Sinusoidal oscillators

Objectives

Describe the basic concept of an oscillator Discuss the basic principles of operation of an

oscillator Analyze the operation of RC oscillators

Introduction

Oscillator is an electronic circuit that generates a periodic waveform on its output without an external signal source. It is used to convert dc to ac.

Oscillators are circuits that produce a continuous signal of some type without the need of an input.

These signals serve a variety of purposes. Communications systems, digital systems

(including computers), and test equipment make use of oscillators

Introduction

An oscillator is a circuit that produces a repetitive signal from a dc voltage.

The feedback oscillator relies on a positive feedback of the output to maintain the oscillations.

The relaxation oscillator makes use of an RC timing circuit to generate a nonsinusoidal signal such as square wave

Sine wave

Square wave

Sawtooth wave

Types of oscillators

1. RC oscillators Wien Bridge Phase-Shift

1. LC oscillators Hartley Colpitts Crystal

1. Unijunction / relaxation oscillators

Feedback Oscillator Principles

When switch at the amplifier input is open, no oscillation occurs.

Consider Vi,, results in Vo=AVi (after amplifier stage) and Vf = β(AVi) (after feedback stage)

Feedback voltage Vf = β(AVi) where βA is called loop gain.

In order to maintain Vf = Vi , βA must be in the correct magnitude and phase.

When the switch is closed and Vi is removed, the circuit will continue operating since the feedback voltage is sufficient to drive the amplifier and feedback circuit, resulting in proper input voltage to sustain the loop operation.

Feedback circuit used as an oscillator

Basic principles for oscillation

An oscillator is an amplifier with positive feedback.

A

β

V e

V f

V sV o

+

(1) fse VVV +=(2) of βVV =

( ) ( ) (3) osfseo βVVAVVAAVV +=+==


The closed loop gain is:

( ) ( )osfs

eo

βVVAVVA

AVV

+=+==

oso VAAVV β+=( ) so AVVA =− β1

( )Aβ

A

V

VA

s

of −

=≡1


In general A and β are functions of frequency and thus may be written as;

is known as loop gain

( ) ( ) ( )( ) ( )sβsA1

sAs

V

VsA

s

of −

==

( ) ( )sβsA


Writing the loop gain becomes;

Replacing s with jω

and

( ) ( ) ( )ss βAsT =

( ) ( )( )sT1

sAsA f −

=

( ) ( )( )jωT1

jωAjωA f −

=

( ) ( ) ( )jωβjωAjωT =


At a specific frequency f0

At this frequency, the closed loop gain;

will be infinite, i.e. the circuit will have finite output for zero input signal - oscillation

( ) ( ) ( ) 1000 == jωβjωAjωT

( ) ( )( ) ( )00

00 jωβjωA1

jωAjωA f −

=


Thus, the condition for sinusoidal oscillation of frequency f0 is;

This is known as Barkhausen criterion. The frequency of oscillation is solely determined by

the phase characteristic of the feedback loop – the loop oscillates at the frequency for which the phase is zero.

( ) ( ) 100 =jωβjωA


The feedback oscillator is widely used for generation of sine wave signals.

The positive (in phase) feedback arrangement maintains the oscillations.

The feedback gain must be kept to unity to keep the output from distorting.


In phase

Noninverting amplifier

V f V oA v

Feedback circuit

Design Criteria for Oscillators

1. The magnitude of the loop gain must be unity or slightly larger

– Barkhaussen criterion2. Total phase shift,φ of loop gain must be 0 ° or 360°

1=Aβ

RC Oscillators

RC feedback oscillators are generally limited to frequencies of 1 MHz or less.

The types of RC oscillators that we will discuss are the Wien-bridge and the phase-shift

Wien-bridge Oscillator

It is a low frequency oscillator which ranges from a few kHz to 1 MHz.

The Wien-bridge oscillator schematic drawn in two different but equivalent ways

Oscillator Output Gain and Phase Shift


The loop gain for the oscillator is;

where;

and;

( ) ( ) ( )

+

+==

sp

p

ZZ

Z

R

RsβsAsT

1

21

sRC

RZ p +

=1

sC

sRCZ s

+= 1


Hence;

Substituting for s;

For oscillation frequency f0;

( ) ( )

++

+=

RC/jRCjR

RjT

001

20 13

11

ωωω

( ) ( )

++

+=

/sRCsRCR

RsT

13

11

1

2

( ) ( )

++

+=

RC/jRCjR

RjT

ωωω

13

11

1

2


Since at the frequency of oscillation, T(jω) must be real (for zero phase condition), the imaginary component must be zero;

Which gives us;

01

00 =+

RCjRCj

ωω

RC

10 =ω


From the previous equation;

the magnitude condition is;

or

+=

3

111

1

2

R

R

( ) ( )

++

+=

RC/jRCjR

RjT

001

20 13

11

ωωω

21

2 =R

R

To ensure oscillation, the ratio R2/R1 must be slightly greater than 2.


With the ratio;

then;

K = 3 ensures the loop gain of unity – oscillation K > 3 : growing oscillations K < 3 : decreasing oscillations

21

2 =R

R

311

2 =+≡R

RK

T i me

0 s 0 . 2 ms 0 . 4 ms 0 . 6 ms 0 . 8 ms 1 . 0 msV( R5 : 2 )

- 4 . 0 V

0 V

4 . 0 VG = 3

T i me

0 s 0 . 2 ms 0 . 4 ms 0 . 6 ms 0 . 8 ms 1 . 0 msV( R5 : 2 )

- 4 . 0 V

0 V

4 . 0 V

G = 2.9

T i me

0 s 1 0 0 u s 2 0 0 u s 3 0 0 u s 4 0 0 u s 5 0 0 u s 6 0 0 u sV( R5 : 2 )

- 2 0 V

0 V

2 0 V

G = 3.05

Ideal vs. Non-Ideal Op-Amp

Red is the ideal op-amp. Green is the 741 op-amp.

T i me

0 s 0 . 2 ms 0 . 4 ms 0 . 6 ms 0 . 8 ms 1 . 0 msV ( R1 : 2 ) V( R5 : 2 )

- 4 . 0 V

0 V

4 . 0 V

Start-Up Conditions

-Initially, the closed-loop gain of the amplifier itself must be more than 3 until the output signal builds up to a desired level.

-Ideally, the gain of the amplifier must then decrease to 3 so that the total gain around the loop is 1 and the output signal stays at the desired level, thus sustaining oscillation.

- This is illustrated in Figure on next slide.

In order to keep the oscillations constant, Hewlett Packard put a positive temperature co-effient lamp in the circuit at grounding resistor.The resistance of the lamp is strongly dependent on the temperature of the filament of the bulb. If the amplitude is too high, the current becomes large and the resistance of the lamp increases, thereby reducing the gain. If the amplitude is low, the lamp cools, the resistance decreases, and the loop gain increases.

The feedback fraction at fR in this circuit is one-third:

A must be > 3 for oscillations to start. After that, A must be reduced to avoid driving the op amp to VSAT.

in

out

B = inout =

1

3

R2 ≅ 2R1

R1

A = 1 +R2

R1

One solution is a positivetemperature coefficient

device here to decrease gain.

After theoscillationsstart, the

lamp heatsto reducegain andclipping.

R

Vout

C

RL

2R1

Tungstenlamp

C R

R1

Vout

time

Making the Oscillations Steady

Add a diode network to keep circuit around G = 3

If G = 3, diodes are off


When output voltage is positive, D1 turns on and R9 is switched in parallel causing G to drop


When output voltage is negative, D2 turns on and R9 is switched in parallel causing G to drop

Phase-Shift Oscillator

Phase-shift oscillator

The phase shift oscillator utilizes three RC circuits to provide 180º phase shift that when coupled with the 180º of the op-amp itself provides the necessary feedback to sustain oscillations.


vi v1 v1

v2v2 v3

voC

CC

R R

RR2

ivsRC

sRCv

+=11

ivsRC

sRCv

2

2 1

+=

ivsRC

sRCv

3

3 1

+=

3

3

1)(

+==

sRC

sRCs

v

v

i

β

R

R

v

vsA o 2

3

)( ==


Loop gain, T(s):

Set s=jw

3

2

1)()()(

+

==

sRC

sRC

R

RssAsT β

[ ] [ ]222222

22

3

2

331

))(()(

1)(

CRRCjCR

RCRCj

R

RjT

RCj

RCj

R

RjT

ωωωωωω

ωωω

−+−

−=

+

=


To satisfy condition T(jwo)=1, real component must be zero since the numerator is purely imaginary.

the oscillation frequency:

Apply wo in equation:

To satisfy condition T(jwo)=1

031 222 =− CRω

RC3

10 =ω

[ ]

−=

−+

−=

8

1

)3/1(3)3/(0

)3/1)(3/()( 22

R

R

j

j

R

RjT oω

82 =R

R The gain greater than 8, the circuit will spontaneously begin oscillating & sustain

oscillations

62

1

RCf

π=

where β = 1/29 and the phase-shift is 180o

For the loop gain βA to be greater than unity, the gain of the amplifier stage must be greater than 29.

If we measure the phase-shift per RC section, each section would not provide the same phase shift (although the overall phase shift is 180o). In order to obtain exactly 60o phase shift for each of three stages, voltage follower stages would be needed for each RC section.

when voltage follower is not used b/w RC stages

RCfo

62

1

π= 292 =

R

R The gain must be at least 29 to maintain the

oscillations

LC Oscillators

Use transistors and LC tuned circuits or crystals in their feedback network.

For hundreds of kHz to hundreds of MHz frequency range.

Examine Hartley, Colpitts and crystal oscillator.

Hartley oscillator Hartley oscillator was invented in 1915 by the American

engineer Ralph Hartley while he was working for the Western Electric company. The original design was tube based and he got a patent for it in the year 1920.

In Hartley oscillator the oscillation frequency is determined by a tank circuit comprising of two inductors and one capacitor. The inductors are connected in series and the capacitor is connected across them in parallel.

Hartley oscillators are commonly used in radio frequency (RF) oscillator applications and the recommended frequency range is from 20KHz to 30MHz.

In the circuit diagram resistors R1 and R2 give a potential divider bias for the transistor Q1.

Ce is the emitter by pass capacitor, which by-passes the amplified AC signals. If the emitter by-pass capacitor not there, the amplified ac voltages will drop across Re and it will get added on to the base-emitter voltage of Q1 and will disrupt the biasing conditions.

Cin is the input DC decoupling capacitor while Cout is the output DC decoupling capacitor. The task of a DC decoupling capacitor is to prevent DC voltages from reaching the succeeding stage. Inductor L1, L2 and capacitor C1 forms the tank circuit.

When the power supply is switched ON the transistor starts conducting and the collector current increases. As a result the capcitor C1 starts charging and when the capacitor C1 is fully charged it starts discharging through coil L1. This charging and discharging creates a series of damped oscillations in the tank circuit and it is the key.

The oscillations produced in the tank circuit is coupled (fed back) to the base of Q1 and it appears in the amplified form across the collector and emitter of the transistor. The output voltage of the transistor (voltage across collector and emitter) will be in phase with the voltage across inductor L1. Since the junction of two inductors is grounded, the voltage across L2 will be 180° out of phase to that of the voltage across L1.

The voltage across L2 is actually fed back to the base of Q1. the feed back voltage is 180° out of phase with the transistor and also the transistor itself will create another 180° phase difference. So the total phase difference between input and output is 360° and it is very important condition for creating sustained oscillations.

Frequency of the Hartley oscillator.

The frequency “F” of a Hartley oscillator can be expressed using the equation;

C is the capacitance of the capacitor C1 in the tank circuit.L = L1+L2, the effective series inductance of the inductors L1 and L2 in the tank circuit.

Here the coils L1 and L2 are assumed to be winded on different cores. If they are winded on a single core then L=L1+L2+2M where M is the mutual inductance between the two coils.

Colpitts Oscillator

Colpitts oscillator was invented by American scientist Edwin Colpitts in 1918. It is another type of sinusoidal LC oscillator which has a lot of applications. The Colpitts oscillator can be realized using transistors, FETs or op-amp.

In Colpitts oscillator the tank circuit consists of two capacitors in series and an inductor connected in parallel to the serial combination. The frequency of the oscillations are determined by the value of the capacitors and inductor in the tank circuit.

Collpitts oscillator is generally used in RF applications and the typical operating range is 20KHz to 300MHz.

In Colpitts oscillator, the capacitive voltage divider setup in the tank circuit works as the feed back source and this arrangement gives better frequency stability.

In the circuit diagram resistors R1 and R2 gives a voltage divider biasing to the transistor. Resistor R4 limits the collector current of the transistor.

Cin is the input DC decoupling capacitor while Cout is the output decoupling capacitor. Ce is the emitter by-pass capacitor. Job of the emitter by-pass capacitor is to by-pass the amplified AC signals from dropping across Re.

If the emitter by-pass capacitor is not there, the amplified AC signal would have dropped across Re and it may have altered the DC biasing conditions of the transistor and the result will be reduced gain.

Capacitors C1, C2 and inductor L1 forms the tank circuit. Feedback to the base of transistor is taken from the junction of Capacitor C2 and inductor L1 in the tank circuit.

When power supply is switched ON, capacitors C1 and C2 starts charging. When they are fully charged they starts discharging through the inductor L1. When the capacitors are fully discharged, the electrostatic energy stored in the capacitors gets transferred to the inductor as magnetic flux.

The inductor starts discharging and capacitors gets charged again. This transfer of energy back and forth between capacitors and inductor is the basis of oscillation.

Voltage across C2 is phase opposite to that of the voltage across the C1 and it is the voltage across C2 that is fed back to the transistor.

The feedback signal at the base of transistor appears in the amplified form across the collector and emitter of the transistor.

The energy lost in the tank circuit is compensated by the transistor and the oscillations are sustained.

The tank circuit produces 180° phase shift and the transistor itself produces another 180° phase shift.That means the input and output are in phase and it is a necessary condition of positive feedback for maintaining sustained oscillations.

The frequency of oscillations of the Colpitts oscillator can be determined using the equation.

Where L is the inductance of the inductor in the tank circuit and C is the effective capacitance of the capacitors in the tank circuit.

If C1 and C2 are the individual capacitance, then the effective capacitance of the serial combination C= (C1C2)/(C1+C2). By using ganged variable capacitors in place of C1 and C2, the Colpitts oscillator can be made variable.

Advantages of Colpitts oscillator.

Main advantage of Colpitts oscillator over Hartley oscillator is the improved performance in the high frequency region. This is because the capacitors provide a low reactance path for the high frequency signals and thus the output signals in the high frequency domain will be more sinusoidal. Due to the excellent performance in the high frequency region, the Colpitts oscillator can be even used in microwave applications.

Crystal Oscillator

Crystal Oscillator Most communications and digital applications require the

use of oscillators with extremely stable outputextremely stable output. Crystal oscillators are invented to overcome the output fluctuationoutput fluctuation experienced by conventional oscillators.

Crystals used in electronic applications consist of a quartz wafer held between two metal plates and housed in a package as shown in Fig. (a) and (b).

In crystal oscillators, the usual electrical resonant circuit is replaced by a mechanically vibrating crystal. The crystal (usually quartz) has a high degree of stability in holding constant at whatever frequency the crystal is originally cut to operate.

The crystal oscillators are, therefore, used whenever great stability is needed, for example, in communication transmitters, and receivers, digital clocks etc.

A quartz crystal exhibits a very important property known as piezo-electric effect.

Crystal Oscillator

Piezoelectric Effect The quartz crystal is made of silicon oxide (SiO2) and

exhibits a property called the piezoelectricpiezoelectric

When a alternating voltage is applied across the crystal, it vibrates at the frequency of the applied voltage.

The thinner the crystal, higher its frequency of vibration. This phenomenon is called piezoelectric effect.

Crystal Oscillator

Characteristic of Quartz Crystal

The crystal can have two resonant frequencies;

One is the series resonance frequency f1 which occurs when XL = XC. At this frequency, crystal offers a very low impedance to the external circuit where Z = R.

The other is the parallel resonance (or antiresonance) frequency f2 which occurs when reactance of the series leg equals the reactance of CM. At this frequency, crystal offers a very high impedance to the external circuit

R

L

C

CM

Since, in series resonance, the crystal impedance is smallest, it can cause the crystal to provide the largest positive feedback.

CM

R

L

C

CM

Crystal Pierce Oscillator To excite a crystal for operation in the

seriesresonant mode it may be connected as a series element in a feedback path, as shown in figure.

In this mode of operation the crystal impedance is the smallest and the amount of positive feedback is the largest.

Resistor R1, R2 and RE provide a voltagedivider stabilized dc bias circuit, the capacitor CE provides ac bypass of the emitter resistor Re and the radiofrequency coil (RFC) provides for dc bias while decoupling any ac signal on the power lines from affecting the output signal.

The coupling capacitor Cc has negligible impedance at the circuit operating frequency but blocks any dc between collector and base. The resulting circuit frequency of oscillations is set by the series resonant frequency of the crystal. Variations in supply voltage, transistor parameters, etc. have no effect on the circuit operating frequency which is held stabilized by the crystal. The circuit frequency stability is set by the crystal frequency stability, which is good.

Colpitts Quartz Crystal Oscillator The design of a Crystal Oscillator is very

similar to the design of the Colpitts Oscillator except that the LC tank circuit that provides the feedback oscillations has been replaced by a quartz crystal.

These types of Crystal Oscillators are designed around the common emitter amplifier stage of a Colpitts Oscillator. The input signal to the base of the transistor is inverted at the transistors output. The output signal at the collector is then taken through a 180o phase shifting network which includes the crystal operating in a series resonant mode.

The output is also fed back to the input which is “inphase” with the input providing the necessary positive feedback.

Resistors, R1 and R2 bias the resistor in a Class A type operation while resistor Re is chosen so that the loop gain is slightly greater than unity.

The circuit diagram of the Colpitts Crystal Oscillator circuit shows that capacitors, C1 and C2 shunt the output of the transistor which reduces the feedback signal.

The output amplitude should be kept low in order to avoid excessive power dissipation in the crystal otherwise could destroy itself by excessive vibration.

Engineering

Sinusoidal oscillators