PCIRF_5_MIXER_3

7/29/2019 PCIRF_5_MIXER_3

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EECS 142

Lecture 19: I/Q Mixers; BJT Mixers

Prof. Ali M. Niknejad

University of California, Berkeley

Copyright c 2008 by Ali M. Niknejad

A. M. Nikne ad Universit of California Berkele EECS 142 Lecture 19 . 1/14


2/14

BJT with Large Sine Drive

vi

VA

IC

vi = Vi cost

IC = ISe

vBEVt

vBE = VA + Vi cost

Consider a bipolar device driven with a large sine signal.

This occurs in many types of non-linear circuits, such asoscillators, frequency multipliers, mixers and class C amplifiers.



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BJT Collector Current

The collector current can be factored into a DC bias term and aperiodic signal

IC = ISeVA

Vte

ViV

t

cost

IC = ISeaeb cost

Where the normalized bias is a = Va/Vt and the normalized

drive signal is b = Vi/Vt.Since IC is a periodic function, we can expand it into a Fourier

Series. Note that the Fourier coefficients of eb cost are modifiedBessel functions In(b)

eb cost = I0(b) + 2I1(b)cost + 2I2(b)cos2t +



4/14

BJT DC Current

Assume that the bias current of the amplifier is stabilized. Then

IC = ISeaI0(b) IQ

1 +2I1(b)

I0(b)

cost +2I2(b)

I0(b)

cos2t +

IC = ISeaeb cost

= IQI0(b)eb cost

1 2 3 4 5

1

2

3

4

5

6

IC

IQ



5/14

Collector Current Waveform

-0.4 -0.2 0 0.2 0.4

1

2

3

4

5

6

7

8

IC

IQ

b = .5

b = 4

b = 10

With increasing input drive, the current waveform becomes

peaky. The peak value can exceed the DC bias by a largefactor.



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Harmonic Current Amplitudes

2.5 5 7.5 10 12.5 15 17.5 20

0.2

0.4

0.6

0.8

1

In(b)

I0(b)

b

n = 1

n= 2

n = 3

n = 4

n = 5

n = 6

n = 7

The BJT output spectrum is rich in harmonics.



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Small-Signal Region

0.05 0.1 0.15 0.2

0.002

0.004

0.006

0.008

0.01

In(b)

I0(b)

b

n = 1

n = 2

n = 3

If we zoom in on the curves to small b values, we enter thesmall-signal regime, and the weakly non-linear behavior ispredicted by our power series analysis.



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BJT with Stable Bias

Neglecting base current ( 1), the voltage at the base isgiven by

R1

R2 RE CE

+vi

VA

VA =R2

R1 + R2VCC

VE = V

A VBE

IQ =VERE

=VA VBE

REA. M. Nikne ad Universit of California Berkele EECS 142 Lecture 19 . 8/14


9/14

Differential Pair with Sine Drive

The large signal equation for IC1 is given by

IC1 + IC2 = IEE

VBE1 VBE2 = Vi = Vt ln

IC1IC2

+Vi2

Vi2

+

IEE

IC1 =IEE

1 + evi/Vt

vi = Vi cost

IC1 =IEE

1 + eb cost



10/14

Diff Pair Waveforms

1 2 3 4 5 6

0.2

0.4

0.6

0.8

1

IC1

IEE

b = 1

b = 3b = 1 0

For large b, the waveform approaches a square wave

ICIEE

=1

2+

2

cost

1

3cos3t +

1

5cos5t +



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Diff Pair Harmonic Currents

2.5 5 7.5 10 12.5 15 17.5 20

0.1

0.2

0.3

0.4

0.5

0.6

b

2

n = 1

n = 3

n = 5

n = 7

Normalized Harmonic Currents

(negative)

(negative)

As expected, the ideal differential pair does not produce anyeven harmonics.



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Mixer Analysis

As we have seen, a mixer has three ports, the LO, RF, and IFport.

Assume that a circuit is pumped with a periodic large signal atthe LO port with frequency 0.

From the RF port, though, assume we apply a small signal atfrequency s.

Since the RF input is small, the circuit response should belinear (or weakly non-linear). But since the LO port changes theoperating point of the circuit periodically, we expect the overallresponse to the RF port to be a linear time-varying response

io(t) = g(t)vin



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Mixer Assumptions

gm(t)

The transconductance varies periodically and can be expandedin a Fourier series

g(t) = g0 + g1 cos0t + g2 cos20t +

Applying the input vin = V1 cosst

io(t) = (g0 + g1 cos0t + g2 cos20t + ) V1 cosst



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Mixer Output Signal

Expanding the product, we have

io(t) = g0V1 cosst+g1

2

V1 cos(0s)t+g2

2

V1 cos(20s)t+

The first term is just the input amplified. The other terms are alldue to the mixing action of the linear time-varying periodiccircuit.

Lets say the desired output is the IF at 0 s. The conversiongain is therefore defined as

gconv =|IF output current|

|RF input signal voltage|=

g12


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