Objectives

 Describe the basic concept of an oscillator  Discuss the basic principles of operation of an oscillator  Analyze the operation of RC and LC oscillators  Describe the operation of the basic relaxation oscillator circuits 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 Oscillators

Oscillation: an effect that repeatedly and regularly fluctuates about the mean value

Oscillator: circuit that produces oscillation

Characteristics: wave-shape, frequency, amplitude, distortion, stability

4 Application of Oscillators

Oscillators are used to generate signals, e.g.  Used as a local oscillator to transform the RF signals to IF signals in a receiver;  Used to generate RF carrier in a transmitter  Used to generate clocks in digital systems;  Used as sweep circuits in TV sets and CRO.

5 Oscillators

 Oscillators are circuits that generate periodic signals  An oscillator converts DC power from the power supply into AC signal power spontaneously - without the need for an AC input source

Figure 9.67 Repetitive ramp waveform. 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 2. LC oscillators  Hartley  Colpitts  Crystal 3. Unijunction / relaxation oscillators Linear Oscillators

Figure 9.68 A linear oscillator is formed by connecting an amplifier and a feedback network in a loop. Integrant of Linear Oscillators

+ V Vs  Amplifier (A) Vo + Positive Vf Frequency-Selective Feedback Network () Feedback

For sinusoidal input is connected “Linear” because the output is approximately sinusoidal

A linear oscillator contains: - a frequency selection feedback network - an amplifier to maintain the loop gain at unity

Ref:06103104HKN 10 EE3110 Oscillator Basic Linear Oscillator

+ V Vs  A(f) Vo +

Vf SelectiveNetwork (f)

and Vo  AV  A(Vs V f ) V f  Vo V A  o  Vs 1 A

If Vs = 0, the only way that Vo can be nonzero is that loop gain A=1 which implies that

| A |1 (Barkhausen Criterion) A  0

Ref:06103104HKN 11 EE3110 Oscillator Basic principles for oscillation

 An oscillator is an amplifier with positive feedback.

V e V o V s A + V f  Ve Vs V f (1)

V f  βVo (2)

Vo  AVe  AVs V f  AVs  βVo  (3) Basic principles for oscillation

Vo  AVe

 AVs V f  AVs  βVo 

Vo  AVs  AVo

1 A Vo  AVs  The closed loop gain is:

Vo A Af   Vs 1 Aβ Basic principles for oscillation

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

Vo As Af s  s  Vs 1  Asβs Asβs is known as loop gain Basic principles for oscillation

 Writing Ts Asβs the loop gain becomes; As A s  f 1Ts

 Replacing s with j A jω A  jω  f 1 T jω  and Tjω Ajωβjω Basic principles for oscillation

 At a specific frequency f0

T jω0   A jω0 β jω0   1

 At this frequency, the closed loop gain;

A jω0  Af  jω0   1  A jω0 β jω0  will be infinite, i.e. the circuit will have finite output for zero input signal - oscillation Basic principles for oscillation

 Thus, the condition for sinusoidal oscillation of

frequency f0 is;

A jω0 β jω0   1

 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. Barkhausen Criterion – another way

Figure 9.69 Linear oscillator with external signal Xin injected. Barkhausen Criterion How does the oscillation get started?

Noise signals and the transients associated with the circuit turning on provide the initial source signal that initiate the oscillation Practical Design Considerations

Usually, oscillators are designed so that the loop gain magnitude is slightly higher than unity at the desired frequency of oscillation This is done because if we designed for unity loop gain magnitude a slight reduction in gain would result in oscillations that die to zero The drawback is that the oscillation will be slightly distorted (the higher gain results in oscillation that grows up to the point that will be clipped) Basic principles for oscillation

 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. Basic principles for oscillation

In phase

V f V o A v

Noninverting amplifier

Feedback circuit Design Criteria for Oscillators

1. The magnitude of the loop gain must be unity or slightly larger Aβ  1 – Barkhaussen criterion

2. Total phase shift, of the loop gain mus t be Nx360° where N=0, 1, 2, … 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. Another way

Figure 9.70 Typical linear oscillator. Wien-bridge Oscillator

 The loop gain for the oscillator is;  R  Z  T s  A s β s  1 2  p            R1  Z p  Z s   where; R Z  p 1 sRC

 and; 1 sRC Z  s sC Wien-bridge Oscillator

 Hence;

 R2  1  Ts  1    R1 3  sRC  1/sRC   Substituting for s;

 R2  1  T j  1    R1 3 jRC  1/jRC 

 For oscillation frequency f0  [0 ]  R  1   2  T j0   1    R1 3 j0RC  1/j0RC  Wien-bridge Oscillator

 Since at the frequency of oscillation, T(j) must be real (for zero phase condition), the imaginary component must be zero; 1 j0 RC   0 j0 RC

 which gives us – [how?! – do it now] 1   0 RC 1 j0 RC   0 j0 RC 1  j0 RC   j0 RC 2  ( j0 RC )  1 2 2  j 0 RC   1 2  1.0 RC   1 2  0 RC  1

 0 RC 1 1    0 RC Wien-bridge Oscillator

 From the previous eq. (for oscillation frequency f0),  R  1   2  T j0   1    R1 3 j0RC  1/j0RC   the magnitude condition is;

 R2  1   R2  1 R2 1 1    1     31  2  R1  3 0   R1  3 R1

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

 With the ratio; R 2  2 R1

 then; R K 1 2  3 R1 K = 3 ensures the loop gain of unity – oscillation  K > 3 : growing oscillations  K < 3 : decreasing oscillations Wien-Bridge Oscillator – another way R 2  2 R1

 R2  2R1

non inverting, R gain  A 1 2 1 2  3 R1

Wien-bridge oscillator. Wien-Bridge oscillator output

Figure 9.75 Example of output voltage of the oscillator. Wien Bridge Oscillator 1 1 Frequency Selection Network Let and X C1  X C 2  C1 C2 Z1

Z1  R1  jXC1 R1 C1 Z2 1  1 1   jR2 X C 2 Z2      Vi C2 R2 Vo  R2  jXC 2  R2  jXC 2

Therefore, the feedback factor,

V Z ( jR X / R  jX )   o  2  2 C 2 2 C 2 Vi Z1  Z2 (R1  jXC1)  ( jR2 X C 2 / R2  jXC 2 )  jR X   2 C 2 (R1  jXC1)(R2  jXC 2 )  jR2 X C 2

Ref:06103104HKN 36 EE3110 Oscillator  can be rewritten as: R X   2 C 2 R1 X C 2  R2 X C1  R2 X C 2  j(R1R2  X C1 X C 2 ) For Barkhausen Criterion, imaginary part = 0, i.e.,

0.34

R1R2  X C1X C2  0  0.32 1 1 0.3 0.28 =1/3 or R1R2  0.26 C1 C2 0.24

  1/ R1R2C1C2 factor Feedback 0.22 0.2 Supposing, f(R=Xc) 1 R1=R2=R and XC1= XC2=XC, 0.5 Phase=0

0

RX C Phase   2 2 3RX C  j(R  X C ) -0.5

-1 Frequency

37 1 Example By setting   , we get RC Imaginary part = 0 and 1 Rf   3 R1 Due to Barkhausen Criterion, 

Loop gain Av=1 where +

Av : Gain of the amplifier C R Vo

R Z1 R f C Av 1 Av  3 1 Z2 R1 R Therefore, f  2 Wien Bridge Oscillator R1

38 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.  The gain must be at least 29 to maintain the oscillations.  The frequency of resonance for the this type is similar to any RC circuit oscillator: 1 f  r 2 6RC Phase-Shift Oscillator

R2 C R C v2 C v1 v2 v3 vi v1 vo

R R v R A(s)  o  2 v R 3 3  sRC   sRC  v  v v1   vi 3   i 1 sRC  1 sRC  2 3  sRC  v3  sRC  v2    vi  (s)    1 sRC  vi 1 sRC  Phase-Shift Oscillator

 Loop gain, T(s): 3  R  sRC  T (s)  A(s) (s)   2    R 1 sRC   Set s=jw 3  R2  jRC  T ( j)      R 1 jRC   R  ( jRC )(RC )2 T ( j)   2   R  1 3 2 R2C 2  jRC 3 2 R2C 2  Phase-Shift Oscillator

 To satisfy condition T(jwo)=1, real component must be zero since the numerator is purely imaginary. 13 2R2C 2  0  the oscillation frequency: 1   0 3RC  Apply wo in equation:

 R2  ( j / 3)(1/ 3)  R2  1  T( jo )         R  0  ( j / 3)3 (1/ 3)  R  8  To satisfy condition T(jwo)=1 R The gain greater than 8, the circuit will 2  8 spontaneously begin oscillating & sustain R oscillations Phase-Shift Oscillator

1 R2 The gain must be at least fo   29 29 to maintain the 2 6RC R 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 Colpitts, Hartley and crystal oscillator. Colpitts Oscillator

 The Colpitts oscillator is a type of oscillator that uses an LC circuit in the feed-back loop.  The feedback network is made up of a pair of tapped

capacitors (C1 and C2) and an inductor L to produce a feedback necessary for oscillations.  The output voltage is

developed across C1.  The feedback voltage is

developed across C2. Colpitts Oscillator

 KCL at the output node: V V V o  o  g V  o  0 - Eq (1) 1 m gs 1 R sL  sC1 sC2  voltage divider produces:  1    sC V   2 V gs  1  o   sL  - Eq (2)  sC2   substitute eq(2) into eq(1):

 2  1  Vo gm  sC2  1 s LC2   sC1   0   R  Colpitts Oscillator

 Assume that oscillation has started, then Vo≠0 2 3 s LC2  1  s LC1C2   sC1  C2   gm    0 R  R   Let s=jω  1 2LC   2  2  gm     jC1  C2  LC1C2  0  R R   both real & imaginary component must be zero  Imaginary component: 1 o  - Eq (3)  C1C2  L   C1  C2  Colpitts Oscillator

 both real & imaginary component must be zero  Imaginary component:  2LC 1 2  g  - Eq (4) R m R  Combining Eq(3) and Eq(4):

C2  gm R C1  to initiate oscillations spontaneously:  C   2  gm R     C1  Hartley Oscillator

 The Hartley oscillator is almost identical to the Colpitts oscillator.  The primary difference is that the feedback network of the Hartley oscillator uses tapped

inductors (L1 and L2) and a single capacitor C. Hartley Oscillator

 the analysis of Hartley oscillator is identical to that Colpitts oscillator.  the frequency of oscillation: 1 o  L1  L2 C Crystal Oscillator

 Most communications and digital applications require the use of oscillators with extremely stable output. Crystal oscillators are invented to overcome the output fluctuation experienced by conventional oscillators.  Crystals used in electronic applications consist of a quartz wafer held between two metal plates and housed in a a package as shown in Fig. 9 (a) and (b). Crystal Oscillator

 Piezoelectric Effect

 The quartz crystal is made of silicon oxide (SiO2) and exhibits a property called the piezoelectric

 When a changing an alternating voltage is applied across the crystal, it vibrates at the frequency of the applied voltage. In the other word, the frequency of the applied ac voltage is equal to the natural resonant frequency of the crystal.

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

 Characteristic of Quartz Crystal R  The crystal can have two resonant frequencies; L CM  One is the series resonance frequency f1 which occurs when XL = XC. At this C 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 Crystal Oscillator

 The crystal is connected as a series element in the feedback path from collector to the base so that it is excited in the series-resonance mode

BJT FET Crystal Oscillator

 Since, in series resonance, crystal impedance is the smallest that causes the crystal provides the largest positive feedback.

 Resistors R1, R2, and RE provide a voltage-divider stabilized dc bias circuit. Capacitor CE provides ac bypass of the emitter resistor, RE to avoid degeneration.  The RFC coil provides dc collector load and also prevents any ac signal from entering the dc supply.

 The coupling capacitor CC has negligible reactance at circuit operating frequency but blocks any dc flow between collector and base.  The oscillation frequency equals the series-resonance frequency of the crystal and is given by: 1 fo  2 LCC Unijunction Oscillator

 The unijunction transistor can be used in what is called a relaxation oscillator as shown by basic circuit as follow.  The unijunction oscillator provides a pulse signal suitable for digital-circuit applications. UJT

 Resistor RT and capacitor CT are the timing components that set the circuit oscillating rate Unijunction Oscillator

 Sawtooth wave appears at the emitter of the transistor. This wave shows the gradual increase of capacitor voltage Unijunction Oscillator

 The oscillating frequency is calculated as follows: 1 fo  RT CT ln 1/1

 where, η = the unijunction transistor intrinsic stand- off ratio  Typically, a unijunction transistor has a stand-off ratio from 0.4 to 0.6