Università degli Studi di Roma Tor Vergata Dipartimento di Ingegneria Elettronica

Analogue Electronics

Paolo Colantonio A.A. 2015-16 Introduction

• A generic system can be represented as following:

• The output signal is related to the input signal by:

• A is the open‐ • G is the overall or closed‐loop gain • AB is the loop gain

P. Colantonio – Analogue Electronics A.A. 2015/16 2|21 Introduction

• Consider the expression of the Gain

• If AB=‐1, then the gain becomes infinite  this represents the condition for oscillation

• The requirements for oscillation are described by the Barkhausen criterion: 1. The magnitude of the loop gain AB must be 1 2. The phase shift of the loop gain AB must be 180°, or 180° plus an integer multiple of 360°

P. Colantonio – Analogue Electronics A.A. 2015/16 3|21 Stability

• The gain of a real has not only a magnitude, but also a phase angle (both variable in frequency) • a phase shift of 180° represents an inversion and so the gain changes polarity • this can turn into positive feedback

• The gain of all real falls at high frequencies and this also produces a phase shift • All multi‐stage amplifier will produce 180 of phase shift at some frequency • To ensure stability we must ensure that the Barkhausen conditions for oscillation are not met  to guarantee this we must ensure that the gain falls below unity before the phase shift reaches 180

P. Colantonio – Analogue Electronics A.A. 2015/16 4|21 Gain and phase margins

• In order to check the stability of a feedback circuit, it is required to analyze the frequency behavior of the Loop Gain AB

• Gain margin is the amount (in dB) by which the loop gain (AB) is less than 0dB when the phase reaches 180°

• Phase margin is the angle by which the phase is less than 180° when the loop gain falls to unity (0dB)

P. Colantonio – Analogue Electronics A.A. 2015/16 5|21 Unintended feedback

• Stability can be affected by unintended feedback within a circuit • This might be caused by stray capacitance or stray inductance • If these produce positive feedback they can cause instability • a severe problem in high‐frequency applications • must be tackled by careful design

P. Colantonio – Analogue Electronics A.A. 2015/16 6|21 Oscillators

• The positive feedback can be intentionally used to realize oscillators. • There are two types of oscillators, depending on the output signal waveform generate: • If the output signal is a sinusoidal waveform, the oscillator is called sinusoidal oscillator • If the output signal is pulsed, the oscillator is called relaxing oscillator

XYA 

• If for a determined frequency we fulfill the condition AB=1 • Then X=Y and we can utilize the circuit response as an excitation (on behalf of the external ones), by closing the loop

Barkhausen conditions

P. Colantonio – Analogue Electronics A.A. 2015/16 7|21 Oscillators

XYA 

• In order to realize an oscillator at frequency f0, the following two conditions must be fulfilled 1. The phase shift along the loop must be zero 2. The loop gain must be (theoretically*) equal to 1

• The phase shift can be partitioned between the amplifier (A) and the feedback network (B) in different ways. • Moreover, to satisfy the condition Im[AB]=0, in the loop must be present at least 2 reactive elements (eventually including parasitic elements of the amplifier)

* in actual oscillator the condition realized is |AB|>1 to avoid that circuit parametric variation could extinguish the oscillations. The nonlinearities of the active elements will automatically reduce the oscillating amplitude in order to fulfill the Barkhausen conditions

P. Colantonio – Analogue Electronics A.A. 2015/16 8|21 RC or phase‐shift oscillator

• Consider an inverting amplifier, thus introducing a phase shift of 180° • In order to realize an oscillator, we need to add a network (B) to realize a further phase shift of 180° • one simple way of producing a phase shift of 180° is to use an RC ladder network

P. Colantonio – Analogue Electronics A.A. 2015/16 9|21 RC or phase‐shift oscillator

Z B Z A Z - 1 1 1 +++

v1 Avv1 Z2 Z2 Z2 v0v1 ---

• Since each RC network will introduce a phase shift less than 90°, at least 3 stages are required

P. Colantonio – Analogue Electronics A.A. 2015/16 10|21 RC or phase‐shift oscillator

B A Z Z Z - 1 1 1 +++

v1 Avv1 Z2 Z2 Z2 v0v1 ---

• Replacing the expressions for ZA and ZB, it follows:

• Since Z1 and Z2 are reactive elements (only one, not both!), it follows that the imaginary part is given by the odd terms

P. Colantonio – Analogue Electronics A.A. 2015/16 11|21 RC or phase‐shift oscillator

Z B Z A Z - 1 1 1 +++

v1 Avv1 Z2 Z2 Z2 v0v1 ---

• In both cases

P. Colantonio – Analogue Electronics A.A. 2015/16 12|21 RC or phase‐shift oscillator

• The complete oscillator could be

P. Colantonio – Analogue Electronics A.A. 2015/16 13|21 Wien‐bridge oscillator

• Consider the following network (Wien‐bridge)

• If R1=R2=R and C1=C2=C, it follows

• The network produces a phase‐shift of 0° at a single frequency

• The gain is

P. Colantonio – Analogue Electronics A.A. 2015/16 14|21 Wien‐bridge oscillator

• The Wien‐bridge network can be combined with a non inverting amplifier

P. Colantonio – Analogue Electronics A.A. 2015/16 15|21 3‐points oscillator

• Oscillators containing both L and C can be considered 3‐points oscillators

• If all the impedances are purely reactive, then it will be possible to fulfill Im[AB]=0 but it will not satisfied the condition Re[AB]=1

• Thus it is mandatory that at least one impedance is Zi=Ri+jXi

P. Colantonio – Analogue Electronics A.A. 2015/16 16|21 3‐points oscillator: case A

+

j X3

V

R1 j X1 gmV j X2

-

Barkhausen conditions

P. Colantonio – Analogue Electronics A.A. 2015/16 17|21 3‐points oscillator: case B

+

j X3

V j X1

R2 j X2 gmV

-

Barkhausen conditions

P. Colantonio – Analogue Electronics A.A. 2015/16 18|21 Colpitts oscillator

L L

+ gm Vgs

V C C2 C1 g rd 2 s C1 -

• The Barkhausen condition (and thus the oscillating conditions) are fulfilled if

P. Colantonio – Analogue Electronics A.A. 2015/16 19|21 Hartley oscillator

C C

vb’e rbb’ rb’c

L 2 gmvb’ e L L1 1 r L2 rb’e ce

• Neglecting the resistances roe, rb’c and rbb’

P. Colantonio – Analogue Electronics A.A. 2015/16 20|21 Crystal oscillators

• The frequency stability is determined by the ability of the circuit (feedback) to select a particular frequency • In tuned circuits this is described by the quality factor, Q • Piezoelectric crystals act like resonant circuits with a very high Q –as high as 100,000

P. Colantonio – Analogue Electronics A.A. 2015/16 21|21