oscillators communication circuits prof. m. kamarei

Oscillators

Communication Circuits

Prof. M. Kamarei

M. Kamarei, Spring 2009

Oscillators 2

Contents

● Introduction● General Considerations● Ring Oscillators● LC Oscillators● Other Oscillators● Voltage-Controlled Oscillators


Oscillators 3

Introduction

● An oscillator is an autonomous circuit that converts DC power into a periodic waveform.

● Oscillators are extensively used in both receive and transmit paths. They are used to provide the local oscillation for the mixers for up and down conversion.


Oscillators 4

Contents

● Introduction● General Considerations

– Oscillator As Feedback System– Barkhausen Criteria– One-Port View of Resonator-Based Oscillators

● Ring Oscillators● LC Oscillators● Other Oscillators● Voltage-Controlled Oscillators


Oscillators 5

Oscillator As Feedback System

● General feedback system

● We assume s=jω

● If there is a frequency in which H(jω0)=1180º then

)(1

)()(

sH

sHs

V

V

in

out

)(1

)()(

jH

jHj

V

V

in

out

)( 0jV

V

in

out


Oscillators 6

How a circuit can oscillate?

● Input noise component at ω0 experiences unity gain and 180º phase shift in H(s). Then it is subtracted from input and gives larger difference.

● The circuit continues regenerating and allows the ω0 to grow.


Oscillators 7

|H(jω0)|<1 |H(jω0)|>1

VX : Diverged

How a circuit can oscillate?

● Following the signal around the loop over several cycles:


Oscillators 8

Barkhausen Criteria

● If a negative feedback circuit satisfies these two conditions, the circuit may oscillate at ω0 .

● First criterion– We typically choose the loop gain to be at least two or three

times more than the required value to ensure oscillation in presence temperature and process variations.

180)(

1)(

jH

jH


Oscillators 9

Barkhausen Criteria

● Second criterion– 180º phase shift criterion is equivalent to totally 360º loop phase

shift.

– (a): 180º phase shift is provided by negative feedback and another 180º by amplifier.

– (b): 360º phase shift is totally provided by amplifier.– (c): 2nπ phase shift function same as (b)


Oscillators 10

One-Port View of Resonator-Based Oscillators

● Convenient for intuitive analysis● Here we seek to cancel out loss in tank with a negative

resistance element– To achieve sustained oscillation, we must have:


Oscillators 11

One-Port View of Resonator-Based Oscillators

● Simple tank stimulated with current impulse– Decaying oscillation– In each cycle energy is lost in the resistor.

● Canceling out RP with a negative resistance –RP

– Stable oscillation


Oscillators 12

Contents

● Introduction● General Considerations● Ring Oscillators

– Three-Stages Ring Oscillator– Natural Response– Some Other Implementations

● LC Oscillators● Other Oscillators● Voltage-Controlled Oscillators


Oscillators 13

Simple Ring Oscillator

● Ring oscillator consists of some gain stages in a loop.

● Dose a single common-source stage oscillate in loop?

Max. Phase Shift = 180º + 90º < 360º

Common-source negative gain

Max. Phase shift of the output single pole

No Oscillation


Oscillators 14

Three-Stages Ring Oscillator

● Negative feedback provides 180º phase shift.

● 3 single pole gain stages can provide 0 to 270º phase shift

● Phase criterion → Oscillation frequency

00

1 360tan603180360)(

oscoscjH

Negative feedback

Each stage phase delay


Oscillators 15


● Gain criterion → The minimum gain of stages

● Phase delay between 2 outputs:– 180º-60º=120º

● The ability to generate multiple phases


Oscillators 16


● Amplitude limiting

Poles:

Neglecting s1


Oscillators 17

Three-Stages Ring Oscillator (Root Locus)

● A0=2– Pure Complex poles → Stable oscillation

● A0<2– Circuit is stable (LHP poles) → Damped oscillation

● A0>2– Circuit is unstable (RHP poles) → Oscillation amplitude grows


Oscillators 18

Natural Response

● Given a transfer function

● The total response of the system can be partitioned into the natural response and the forced response

● where f2(si(t)) is the forced response whereas the first term f1() is the natural response of the system, even in the absence of the input signal. The natural response is determined by the initial conditions of the system.


Oscillators 19

Real LHP Poles

● Stable systems have all poles in the left-half plane (LHP).

● Consider the natural response when the pole is on the negative real axis, such as s1 for our examples.

● The response is a decaying exponential that dies away with a time-constant determined by the pole magnitude.


Oscillators 20

Complex Conjugate LHP Poles

● Since s2,3 are a complex conjugate pair

● We can group these responses since a3 = a*

2

into a single term

● When the real part of the complex conjugate pair σ is negative, the response also decays exponentially.


Oscillators 21

Complex Conjugate Poles (RHP)

● When σ is positive (RHP), the response is an exponential growing oscillation at a frequency determined by the imaginary part ω0.

● Thus we see for any amplifier with three identical poles, if feedback is applied with loop gain A0

3 > 23 = 8, the amplifier will oscillate.


Oscillators 22

Frequency Domain Perspective

● In the frequency domain perspective, we see that a feedback amplifier has a transfer function

● If the loop gain a0 f = 8, then we have with purely imaginary poles at a frequency ωosc = √3×ω0 where the transfer function a(jωosc)f = -1 blows up. Apparently, the feedback amplifier has infinite gain at this frequency.


Oscillators 23

Oscillation Build Up

● In a real oscillator, as the oscillation amplitude increases, the stages in the signal path experience nonlinearity and eventually saturation, limiting the maximum amplitude.

● We may say the poles being in the RHP and finally move to imaginary axis to stop

the growth.● The circuit must spend enough time in saturation so that the average loop gain is still equal to unity.


Oscillators 24

Some Other Implementations

● Differential implementation


Oscillators 25


● CMOS inverter ring oscillator


Oscillators 26


● Five-stages single-ended ring oscillator

● Four-stages differential ring oscillator


Oscillators 27

Contents

● Introduction● General Considerations● Ring Oscillators● LC Oscillators

– RLC Circuits

– A Simple Negative Resistance Oscillator

– Differential Negative Resistance Oscilator

– Colpitts Oscillator

– Clapp Oscillator

– Hartley Oscillator

– CE and CC Oscillators

– Pierce Oscillator

● Other Oscillators● Voltage-Controlled Oscillators


Oscillators 28

RLC Circuits

● Resonance of LC tank

● The impedance of the circuit goes to infinity at resonance frequency

● Q of the tank in infinite.

(Q is the quality factor of the tank and defined as energy stored, divided by energy dissipated per cycle)

Ideal Tank

jC

jLjZ1

||

LCjC

jLjZ resres

resres

10

1


Oscillators 29

RLC Circuits

● Realistic inductors suffer from resistive components.

● Impedance peaks at resonance frequency.● Q in finite.

Realistic Tank


Oscillators 30

Series RL to Parallel RL

● In narrow frequency range it is possible to convert circuit to parallel configuration.

● For the two impedances to be equivalent:

L1ω/Rs=Q typically more than 3


Oscillators 31

A Simple RLC Model

● RP is typically higher than RS

● CP is equal to CS

● In resonance– frequency the tank reduces to a simple resistor– Phase difference between voltage and current is zero

● For ω<ω1

– Inductive behavior, Positive phase

● For ω>ω1

– Capacitive behavior, Negative phase


Oscillators 32

A Simple Negative Resistance OSC

● How can a circuit provide negative resistance?– Positive feedback circuits with

sufficient gain can provide it

Small signal model

If gm1=gm2=gm


Oscillators 33

A Simple Negative Resistance Osc.

● Adding a tank to the negative resistor, we make an oscillator● For oscillation build-up

– RP-2/gm 0

● Redrawing the oscillator● Converting to differential negative resistance oscillator

Differential


Oscillators 34

Differential Negative Resistance Osc.

● This type of oscillator structure is quite popular in current CMOS implementations– Advantages

● Simple topology

● Differential implementation (good for feeding differential circuits)

● Good phase noise performance can be achieved


Oscillators 35

Analysis of Negative Resistance Osc. (Step 1)

● Derive a parallel RLC network that includes the loss of the tank inductor and capacitor– Typically, such loss is dominated by series resistance in the

inductor


Oscillators 36


● Split oscillator circuit into half circuits to simplify analysis– Leverages the fact that we can approximate Vs as being

incremental ground (this is not quite true, but close enough)

● Recognize that we have a diode connected device with a negative transconductance value– Replace with negative resistor

● Note: Gm is large signal transconductance value


Oscillators 37


● Design tank components to achieve high Q– Resulting Rp value is as large as possible

● Choose bias current (Ibias) for large swing (without going far into Gm saturation)– We’ll estimate swing as a function of Ibias shortly

● Choose transistor size to achieve adequately large gm1

– Usually twice as large as 1/Rp1 to guarantee startup


Oscillators 38

Calculation of Oscillator Swing: Max. Sinusoidal Oscillation

● If we assume the amplitude is large, Ibias is fully steered to one side at the peak and the bottom of the sinusoid:


Oscillators 39

Calculation of Oscillator Swing: Square wave Oscillation

● If amplitude is very large, we can assume I1(t) is a square wave– We are interested in determining fundamental component

● (DC and harmonics filtered by tank)

– Fundamental component is

– Resulting oscillator amplitude


Oscillators 40

Variations on a Theme

● Biasing can come from top or bottom● Can use either NMOS, PMOS, or both for transconductor

– Use of both NMOS and PMOS for coupled pair would appear to achieve better phase noise at a given power dissipation


Oscillators 41

Colpitts Oscillator

● Carryover from discrete designs in which single-ended approaches were preferred for simplicity– Achieves negative resistance with only one transistor– Differential structure can also be implemented, though

● Good phase noise can be achieved, but not apparent there is an advantage of this design over negative resistance design for CMOS applications


Oscillators 42

Analysis of Cap Transformer used in Colpitts

● Voltage drop across RL is reduced by capacitive voltage divider– Assume that impedances of caps are less than RL at resonant

frequency of tank (simplifies analysis)● Ratio of V1 to Vout set by caps and not RL

● Power conservation leads to transformer relationship


Oscillators 43

Simplified Model of Colpitts

● Purpose of cap transformer– Reduces loading on tank– Reduces swing at source node (important for bipolar version)

● Transformer ratio set to achieve best noise performance


Oscillators 44

Design of Colpitts Oscillator

● Design tank for high Q

● Choose bias current (Ibias) for large swing (without going far into Gm saturation)

● Choose transformer ratio for best noise– Rule of thumb: choose N = 1/5 according to Tom Lee

● Choose transistor size to achieve adequately large gm1


Oscillators 45

Calculation of Oscillator Swing as a Function of Ibias

● I1(t) consists of pulses whose shape and width are a function of the transistor behavior and transformer ratio – Approximate as narrow square wave pulses with width W

– Fundamental component is

– Resulting oscillator amplitude


Oscillators 46

Clapp Oscillator

● Same as Colpitts except that inductor portion of tank is isolated from the drain of the device– Allows inductor voltage to achieve a larger amplitude without

exceeded the max allowable voltage at the drain● Good for achieving lower phase noise


Oscillators 47

Simplified Model of Clapp Oscillator

● Looks similar to Colpitts model– Be careful of parasitic resonances!


Oscillators 48

Hartley Oscillator

● Same as Colpitts, but uses a tapped inductor rather than series capacitors to implement the transformer portion of the circuit– Not popular for IC implementations due to the fact that

capacitors are easier to realize than inductors


Oscillators 49

Simplified Model of Hartley Oscillator

● Similar to Colpitts, again be wary of parasitic resonances


Oscillators 50

CE and CC Oscillators

● If we ground the emitter, we have a new oscillator topology, called the Pierce Oscillator. Note that the amplifier is in CE mode, but we don’t need a transformer!

● Likewise, if we ground the collector, we have an emitter follower oscillator.

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

C1

C2 C1

C2


Oscillators 51

● If we assume that the current through C1,2 is larger than the collector current (high Q), then we see that the same current flows through both capacitors. The voltage at the input and output is therefore

Pierce Oscillator

11

1

CjIvo

21

1

CjIvi 2

1

C

Cn

v

v

i

o


Oscillators 52

Pierce Oscillator Bias

● A current source or large resistor can bias the Pierce oscillator.

● Since the bias current is fixed, the same large signal oscillator analysis applies.


Oscillators 53

Contents

● Introduction● General Considerations● Ring Oscillators● LC Oscillators● Other Oscillators

– Wien-Bridge Oscillator– Phase-Shift Oscillator– Crystal Oscillators– Quadrature Oscillator

● Voltage-Controlled Oscillators


Oscillators 54

Wien-Bridge Oscillator

● By breaking the loop at the positive input, the return signal is calculated as:

● When ω=2πf=1/RC, the feedback is in phase, and the gain is 1/3.→ Oscillation requires an amplifier with

the gain of 3.

.1

3

11

3

11

1

1

RCRCj

RCsRCs

CsR

RCsR

RCs

R

V

V

OUT

RETURN


Oscillators 55

Wien-Bridge Oscillator

● When RF = 2RG, amplifier gain is 3 and oscillation occurs at f = 1/2πRC

● Example:– [TI, Analog Applications Journal, 2000]

– f = 1.59 kHz


Oscillators 56

Phase-Shift Oscillator

● With one op amp● Assuming that the phase-shift sections are independent

of each other:3

1

1

RCsAA


Oscillators 57


● When the phase shift of each section is -60º, the loop phase shift is -180º. – This occurs when ω=2πf=1.732/RC (since tan(60º) ≈ 1.732)– The magnitude of β is (1/2)3, so the gain A must be 8.

● However in practice, the RC sections load each other and this results in discrepancies from the simple calculated gain and frequency equations.

● Besides, now op amps are inexpensive and small. So the single op amp phase-shift oscillator is losing popularity.


Oscillators 58


● Buffered Phase-Shift Oscillator– The buffers prevent the RC sections from loading each other,

hence closer performance to the simple calculated gain and frequency.

● Note: output is a high impedance node, so a high impedance load is mandated to prevent loading and phase shifting.


Oscillators 59

Crystal Oscillators

● The most common non-RLC oscillator is made of quartz.● Quartz is a piezoelectric material and thus it exhibits a

reciprocal transduction between mechanical strain and electric charge. – Exceptional electrical and mechanical stability– Crystals with very low temperature coefficients are feasible– Q values in the range of 104 - 106

● Electrical model:contacts and lead wires

mechanical energy storage

loss (RS ≈ k/f0)


Oscillators 60

Crystal Oscillators

● In upper fig. the crystal is used in its series resonant mode to close the feedback loop at the desired frequency as in a Colpitts oscillator.

● The inductance across the crystal is frequently needed in practical designs to prevent unwanted off-frequency oscillations due to feedback provided by the crystal’s parallel capacitance (C0). The inductance resonates out C0.

● The modified scheme is shown in bottom.– This topology is useful if one terminal of the

crystal must be grounded.

Modified


Oscillators 61

Crystal Oscillators

● The MOS Pierce oscillator is a popular crystal oscillator.● The crystal behaves like an inductor in the frequency

range above the series resonance of the fundamental tone (below the parallel resonance due to the parasitic capacitance).

● The crystal is a mechanical resonators. Other structures, such as poly MEMS combs, disks, or beams can also be used.


Oscillators 62

Quadrature Oscillator (with feedback loop)

● Coupling two oscillators in such a way to make them oscillate with 90° difference in phase

● Measuring the phase difference and controlling the bias current leads to precise 90° phase shift


Oscillators 63

Contents

● Introduction● General Considerations● Ring Oscillators● LC Oscillators● Other Oscillators● Voltage-Controlled Oscillators

– Performance Parameters of VCOs– Mathematical Model of VCOs– PN Junction Varactors– Varactor Tuned LC Oscillator– Clapp Varactor Tuned VCO– Varactor Tuned Differential Oscillator– MOS Varactors


Oscillators 64

Voltage-Controlled Osc.

● In most applications we need to tune the oscillator to a particular frequency electronically.

● A voltage controlled oscillator (VCO) has a separate “control” input where the frequency of oscillation is a function of the control signal Vc


Oscillators 65

Performance Parameters of VCOs

● Center Frequency● Tuning Range

– The tuning range of the VCO is given by the range in frequency normalized to the average frequency

– A typical VCO might have a tuning range of 20% 30% to cover a band of frequencies (over process and temperature)

● Tuning Linearity– Higher sensitivity in some regions


Oscillators 66

Performance Parameters of VCOs

● Output Amplitude– Larger oscillation amplitude → Less sensitive to noise– Amplitude trades with power dissipation, supply voltage, and

tuning range

● Power Dissipation– Trade-offs between speed, power dissipation, and noise

● Supply and Common-Mode Rejection– Single-ended forms are more sensitive– Noise may coupled to VCO control line and modulates carrier

● Output Signal Purity– Jitter– Phase Noise


Oscillators 67

Mathematical Model of VCOs

● Consider two waveforms

● V2(t) accumulate phase faster than V1(t)

● The faster the phase of a waveform varies, the higher the frequency of the waveform


Oscillators 68

Mathematical Model of VCOs

● Frequency can be defined as:

● Since for a VCO, ωout = ω0 + KVCOVcont we have:

● If a VCO is placed in PLL, the only term KVCO ∫Vcont dt is important

s

KVCO ext 0cos exoutVcontV


Oscillators 69

An Example of the Model

● The Control line of a VCO senses a rectangular signal toggling between V1 and V2 and period Tm.


Oscillators 70

PN Junction Varactors

● The most common way to control the frequency is by using a reverse biased PN junction

● The small signal capacitance is given by

● The above formula is easily derived by observing that a positive increment in reverse bias voltage requires an increment of growth of the depletion region width. Since charge must flow to the edge of the depletion region, the structure acts like a parallel plate capacitors for small voltage perturbations.


Oscillators 71

PN Junctions Doping

● Depending on the doping profile, one can design capacitors with n = ½ (abrupt junction), n = ⅓ (linear grade), and even n = 2.

● The n = 2 case is convenient as it leads to a linear relationship between the frequency of oscillation and the control voltage


Oscillators 72

Varactor Tuned LC Oscillator

● The DC point of the varactor is isolated from the circuit with a capacitor. For a voltage V < VCC, the capacitor is reverse biased (note the polarity of the varactor). If the varactor is flipped, then a voltage larger than VCC is required to tune the circuit.


Oscillators 73

Clapp Varactor Tuned VCO

● Similar to the previous case the varactor is DC isolated● In all varactor tuned VCOs, we must avoid forward

biasing the varactor since it will de-Q the tank


Oscillators 74

Varactor Tuned Differential Osc.

● Two varactors are used to tune the circuit. In many processes, though, the n side of the diode is “dirty” due to the large well capacitance and substrate connection resistance.


Oscillators 75

Varactor Tuned Diff Osc.

● By reversing the diode connection, the “dirty” side of the PN junction is connected to a virtual ground. But this requires a control voltage Vc > Vdd. We can avoid this by using capacitors.

● The resistors to ground simply provide a DC connection to the varactor n terminals.


Oscillators 76

MOS Varactors

● The MOS capacitor can be used as a varactor. The capacitance varies dramatically from accumulation Vgb < Vfb to depletion. In accumulation majority carriers (holes) form the bottom plate of a parallel plate capacitor.

● In depletion, the presence of a depletion region with dopant atoms creates a non-linear capacitor that can be modeled as two series capacitors (Cdep and Cox), effectively increasing the plate thickness to tox + tdep


Oscillators 77

MOS Varactor (Inversion)

● For a quasi-static excitation, thermal generation leads to minority carrier generation. Thus the channel will invert for VGB > VT and the capacitance will return to Cox.

● The transition around threshold is very rapid. If a MOSFET MOS-C structure is used (with source/drain junctions), then minority carriers are injected from the junctions and the high-frequency capacitance includes the inversion transition.


Oscillators 78

Accumulation Mode MOS Varactors

● The best varactors are accumulation mode electron MOS-C structures. The electrons have higher mobility than holes leading to lower series resistance, and thus a larger Q factor.

● Notice that this structure cannot go into inversion at high frequency due to the limited thermal generation of minority carriers.

oscillators communication circuits prof. m. kamarei

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