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Summary of Last Lecture

vovi

Last lecture we analyzed the small-signal behavior ofthe above circuit. We found that the closed-loop gain is

given by

H(s) =gmRs

LR

1 + sLR

(1 A) + s2LC

A. M. Niknejad University of California, Berkeley EECS 142 Lecture 22 p. 2/23

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Review: Role of Loop Gain

The behavior of the circuit is determined largely by A,the loop gain at DC and resonance. When A = 1, thepoles of the system are on the j axis, corresponding

to constant amplitude oscillation.

When A < 1, the circuit oscillates but decays to aquiescent steady-state.

When A > 1, the circuit begins oscillating with anamplitude which grows exponentially. Eventually, wefind that the steady state amplitude is fixed.


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Steady-State Analysisstart-up region

steady-state region

To find the steady-state behavior of the circuit, we will

make several simplifying assumptions. The mostimportant assumption is the high tank Q assumption(say Q > 10), which implies the output waveform vo issinusoidal.

Since the feedback network is linear, the inputwaveform vi = vo/n is also sinusoidal.

We may therefore apply the large-signal periodicsteady-state analysis of the BJT to the oscillator.


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Steady-State Waveforms

vo

vi

VCC

VBE,Q

IQ

The collector current is not sinusoidal, due to the largesignal drive.

The output voltage,though, is sinusoidal and given by

vo I1ZT(1) = GmZTvi


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Steady State Equations

But the input waveform is a scaled version of the output

vo = GmZTvon

=GmZT

nvo

The above equation implies that

GmZT

n 1

Or that the loop gain in steady-state is unity and thephase of the loop gain is zero degrees (an exactmultiple of 2)

GmZT

n

1

GmZT

n 0


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Large Signal Gm

Recall that the small-signal loop gain is given by

|A| =

gmZTn

Which implies a relation between the small-signalstart-up transconductance and the steady-state

large-signal transconductance

gmGm

= A

Notice that gm and A are design parameters under ourcontrol, set by the choice of bias current and tank Q.The steady state Gm is therefore also fixed by initialstart-up conditions.


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Large Signal Gm (II)

0.2

0.4

0.6

0.8

1

2 4 181614121086 20

Gm

gm(b)

b

To find the oscillation amplitude we need to find theinput voltage drive to produce Gm.

For a BJT, we found that under the constraint that the

bias current is fixed

I1 =2I1(b)

I0(b)IQ = Gmvi = Gmb

kT

q


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Large Signal Gm (III)

Thus the large-signal Gm is given by

Gm =2I1(b)

bI0(b)

qIQ

kT=

2I1(b)

bI0(b)gm

Gmgm

=2I1(b)

bI0(b)


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Stability (Intuition)

Heres an intuitive argument for how the oscillatorreaches a stable oscillation amplitude. Assume thatinitially Al > 1 and oscillations grow. As the amplitude ofoscillation increases, though, the Gm of the transistor

drops, and so effectively the loop gain drops.As the loop gain drops, the poles move closer to the jaxis. This process continues until the poles hit the j

axis, after which the oscillation ensues at a constantamplitude and A = 1.


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Intuition (cont)

To see how this is a stable point, consider whathappens if somehow the loop gain changes. If the loopgain changes to A + ||, then we already see that the

system will roll back. If the loop gain drops below unity,A ||, then the poles move into the LHP andamplitude of oscillation will begin to decay.

As the oscillation amplitude decays, the Gm increasesand this causes the loop gain to grow. Thus the systemalso rolls back to the point where A = 1.


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BJT Oscillator Design

Say we desire an oscillation amplitude of v0 = 100mV ata certain oscillation frequency 0.

We begin by selecting a loop gain A > 1 with sufficientmargin. Say A = 3.

We also tune the LC tank to 0, being careful to includethe loaded effects of the transistor (ro, Co, Cin, Rin)

We can estimate the required first harmonic currentfrom

I0 =vo

R

T


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Design (cont)

This is an estimate because the exact value of RT is notknown until we specify the operating point of thetransistor. But a good first order estimate is to neglect

the loading and use R

T

We can now solve for the bias point from

I1 = 2I1(b)I0(b)IQ

b is known since its the oscillation amplitude normalized

to kT/q and divided by n. The above equation can besolved graphically or numerically.

Once IQ is known, we can now calculate R

T and iterate

until the bias current converges to the final value.


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Squegging

Squegging is a parasitic oscillation in the bias circuitryof the amplifier.

It can be avoided by properly sizing the emitter bypasscapacitance

CE nCT


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Common Base Oscillator

vovi

Another BJT oscillator uses the common-basetransistor. Since there is no phase inversion in theamplifier, the transformer feedback is in phase.

Since we dont need phase inversion, we can use asimpler feedback consisting of a capacitor divider.


C l i O ill

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Colpitts Oscillator

The cap divider works at higher frequencies. Under the

cap divider approximation f C1C1 + C2

= 1n

n = 1 +C2

C1

C2

includes the loading from the transistor and currentsource.


C l itt Bi

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Colpitts Bias

Since the bias current is held constant by a currentsource IQ or a large resistor, the analysis is identical to

the BJT oscillator with transformer feedback. Note theoutput voltage is divided and applied across vBE just as

before.


C l itt F il

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Colpitts Family

If we remove the explicit ground connection on theoscillator, we have the template for a generic oscillator.

It can be shown that the Colpitts family of oscillatornever squegg.


CE d CC O ill t

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CE and CC Oscillators

If we ground the emitter, we have a new oscillatortopology, called the Pierce Oscillator. Note that theamplifier is in CE mode, but we dont need atransformer!

Likewise, if we ground the collector, we have an emitterfollower oscillator.

A fraction of the tank resonant current flows throughC1,2. In fact, we can use C1,2 as the tank capacitance.


Pi O ill t

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Pierce Oscillator

I1

If we assume that the current through C1,2 is larger than

the collector current (high Q), then we see that thesame current flows through both capacitors. Thevoltage at the input and output is therefore

vo = I11

jC1 vi = I1

1

jC2

orvo

vi = n =

C1

C2


Pi Bi

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Pierce Bias

A current source or large resistor can bias the Pierceoscillator.

Since the bias current is fixed, the same large signaloscillator analysis applies.


C C ll t O ill t

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Common-Collector Oscillator

Note that the collector can be connected to a resistorwithout changing the oscillator characteristics. In fact,

the transistor provides a buffered output for free.


Clapp Oscillator

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Clapp Oscillator

CB

C1

C2

RB

The common-collector oscillator shown above uses alarge capacitor CT to block the DC signal at the base.

RB is used to bias the transistor.If the shunt capacitor CT is eliminated, then thecapacitor CB can be used to resonate with L and theseries combination of C1 and C2. This is a seriesresonant circuit.


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