal
FIGURE 7.9 Variation of FET transconductance versus gate-to-source voltage.
FIGURE 7.10 Circuit for a single-ended FET mixer.
Our analysis of the mixer of Figure 7.10 follows the original work described in reference [6]. The
simplified equivalent circuit is shown in Figure 7.11 and is based on the standard unilateral
equivalent circuit for a FET. The RF and LO input voltages are given in (7.22) and (7.23). Let Zg
= Rg + jXg be the Thevenin source impedance for the RF input port, and ZL = RL + JX^ be the
Thevenin source impedance at the IF output port. These impedances are complex to allow us to
conjugately match the input and output ports for maximum power transfer. The LO port has a
real generator impedance of Zo, since we are not concerned with maximum power transfer for
the LO signal.
As we did for the large-signal analysis of the diode mixer, we express the LO pumped FET
transconductance as a Fourier series in terms of harmonics of the LO signal:
Because we do not have an explicit formula for the transconductance, we cannot calculate directly
the Fourier coefficients of (7.48), but must rely on measurements for these values, As in the case
of the switching model, the desired down-conversion result is due to the n = 1 term of the Fourier
series, so we only need the g\ coefficient. Measurements typically give a value in the range of 10
mS for g\.
The conversion gain of the FET mixer can be found as
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H.-R. Chuang, EE NCKU
7-19
where V^f is the IF drain voltage, and the impedances Zg and Z^ are chosen for maximum power
transfer at the RF and IF ports. The RF frequency component of the phasor voltage
FIGURE 7.11 Equivalent circuit for the FET mixer for Figure 7.10
across the gate-to-source capacitance is given in terms of the voltage divider between
Z ,R,SindCgs:
Multiplying the transconductance of (7.48) by ^(r) = V^COSCURF; gives terms oftr form
The down-converted IF frequency component can be extracted from the second ten
of (7.51) using the usual trigonometric identity:
Then the IF component of the drain voltage is, in phasor form,
where (7.50) has been used. Using this result in (7.49) gives the conversion gain (before conjugate
matching) as
We must now conjugately match the RF and IF ports. Thus we let Rg = Ri,X^ = l/oiRpCg. RL = Rd,
and XL = 0, which reduces the above result to
The quantities gt, R^, R,, and Cg, are all parameters of the FET. Practical mixer circuit generally use
matching circuits to transform the FET impedance to 50 ^ for the RF, LC and IF ports.
EXAMPLE 7.2 MIXER CONVERSION GAIN
A single-ended FET mixer is to be designed
A single-ended FET mixer is to be designed for a wireless local area network receiver operating at
2.4 GHz. The parameters of the FET are: Rd = 300 ?, R, = 100, Cgs =0.3 pF, and g\ = 10 mS.
Calculate the maximum possible conversion gain.
Solution
This is a straightforward application of the formula for conversion gain given
in (7.54):
Note that this value does not include losses due to the necessary impedance matching networks.
FIGURE 7.12 A dual-gate FET mixer.
Other FET Mixers
There are several variations of mixer circuits that can be implemented using
FET. Figure 7.12 shows a single-ended mixer using a dual-gate FET, where the RF and LO inputs
are applied to separate gates of the PET. This provides a high degree of RF-LO isolation
generally an inferior noise figure relative to the transconductance mixer of Figure 7.10.
Another configuration is shown in Figure 7.13, using two FETs in a differential amplifier
configuration. The balun (balanced-to-unbalanced) networks on the LO and IP ports provide a
transition between a two-wire line that is balanced with respect to ground and a single line that is
unbalanced relative to ground. Baluns may be implemented with center-tapped transformers, or
with 180 hybrid junctions.
The differential mixer operates as an altenating switch, with the LO turning the top two
FETs on and off on alternate cycles of the LO. These PETs are biased slightly above pinch-off so
each PET will be conducting for slightly more than half of each LO cycle: Thus, one of the upper
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FETs is always conducting, and the lower FET will remain in saturation The RF and LO ports
should each be impedance matched. The IF output circuit must provide a return path to ground
for the LO signal.
An extension of the differential FET mixer is the Gilbert cel1 mixer show in Figure
7.14 .This mixer us differential FET mixer stages to form a double balanced mixer. This circuit
achieves high RF-LO isolation and a high dynamic range. It also cancels all even-order
intermodulation products. This circuit is very popular for wireless integrated circuits.
FIGURE 7.13 A differential FET mixer.
FIGURE 7.14 A Gilbert cell mixer.
7.4 OTHER MIXER CIRCUITS
The single-ended diode and FET mixers discussed above provide frequency conversion, but often
have poor RF input matching and RF-LO isolation. This reduces the performance of wireless
systems, but fortunately it is possible to improve these characteristics by combining two or more
single-ended mixers with hybrid junctions.
Balanced Mixers
RF input matching and RF-LO isolation can be improved through the use of a balanced mixer,
which consists of two single-ended mixers combined with a hybrid junction. Figure 7.15 shows
the basic configuration, with either a 90?hybrid (Figure 7.15a), or a 180?hybrid (Figure 7.15b).
As we will see, a balanced mixer using a 90?hybrid junction will ideally lead to a perfect input
match at the RF port over a wide frequency range, while the use of a 180?hybrid will ideally lead
to perfect RF-LO isolation over wide frequency range. In addition, both mixers will reject all
even-order intermodulation products.
Microwave quadrature or ring hybrids [1] can be used to implement balanced mixers, but at lower
frequencies a center-tapped transformer can be used. As shown in Figure 7.16, the secondary of
the transformer provides outputs with a 180?phase shift to the two mixer diodes. The LO signal
is applied to the center tap of the secondary.
The double-balanced mixer of Figure 7.17 uses two hybrid junctions or transformers, and
provides good isolation between all three ports, as well as rejection of all even harmonics of the
RP and LO signals. This leads to very good conversion loss, but less than ideal input matching at
the RF port. The double-balanced mixer also provides a higher third-order intercept point than
either a single-ended mixer or a balanced mixer.
FIGURE 7.15 Balanced mixer circuits, (a) Using a 90?hybrid, (b) Using a 180 hybrid.
FIGURE 7.16 Balanced mixer using a hybrid transformer.
FIGURE 7.17 Double-balanced mixer circuit
The following table summarizes the characteristics of several types of mixers
Small-Signal Analysis of the Balanced Mixer
We can analyze the performance of a balanced mixer using the small-signal approachthat was
used in Section 7.2. Here we will concentrate on the balanced mixer with a 90 hybrid, shown in
Figure 7.15a, and leave the 180?hybrid case as a problem.
As usual, let the RF and LO voltages be defined as
The scattering matrix for the 90?hybrid junction is [1]
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H.-R. Chuang, EE NCKU
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where the ports are numbered as shown in Figure 7.15a. Then the total RF and LO voltages applied
to the two diodes can be written as
Using only the quadratic term from the small-signal diode approximation of (7.20) gives the diode
currents as
where the negative sign on ('2 accounts for the reversed diode polarity, and K is a constantfor the
quadratic term of the diode response. Adding these two currents at the input to the low-pass filter
gives
where the usual trigonometric identities have been used, and &IIF = WRF ?(ULO is the IF
frequency. Note that the DC components of the diode currents cancel upon combining. After
low-pass filtering, the IF output is
as desired.
We can also calculate the input match at the RF port, and the coupling between the RF and LO
ports. If we assume the diodes are matched, and each exhibits a voltage reflection coefficient F at
the RF frequency, then the phasor expression for the reflected RF voltages at the diodes will be
and
(7.61a)
These reflected voltages appear at ports 2 and 3 of the hybrid, respectively, and combine to form
the following outputs at the RF and LO ports:
Thus we see that the phase characteristics of the 90?hybrid lead to perfect cancellation of reflections
at the RF port. The isolation between the RF and LO ports, however, is dependent on the
matching of the diodes, which may be difficult to maintain over a reasonable frequency range.
Image Reject Mixer
We have already discussed the fact that two distinct RF input signals at frequencies &IRF ==
&>LO ?^IF will down-convert to the same IF frequency when mixed with iMLo. These two
frequencies are the upper and lower sidebands of a double-sideband signal. The desired response
can be arbitrarily selected as either the LSB (&JLO ?^w) or the USB ((ULO + 展F) assuming a
positive IF frequency. The image reject mixer, shown in Figure 7.18, can be used to isolate these
two responses into separate output signals. The same circuit can also be used for up-conversion,
in which case it is usually called a single-sideband modulator. In this case, the IF input signal is
delivered to either the LSB or the USB port of the IF hybrid, and the associated single sideband
signal is produced at the RF port of the mixer.
We can analyze the image reject mixer using the small-signal approximation. Let the RF input
signal be expressed as
FIGURE 7.18 Circuit for an image reject mixer.
where Vu and VL represent the amplitudes of the upper and lower sidebands, respectively. Using the
S-matrix given in (7.57) for the 90?hybrid gives the RF voltages at the diodes as
After mixing with the LO signal given in (7.56) and low-pass filtering, the IF inputs to the IF hybrid
are
where K is the mixer constant for the squared term of the diode response. The phasor
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H.-R. Chuang, EE NCKU
7-22
representation of the IF signals of (7.65) is
Combining these voltages in the IF hybrid gives the following outputs
which we see are the separate sidebands of the downconverted input signal of (7.63). These
outputs can be expressed in time-domain form as
which clearly shows the presence of a 90?phase shift between the two sidebands. Aso note that the
image rejection mixer does not incur any additional losses beyond the usual conversion losses of
the single rOection mixer.
A practical difficulty with image rejection mixers is in fabricating a good hybrid at the relatively
low IF frequency. Losses, and hence noise figure, are also usu3ly greater than for a simpler
mixer.
REFERENCES
[1] D. M. Pozar, Microwave Engineering, 2nd edition. Wiley, New York, 1998.
[2] S. Y. Yngvesson, Microwave Semiconductor Devices, Kluwer Academic Publishers, 1991.
[3] K. Chang, Handbook of Microwave and Optical Components, vol. 2, Chapter 2, "Mixers and
Detectors," by E. L. Kollberg, Wiley InterScience, New York, 1990.
[4] C. T. Torrey and C. A. Whitmer, Crystal Rectifiers, MIT Radiation Laboratory Series, vol. 14,
McGraw-Hill, New York, 1948.
[5] S. A. Maas, Microwave Mixers, 2nd edition, Artech House, Dedham, MA, 1993.
[6] R. A. Pucel, D. Masse, and R. Bera, "Performance of GaAs MESFET Mixers at X Band," IEEE
Tram.Microwave Theory and Techniques, vol. MTT-24, pp. 351-360, June 1976.
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3.3 5.7GHz CMOS雙端平衡式混波器設計與製作
為了減少共模雜訊以及clock feed-through透過基板寄生耦合電容對電路的干擾，且增
加LO-IF、LO-RF隔離度、抑制RF和LO訊號的偶次項諧波，故選擇double-balanced mixer
為主要架構(圖3.10)。Double-balanced mixer 架構較single-balanced mixer架構複雜在於混波
器的輸入端需接上一個單端轉雙端的balun，增加電路面積與功率消耗。由於使用在外差
式系統中，LO頻率(5.465~5.525GHz)遠高於IF(280MHz)的情形下，LO洩漏至IF輸出端之
訊號可以輕易被中頻濾波器所衰減，更提高了LO-IF隔離度。
VDD
M7
M8
RL
VIF-
VLO-
RL
M3 M4
M5 M6
VIF+
VLO-
VLO+
VRF+
M1
M2
VRF-
圖3.10 double-balanced mixer電路架構圖
在設計double-balanced mixer時首先決定轉導電晶體大小及偏壓。M1、M2為轉導電晶體，
負責將RF電壓訊號轉為電流訊號，因此其轉導大小為首要考量，轉導要大，偏壓電流勢必要
增大，需要較大的功率消耗，然而較大的直流偏壓卻可獲得較佳的雜訊表現。所以首先考量
轉導電晶體M1 、M2 在偏壓電流各為1mA的條件下，參考[8]，找出最佳雜訊表現之電晶體寬
度，決定轉導電晶體M1、M2大小為各為20m/0.18m。
本地振盪信號VLO+、VLO-為大訊號，輸入M3 、M4、M5、M6閘極端形成開關電晶體負責
切換RF電流訊號，達到混波目的。開關電晶體大小決定了開關特性，通常電晶體寬度越大開
關特性越佳，但是其source端較大的雜散電容容易使RF電流訊號衰減，因此需審慎的選擇
M3~M6 的大小，經過模擬微調後，選擇各為40m/0.18m。已知訊號產生器輸出阻抗為50歐
姆，為了達到最大的功率傳輸，利用on chip電阻與電容再加上bondwire等效電感，將開關電
晶體閘極輸出阻抗匹配至50歐姆。由於本地振盪信號為雙端差模輸入，故利用一rat race ring
將原本單端本地振盪訊號分為雙端差動訊號。
混波器負載部分利用RL 電阻接成diode connected PMOS型式，負載大小由電阻RL 與
PMOS汲極與遠源級間的阻抗所構成。M7、M8電晶體寬度大小取決於負載壓降與負載阻抗，
在相同偏壓電流情況下，為了達到最小壓降，可增大M7、M8電晶體寬度，此時電晶體較容易
操作在低飽和區，易造成線性度因非線性負載而下降。增加M7、M8電晶體寬度的結果也增加
了電晶體本身之寄身效應，導致整體負載阻抗因寄生效應而變小，造成轉換增益下降，取決
於線性度與轉換增益之考量下，選擇M7 、M8電晶體寬度為160m/0.18m。考量量測時，所
使用的量測儀器皆是50歐姆輸入阻抗之負載，為了避免嚴重之負載效應，在混波器核心負載
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輸出端與50歐姆負載端間置入一緩衝放大器，此緩衝放大器由一共源極電晶體所組成，因此
對混波訊號也有放大之功能。
Double-balanced mixer 之電路與電路佈局圖如圖 3.11 所示，輸出為雙端輸出，因
為 mixer 後級接單端中頻濾波器，因此加上 2:1 的 balun 作為雙端轉單端電路[10]。IF
端 balun 使用 TOKO 616PT-1039，insertion loss 為 3dB。Balun 單端輸出為 480MHz 中
頻訊號，利用 off-chip 的晶片電感 5.6nH、並聯電容 7pF 匹配至 50 歐姆。
5.6nH
MLIN
MLIN
IF +
1.8V
IF +
7pF
1.8V
0.8V
IF_OUT
IF -
M8
M7
M10
M9
4.5K
IF -
4.5K
1.1V
M1&M2 ; gate width = 20um/0.18um
LO+
LO+
M3
LO -
M4 M5
M6
M3&M4&M5&M6 ; gate width = 40um/0.18um
M7&M8 ; gate width = 160um/0.18um
M9&M10 ; gate width = 15um/0.18um
0.7V
M1
M2
pad equivalent circuit
0.065pF 625Ω
bondwire equivalent circuit
2nH 0.65Ω
圖3.11 5.7GHz CMOS double-balanced mixer電路及晶片佈局圖
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3.4 模擬與量測結果
Double-balanced mixer設計電路的RF輸入範圍為5.725~5.825GHz，中頻輸出為480MHz頻寬
20MHz，mixer核心部分直流偏壓為1.8V/1.8mA，緩衝放大器部分為各1.8V/1.4mA。在IF
balun 插入損耗為 3dB代入轉換損代入模擬結果，轉換增益約12.76dB，input P1dB 約15dBm，IIP3約-6.9dBm，LO-RF的隔離度皆在42dB左右，LO-IF的隔離度約100dB。
量測上利用FR-4製作測試基板來量測晶片，mixer核心部分直流偏壓量測為1.8V/2mA，緩
衝放大器部分為各1.8V/1mA。RF輸入訊號端與本地振盪埠(LO)的輸入訊號由訊號產生器
經由rat race ring產生差動訊號饋入。混波器量測特性結果為轉換增益約11.06dB，input
P1dB 約-16.4dBm，IIP3約-7.5dBm， LO-RF的隔離度皆在19dB左右，LO-IF的隔離度約
50dB。模擬與量測結果比較如圖3.12、表3.1所示。晶片照片圖、測試板佈局圖及測試板
照片，如圖3.13所示。
3.5 結果與討論
量測結果除了LO-RF隔離度外，在輸入/輸出匹配、轉換增益、P1dB、OIP3與模擬大
致吻合。而在LO-IF隔離度方面，設計之時因為balun沒有model可以模擬，因此利用一個
理想balun加上3dB的insertion loss代入模擬軟體，由於量測使用之IF balun的高頻響應對LO
有額外的衰減因此得到比模擬更佳的LO-IF隔離度。LO-RF隔離度方面模擬、量測差距較
大，理想上single-balanced的架構LO-RF隔離度應該是相當大。原因可能是因為製程偏移
使得M2及M3電晶體些微不對稱或是substrate coupling，造成LO洩漏至RF端。改進之處為
佈局方面可以再將M2及M3電晶體對稱性提高及將substrate coupling問題列入佈局考量。
0
0
5.7GHz double-balanced mixer
measurement
simulation
-5
-5
-10
S22(dB)
S11(dB)
-10
-15
-20
-15
-20
-25
-25
-30
-30
-35
5.5
5.55
5.6
5.65
5.7
5.75
5.8
5.85
5.9
5.95
6
-35
300
5.7GHz double-balanced mixer
measurement
simulation
350
400
RF Frequency (GHz)
450
500
550
600
650
700
IF Frequency (MHz)
(a) RF input return loss
(b) IF output return loss
25
52
LO-IF isolation(dB)
LO-RF isolation (dB)
23
21
19
51
50
49
17
5.7GHz double-balanced mixer
LO-RF isolation
15
5245
5265
5285
5305
5325
5345
5365
LO Frequency (MHz)
(c) LO-RF isolation
2008
5.7GHz double-balanced mixer
LO-IF isolation
48
5245
5265
5285
5305
5325
5345
5365
LO Frequency (MHz)
(d) LO-IF isolation
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H.-R. Chuang, EE NCKU
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15
14
14
Noise Figure(dB)
Conversion Gain (dB)
7-27
12
10
5.7GHz double balanced mixer
measurement
simulation
8
6
-50
-45
-40
-35
-30
-25
-20
-15
13
12
-10
10
470
475
480
485
490
IF Channel Frequency (MHz)
RF Input Power (dBm)
(e)轉換增益及input P1dB
(f)雜訊指數
m1
0
m7
-50
dBm(OUT)
5.7GHz double-balanced mixer
measurement
simulation
11
-100
m1
freq=480.5MHz
dBm(OUT)=-16.921
-150
m7
freq=481.5MHz
dBm(OUT)=-63.246
-200
-250
477
479
481
483
freq, MHz
(g) Two tone test / OIP3模擬
(h) Two tone test / OIP3量測
圖3.12 5.7GHz CMOS double-balanced mixer模擬/量測結果
(a)
(b)
(c)
圖3.13 5.7GHz CMOS double-balanced mixer：
(a)晶片照片圖 (b)測試板照片圖 (c)測試板佈局圖
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表3.1 5.7GHz CMOS double-balanced mixer模擬/量測特性表
5.7GHz CMOS Double-Balanced Mixer (TSMC 0.18m)
RF Frequency Range
5.725~5.825GHz
IF Frequency
480MHz
LO Frequency
5.265~5.325GHz
Vdd
1.8V
Simulation
Measurement
LO Power
-3dBm
-3dBm
Core / Each Buffer Current
1.8/1.4mA
2/1mA
Conversion Gain
12.76dB
11.06dB
RF Input Return Loss
>21dB
>18dB
Output Return Loss
15dB@280MHz
20dB@280MHz
LO-RF Isolation(LO=-3dBm)
42dB
19dB
LO-IF Isolation(LO=-3dBm)
>100dB
>50dB
RF-IF Isolation(RF=-30dBm)
>200dB
Noise Figure
12.4dB
12.8dB
IIP3
-6.9dBm@RF=-28dBm
-7.5dBm@RF=-28dBm
Input P1dB
-15dBm
-16.4dBm
Die size
0.627 x 0.649 mm2
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H.-R. Chuang, EE NCKU
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A
Desired interfers
Image
Channel
Band Select
Filter Response
BPF1
r
im
B
Band Select
Filter
Desired
Channel
BPF1
Desired
Channel
 LO1
Image
im
Desired
Channel
 LO 2
Desired
Channel
 IF 2
2008
f
Channel Select
Filter Response:
BPF3
 IF1
f
E
 IF1
im
D
Desired
Channel
Desired
Channel
Image
r
f
C
r
Band Select
Filter Response
BPF2
f
F
Channel Select
Filter Response:
BPF4
 IF 2
f
Desired
Channel
G
f
H
IF Amp
f
 IF 2
f
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Stage 1
Stage 2
Stage 3
Stage 4
Av 4  15 dB
Stage 5
Stage 6
A p 4  5 dB
A
B
Duplexer
L1  2 dB
IF Amplifier
NF4  12 dB
LNA
C
Image-Reject
Filter
Av 2  15 dB
NF2  12 dB
D
L3  6 dB
E
IF F
Filter
L5  5 dB
LO
G
NF6  10 dB
Figure 6.35 Calculation of noise figure in a cascade of stages
Duplexer
LNA
A
B
Stage Gain (dB)
-2
Voltage
-2
Power
Cumulative
Voltage Gain (dB)
Stage NF (dB)
Cumulative
NF (dB)
Stage IP3
Cumulative IP3
Image-RejectMixer
IF
Filter
Filter
C
D
E
15
15
-6
-6
-2
13
2
2
6
8.79
6.79
20.1
+100 dBm
12 dBm
+100 dBm
-10.6 dBm -12.6 dBm +11 dBm
15
5
7
F
-5
22
12
14.1
IF Amplifier
17
5
10
15
10
+5 dBm 1000 Vrms
700mVrms
+5 dBm
700mVrms
22.1Vrms
Figure 6.36 Level diagranm corresponding to the cascade of Fig. 6.35
2008

H.-R. Chuang, EE NCKU
7-31
Av  15 dB
VDD
Ap  5 dB
SSB NF =10 dB
500 
R1
B
NF=10dB
AIF
LO
Figure 6.32 Cascade of a mixer and an IF amplifier
Spectrum at X
Signal
Band
Image
Band
Thermal
Noise
Rs X
Y
Vin

 Lo
+
 Lo
Spectrum at Y

 IF
Figure 6.17 Folding of RF and image noise into the IF band
2008

H.-R. Chuang, EE NCKU
7-32
1st mixer
Filter # 1
Filter # 2
RF
amplifier
1st IF
stages
Injection
filter
1st local
oscillator
2nd
mixer
~
2nd IF
stages
Detector
1st IF
amplifier
~
2nd local
oscillator
( Tx )
( Rx )
For a transceiver, if transnitting and receiving use different frequencies (for example,
celluLOr phone system) a duplexer is used to Separate Tx & Rx
duplexer is a kind of BPF
Hence a duplexer will also act as a BPF
( before RF stage ) for image rejection

VRF t   Vr cos  r t


V Lo t   Vo cos  o t
V  1VRF   2V Lo


depending on combiner
Where to add D.C. bias ?
lumped circuit form
impedance matching is implemmted in the inductor-coie winding.
2008

H.-R. Chuang, EE NCKU
Amplitude at
mixer RF port
7-33
Desired
Signal
f LO
Image
noise
fRF
RF
KTB
IF
Frequency
f
LO
- IFf
f
LO
LO
+ IFf
RF
IF
Amplitude at
mixer RF port
LO
f LO
2f LO
3f LO
KTB
Frequency
fLO - fIF
2fLO - fIF
2fLO + fIF
fLO + fIF
3f LO - fIF
3f LO+ fIF
(e) Dual-Gate MESFET Mixer
MESFET
VDS  3.65V
, I DS  2.5mA
VG1S  0.78V , I G2 S  0.78V
G  8dB
output impedance = 980  j1k
RF & LO signal are combined in dual-gate FET structure
MODEL RR-12
LO 1 to 2 GHz
RF 1 to 2 GHz
IF 0 to 0.64 GHz
Characteristic Min.
Concersion
2008
Typ.
Max. Test Condition

H.-R. Chuang, EE NCKU
7-34
Loss. dB
Phase
Deviation
From 90o
Deg.
Amplitude.
Unbalance.
dB
lsolation, dB
LO-RF
LO-IF
RF-IF
VSWR
RF
LO
IF
1dB
Compression.
dBm
3-rd order
HP. dBm
5
6
4
0.5
18
20
20
25
25
25
1.5
1.5
1.5
2.0
2.0
2.0
All measurements made
in 50 Ohm
system.
RF=1-2GHz
LO=1-2GHz
IF=0.03GHz
PLO = 7 dBm
PRF =10 dBm
5
15
<Conversion Loss of Mixer>
In mixer design several frequencies (RF,LO,IF) and Their harmonics are involved.
Impedance matching design at three ports (RF,LO,IF) is complicated by the above
situation.
Undesired harmonic signal can be
dissipated in resistive termination.
or
blocked with reactive terminetion.
 increase mixer loss
 frequency dependent
An important figure of merit of the mixer
Conversion loss Lc  10 log
avaialable RF input power
IF output power
typical Lc = 5 ~ 8 dB
for passire mixers;a active mixer can have a gain
High local oscillator signal (pump) power can
rgeduce the mixer conversion loss
2008

H.-R. Chuang, EE NCKU
7-35
minimum Lc usually sccurs for 0 ~ 10 dBm LO powers
But (0 ~ 10 dBm) LO power will violate the small signal approximation of the mixer
diode.
precise design requires numerical ssolution of the nonlinear equation theat describes
the diode charactoristics.
 i : IF 
VRF  Vr cos r t   2VrVo sin r   o t  2VrVo sin i t
let  r   o   i
 r   o  i
let  r   o   i
  r   o    i 
VRF
 Vr cos  r t 
 2VrVo sin r   o t   2VrVo sin i t
 fr  f LO  f IF 

 all produce a same IF frequency f IF and cause "image interference"
 fim  f LO  f IF 
Fig. 12 Measured characteristics of a 2.4 GHz bandpass filter (FDK 2450B).
V A  VU cos   t  VL cos   t
o
i
o
i
2
2
 r

V
V
VrB  U cos  o   i t  90 o  L cos  o   i t  90 o
2
2







V A cos  o t  Vi A
IF mixing rB
Vr cos  o t  Vi B



V1  k VU sin  i t  VL sin  i t  VU sin  i t  180 o  VL sin  i t
2

  2kVL sin  i t
Vi A  kVU sin  i t  kVL sin  i t

 B
o
o
Vi  kVU sin  i t  90  kVL sin  i t  90

2008




H.-R. Chuang, EE NCKU
7-36
 Mixer Performance Characteristics


Mixer Conversion Loss : Lc  10log
RF
f RF
IF
f
f
f RF  f LO f RF  f LO
LO
=> stringly depend on LO power level
f
f LO


available RF input power
dB
IF output power
Mixer Noise Figure
混頻器除了射頻(RF)頻帶之雜訊混至中頻(IF)帶外，同時有image頻帶及
贅餘(spurious)頻帶之雜訊混入中頻
Signal
Band
Image
Band
Thermal
Noise

 Lo
f IF
f LO
f RF 2 f LO- f IF 2 f LO+ f IF
Mixer noise

 單旁波帶(SSB: single side band)雜訊指數與雙埠放大器之定義相同
 測量上多採用雙旁波帶(DSB:double side band) 雜訊指數較方便¡
=> 雙旁波帶雜訊指數為單旁波帶雜訊指數的兩倍
Noise Figure Meter
Broad band
Noise Source
Mixer
Pre Amp
LO
Signal Generator
Mixer DSB noise figure measurement
混頻器為雙旁波帶雜訊指數測量
2008

H.-R. Chuang, EE NCKU
Amplitude at
mixer RF port
7-37
Desired
Signal
f LO
fRF
Image
noise
KTB
Frequency
fLO - f IF
2008
fLO + fIF

H.-R. Chuang, EE NCKU
7-38
APPENDIX 3 : SOME APPLICATIONS OF MICROSTRIP CIRCUITS
* Fooks "Microwave Engineering using Microstrip Circuits" (Ch 12)
This part introduces selected practical microstrip circuits or subsystems. It brings
together some of the combination of microwave passive components & active devices
to produce functioning self-contained building blocks, that in turn may be a part of a
complete microwave systems.
FIGURE 11.16 Frequency conversion in a receiver and transmitter.
(a) Down-conversion in a heterodyne. (b) Up-conversion
in a transmitter.
f IF  f RF  f Lo , and a much higher frequency signal, f RF  f Lo (filtered out).
 2-stage mixers  for vetter image frequency rejetion.
Portable communication Receiver BLOck Diagram
<Mixer in Rransmitter>



v1 t   Vr cos  r t  90 o  Vo  Vn  cos  o t  180 o
 Vr sin  r t  Vo  Vn  cos  o t


 conside only quadratic term
o
o 
v2 t   Vr cos  r t  180  Vo  Vn  cos  o t  90  of the diode which givers the
 desired mixer product
 Vr cos  r t  Vo  Vn  sin  o t





混頻器的雜訊指數，根據IEEE的定義，單旁波帶(SSB:single side band)雜訊指數與雙埠放
大器之定義相同，如(2.4)式，但在實際測量上較為困難，若以濾波器阻隔假像頻率(image
grequency)頻帶之雜訊，則濾波器將造成輸入阻抗改變，且增加溫度雜訊。因此在測量上多
採用雙旁波帶(DSB:double side band)雜訊指數較方便，一般的定義，雙旁波帶雜訊指數為
單旁波帶雜訊指數的兩倍。圖2.5說明混頻器雜訊之造成，混頻器除了射頻(RF)頻帶之雜訊混
至中頻(IF)帶外，同時也將假像(image)頻帶及贅餘(spurious)頻帶之雜訊混入中頻，因此混
頻器的雜訊指數將比放大器來得大。圖2.6為測量混頻器雜訊指數之架構，所得為雙旁波帶雜
訊指數。
2008

H.-R. Chuang, EE NCKU
7-39
f IF2
 small & requiring higher Q of the IF filter
=> channel bandwidth
IF
2008

H.-R. Chuang, EE NCKU
7-40
Gr
Gt
Pt
Pr
R
t
t
RF
DC
f
f RF
f
(a) Diode rectifier
t
t
Modulated
RF
Modulation
f
f RF
f
fm
(b) Diode detector
RF
IF
f
f RF
f
f RF  f LO f RF  f LO
LO
f LO
f
(c) Mixer (frequency conversion)
RF
f RF
IF
f
f
f RF  f LO f RF  f LO
LO
f LO
f
For a single-ended mixer, the noise terms will be
2008
1
2
V  V t   V t 
o
2
n
n

1V V
2 o n
t 
H.-R. Chuang, EE NCKU
7-41
Desired
Channel
interfers
Image
A
r
im
im
r
f
Desired
Channel
C
Image
Desired
Channel
B
f
D
Image
r
im
f
F
E
 IF1
 IF 2
f
f
H
G
 IF 2
2008
 IF1
f
f
 IF 2
f

H.-R. Chuang, EE NCKU
7-42
Mixer Nonlinear Simulation by Libra
Since I (V )  I s (e V  1)
 dI dV
Vo
 d 2 I dV 2
 I s e Vo  I o  I s   Gd  R j 1
Vo
R j : junction resistance
 d dI dV  / dV Vo  dGd dV  Vo   2 I s e Vo
  2 I o  I s   Gd  Gd
dI dV V  I s e Vo  I o  I s   Gd  R j 1
o


R j : junction resistance


Since I (V )  I s (e V  1) d 2 I dV 2 Vo  d dI dV  / dV Vo  dGd dV  Vo   2 I s e Vo

  2 I o  I s   Gd  Gd



Baseband
filter
Power
amplifier
Mixer
fM
Antenna
fLo + fM
Up-conversion
(for transmitting)
f LO
Local
oscillator
LNA
fLo + fM
fo = fLo - fM
IF filter
Mixer
fIF + fM
Local
oscillator

IF
amplifier
Detector
fIF
Down-conversion
(receiving)
The band-stop response of the BPF will determine the image-rejection ratio.
2008

H.-R. Chuang, EE NCKU
7-43
Band
Select
Filter
A
BPF1
Image
reject
Filter
LNA
BPF2
B

Channel
Select
Filter
1st
Mixer
C
D
BPF3
Image
Reject
Filter
Mixer
RF
IF
cos0t
E
BPF4
F
IF
Amplifier
G
H
 LO 2
 LO1
LNA
Channel
Select
Filter
2nd
Mixer
Channel
Select
Filter
LO
RF input
VRF  Vr cos r t
LO input
VLo
2008
v1
Diode
1
i1
90o
3 dB hybrid
 Vo  Vn t cosot
+
v2
Diode
2
i2
If output
LP
filters

H.-R. Chuang, EE NCKU
7-44
VU cos(o  i )t 
vr  
VL cos(o  i )t
Mixer A IF
vrA
RF
LO
RF input
vr
vrB RF
90 o
hybrid
Z0
Mixer B
LO
v1
LO
input
3-dB
power
divider
o
v2

2008





 
 
USB
90 o
hybrid
IF
VRF  VRF  VRF  V1  2  j V2  2  VRF  VLo
2
1
2
1
1


  j VRF
 VLo
2  12 Vr  j 12 VLo  j  j 12 Vr  12 VLo
2
2

  jVLo


no Vr reflection, but VLO signal appears at RF port 



VLO  VLO  VLO   jVr
1
2

no VLo reflection, but Vr signal appears at LO port 


VRF  VRF  VRF   V1 2    jV2 2  1 V1  jV2 
1
2
2


 1 1 Vr  jVLO   j 1  jVr  VLO 
2 2
2


 12 Vr  jVLO   j  jVr  VLO 


  jVLO


no Vr reflection, but VLO signal appears at RF port 


 LO
LO
LO
V  V1  V2   jVr

no VLo reflection, but Vr signal appears at LO port 

monolithic quad DMOS FET for mixer application

LSB
IF
out




H.-R. Chuang, EE NCKU
7-45
FET mixer has a gain . (Diode mixer has no gain)
TABLE 11.1 Frequencies and Relative Amplitudes of the Square-Law Output of a
2
Detected AM signal
V t 
Frequency
Relative Amplitude
1  m2 2
0
m
2 m
2m
m2 2
2
1 m 2
m
2 o
2 o   m
2  o   m 
desired demodulated output


2
1 V 2 Gd  2m cos  t  Vo Gd m cos  t


m
m
4 o

 2

1 2
Vo
2 
2
m 4
EXAMPLE 1.4
A diode in as axial-lead package has the following equivalent circuit parameters:
Cp =0.10 pF, Lp=2.0nH, Cj=0.15pF, Rs=10, and I s=0.1A. Calculate and plot the
impedance of this diode from 4 to 14 GHz, for a bias current Io=0 and Io=60A.
Ignore the change in Cj with bias, and assume  =1/25 mV.
solution
From (11.27) the junction resistance for the two bias states is
for I o  0 ,
1
mV  22.5  105 ,
 25
0
.
1
A
 I o  I s 
1
Rj 
 25mV  417.
 I o  I s  60  0.1A
Rj 
for I o  60 A ,
Then the input impedance can be calculated from the equivalent circuit of Figure
11.12; the result is plotted versus frequency on a 50 Smith chart in Figure 11.13.
Diode impedance is frequency dependent!
FIGURE 11.13 Impedance of the diode of Example 11.4 for to bias states, from 4 to
14 GHz.
1
(11.27) R j  Gd1 
 I o  I s 
2008

H.-R. Chuang, EE NCKU
7-46
 FM broadcasting
IF = 10.7MHz
 Cellular phone
IF = 45MHz
IF = 455MHz
DC
bias
Combiner
RF
vi cos(r  o )t
Matching
network
vr cos r t
DC
return
LO
vo cos ot
r ,o
LP filter
r  o
........
Diode
1
RF input
i1
If output
+
LO input
Diode
2
3 dB hybrid
(90 o or 180 o )
i2
LP
filters
Fig.
Band
Select
Filter
Image
reject
Filter
Channel
Select
Filter
Channel
Select
Filter
BPF1
BPF2
BPF3
BPF4
A
C
B
10.19
interfers
interfers Image
Desired
Channel
r
im
f
A
LO1
Band
Select
Filter
Image
reject
Filter
BPF1
BPF2
B
E
D
F
H
G
 LO 2
Channel
Select
Filter
Channel
Select
Filter
BPF3
C
IF
Amplifier
D
 LO1
E
BPF4
F
LO 2
IF
Amplifier
G
H
Desired
Channel
 IF 2
Fig. 6 Measured characteristics of a 2.4-GHz single-ended resistive FET mixer.
2008

H.-R. Chuang, EE NCKU
7-47
Mixer Noise Figure
混頻器除了射頻(RF)頻帶之雜訊混至中頻(IF)帶外，同時有image頻帶及
贅餘(spurious)頻帶之雜訊混入中頻¡
f IF
f LO
f RF 2 f LO- f IF 2 f LO+ f IF
Mixer noise
 單旁波帶(SSB: single side band)雜訊指數與雙埠放大器之定義相同
 測量上多採用雙旁波帶(DSB:double side band) 雜訊指數較方便¡
=> 單旁波帶雜訊指數為單旁波帶雜訊指數的兩倍
Noise Figure Meter
Broad band
Noise Source
Mixer
Pre Amp
LO
Signal Generator
Mixer DSB noise figure measurement
混頻器為雙旁波帶雜訊指數測量
2008

H.-R. Chuang, EE NCKU
7-48
2008

H.-R. Chuang, EE NCKU
7-49
VRF
v1  (Vr
V RF  Vr cosr t
2 )  (  j VLo
i1
Diode 1
Diode 2
LO input
v 2
VLo  Vo coso t
(90 )
2 )  (  j VLo
2)
2)
LP filter
i1
Diode 1
+
Diode 2
LO input
v 2
If output
i IF  2 KV rVo sin i t
i2
LP filter
3 dB hybrid
(90o )
2008
2 )  (V Lo
v1
VLo  Vo coso t
i IF  2 KVrVo sin i t
i2
v 2  (  j Vr
v1  (Vr
V RF  Vr cosr t
If output
LP filter
RF input
VLO
+
3 dB hybrid
o
VRF
LP filter
v1
RF input
VLO
2)
v 2  (  j Vr
2 )  (V Lo
2)

H.-R. Chuang, EE NCKU
7-50
2008

H.-R. Chuang, EE NCKU
7-51
DC
bias
Combiner
RF
IF
vi cos(r  o )t
Matching
network
DC
return
VRF  vr cosr t
VLO  vo cosot
LO
r ,o
r  o
LP filter
........
Single-ended mixer circuit
v  VRF  VLo  Vr cos r t  Vo cos o t
For a diode small-signal approximation
I V   I o  vGd  v 2 Gd 2   ...................
Gd
2
G
 2d
G
 IF output  4d
 v 2 Gd 2  
2008
(note : v  VRF  VLo  Vr cosr t  Vo cosot )
Vr cosr  Vo cosot 2
V 2 cos2 rt  2VrVo cosrt cosot  Vo2 cos2 ot   .......
r
2VrVo cosr  o t   G2d VrVo cosIF t

H.-R. Chuang, EE NCKU
7-52
2008

H.-R. Chuang, EE NCKU
7-53
(a) Single-ended Mixer
Pozar (RF Ch 7)




Mixer in a Transmitter

In a transmitter, a mixer is used to mix with IF signal to up-convert the signal
frequency for efficient radio-wave transmission from antenna.
Baseband filter
fM
Power
amplifier
Mixer
Antenna
fLo + fM
f LO
Up-conversion
(for transmitting)
t
Local oscillator

 f Lo  f IF :Upper Sideband USB 


 f Lo  f IF :Lower Sideband LSB 
Double sideband (DSB)= USB + LSB
=> For a single sideband transmission (SSB)
USB  f Lo  f IF  or LSB  f Lo  f IF 
2008

H.-R. Chuang, EE NCKU
7-54
We use a sideband filter or an image rejection mixer to remove a sideband signal.
( also celled single sideband modulator )
 direct-conversion transmitter
Baseband
I
PA
cos c t
Matching
Network
sin c t
Duplexer
Baseband
Q
 drawback: leakage of PA output to LO
I
PA
LO
BPF
LO

0
0
-5
-5
-10
-10
-15
-15
S22 (dB)
S11 (dB)
Q
-20
-25
-30
-35
-20
-25
-30
-35
measurement
simultaioon
-40
5.6
5.65
5.7
5.75
5.8
5.85
5.9
Frequency (GHz)
-40
400
measurement
simultaioon
420
440
460
480
500
520
540
560
Frequency (MHz)
(a)
(b)
14
Conversion Gain (dB)
12
10
8
6
4
2
0
-40
measuemsnt
simulation
-35
-30
-25
-20
-15
-10
-5
0
input power (dBm)
(c)
2008
(d)

H.-R. Chuang, EE NCKU
7-55
(中正大學電機M.S. Thesis)
f LO
2f LO
3f LO
fLO - fIF
2fLO - fIF
fLO + fIF
2fLO + fIF
3fLO - fIF
3fLO + fIF
 Spurious chart due to LO harmonics
2008

H.-R. Chuang, EE NCKU
7-56
2008

H.-R. Chuang, EE NCKU
7-57
 image rejection filter
 The BPF band-stop response
determines the image-rejection ratio
Desired
Band
Image
Reject
Filter
image

r
LNA
Image Reject
Filter Response
A0 cos  LO t
image
Desired
Band
r
 im

2 IF
A 2.4 GHz bandpass filter
=>passband = 100 MHz
=> insertion loss < 1 dB
RF
f RF
IF
f
f
f RF  f LO
LO
f RF  f LO
f
f LO
(c) Mixer (frequency conversion)
Down-conversion (for receiving)
Mixer
LNA
IF
amplifier
Detector
fIF
f RF  f LO  f M
f RF  f o
fo
(  f LO  f IF )
Local oscillator
2008
IF filter
 f LO  f o  f M
fM
t
t
 f IF  f M

H.-R. Chuang, EE NCKU
7-58
t
t
RF
DC
1st mixer
Filter # 1
f
f RF
f
(a) Diode rectifier
Filter # 2
RF
amplifier
Injection
filter
1st local
oscillator
t

2nd IF
stages
Detector
1st IF
amplifier
~
~
2nd local
oscillator
t
Modulated
RF
Modulation
f
f RF
2nd
mixer
1st IF
stages
fm
(b) Diode detector
f
Then the input to the two mixers through a
o
90 hybrid is
V A  VU cos(   )t  VL cos(   )t
o
IF
o
IF
 r
2
2
RF inputs  
V
V
VrB  U cos[(o   IF )t  90 o ]  L cos[(o   IF )t  90 o ]
2
2

V A
 r



VrB








VU
V
cos[(o   IF )t  90 o ]  L cos[(o   IF )t  90 o ]
2
2
VU
V
sin( o   IF )t  L sin( o   IF )t
2
2
VU
V
cos[(o   IF )t  180 o ]  L cos[(o   IF )t  180 o ]
2
2
V
V
 U cos(o   IF )t  L cos(o   IF )t
2
2
After mixing with an LO signal of cos  o t , the IF outputs of the mixers are
Vi A  kVU sin  IF t  kVL sin  IF t
IF outputs  
Vi B  kVU sin(  IF t  90 o )  kVL sin(  IF t  90 o )

Combining these two signals in the 90o hybrid at the IF output gives
V1 

1
2
(kVU sin  IF t  kVL sin  IF t ) 


o
o
o
o 
[kVU sin(  IF t  90  90 )  kVL sin(  IF t  90  90 )]
k [V
2 U
sin  IF t  VL sin  IF t  VU sin(  IF t  180 o )  VL sin  IF t ]
  2kVL sin  IF t
2008
LSB component

H.-R. Chuang, EE NCKU
7-59

Then the input to the two mixers through a 90o hybrid is
V A  VU cos(   )t  VL cos(   )t
o
IF
o
IF
 r
2
2
RF inputs  
V
V
VrB  U cos[(o   IF )t  90 o ]  L cos[(o   IF )t  90 o ]
2
2


After mixing with an LO signal of cos  o t & lowpass filtered, the IF outputs of
the mixers are
Vi A 

IF outputs  
B
Vi 


1 [ kV
U
2 2
cos  IF t  kVL cos  IF t ]
1 [ kV
U
2 2
cos( IF t  90 o )  kVL cos( IF t  90 o )]
Combining these two signals in the 90o IF hybrid (transformer) at the IF output
gives
 1 [kVU cos  IF t
 kVL cos  IF t
]
2 2

1


V1 

2

 1 [kVU cos( IF t  90 o  90 o )  kVL cos( IF t  90 o  90 o )] 
2 2

 k4 [VU cos  IF t  VU cos( IF t  180 o )  VL cos  IF t  VL cos  IF t ]


0
 k2 VL cos  IF t
LSB component
 1 [ VU cos( IF t  90 o )  VL cos( IF t  90 o )]
2 2

k


V2 

2

 1 [VU cos( IF t  90 o )  VL cos( IF t  90 o )] 
2 2

 k2 VU sin  IF t
2008
USB component

H.-R. Chuang, EE NCKU
7-60

Then the input to the two mixers through a 90o hybrid is
A
v RF



RF inputs  
B
v RF








1
2
1
2
1
2
1
2
V
U
cos(o t   IF t  90 o )  VL cos(o t   IF t  90 o )

VU sin( o  IF )t  VL sin( o  IF )t 
V
U
cos(o t   IF t  180 o )  VL cos(o t   IF t  180 o )

VU cos(o  IF )t  VL cos(o  IF )t 
After mixing with an LO signal of cos  o t & lowpass filtered, the IF outputs of
the mixers are (K = the mixer constant for the squared term of the diode)
A
v IF


IF inputs (to IF hybrid) 
B
v IF 

K
2 2
VLO [VU  VL ] sin  IF t
 K V [V
2 2 LO U
 VL ] cos  IF t
 A  jK
VIF  2 2 VLO VU  VL 
*Phasor representation 
K
B
VIF

VLO VU  VL 

2 2

Combining these two signals in the 90o IF hybrid (transformer) at the IF output
gives
V1   j
A
VIF
V B KVLOVL
 IF 
2
2
2
KVLOVL
cos IF t ,
2
 KVLOVU
v2 ( t ) 
sin IF t ,
2
v1 (t ) 
A
B
jKVLOVU
VIF
VIF
V2  
j

2
2
2
VU cos(o  IF )t
 image
vr ( vDesired
vRF )  
IM Band
V cos(   )t
o
IF
 L
r
 im
vr
RF input
Mixer A IF
A
vRF
B
vRF
Z0
2008
Desired Band
LO
RF
2 IF
90o
hybrid
LO
3-dB
power
divider
LO
input
LSB
v1
r
vLO 
VLO cos(ot )
RF
Mixer B
A
vIF
LPF
IF
v2
90o
image
USB
IF hybrid
(transformer)
B
vIF
LPF

IF
out
 im
H.-R. Chuang, EE NCKU
7-61
I V   I o  vGd  v 2 Gd 2  ...................
(note : v  VRF  VLo  Vr cosr t  Vo cosot )
Gd
Vr cosr t  Vo cosot 2
2
G
 2d (V r2 cos2 r t  2VrVo cosr t cosot  Vo2 cos2 ot )
G
G
 IF output  4d 2VrVo cos r  o t  2d VrVo cosIF t
 v 2 Gd 2 





v1  (Vr
VLO
poor
isolation
v1
V RF  Vr cos r t
2008
2)
LP filter
i1
RF
Diode 1
LO
Diode 2
v 2
V Lo  Vo cos o t
VRF
2 )  (  j VLo
+
i2
i IF  2 KVrVo sin i t
LP filter
3 dB hybrid
(90o )
IF output
v 2  (  j Vr
2 )  (V Lo
2)

H.-R. Chuang, EE NCKU