prof. paolo colantonio a.a. 2011 12...• the requirements for oscillation are described by the...

Prof. Paolo Colantonioa.a. 2011‐12

Analogue ElectronicsProf. Paolo Colantonio 2 | 25

• A generic feedback system can be represented as following:

1o

i

V AGV AB

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

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


• Consider the expression of the Gain1

AGAB

• 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 12. The phase shift of the loop gain AB must be 180°, or 180° plus an integer

multiple of 360°


• The gain of a real amplifier 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 negative feedback into positive feedback

• The gain of all real amplifiers 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


• 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)


• An alternative method of investigating the stability of a circuit is the use of a Nyquistdiagram, which illustrates the relationship between gain and phase in a single plot.

m Re[AB]

Im[AB]

-1 Gm.

log(f)

log(f)

0°

-90°

-180°

(AB)

|AB|dB

Gain margin Gm

Phase margin m


• Amplifier with a single low‐frequency cut‐off and a single high‐frequency cut‐off

• The distance of P from the origin (the magnitude of OP) represents the magnitude of AB and the angle φ represents its phase.

Mid‐band frequencies• the phase of the output is zero and the gain

is simply real.

Low frequencies• the lower cut‐off frequency causes the

magnitude of the gain to fall and produces a positive phase angle.

• For f0 the gain tends to zero and the phase tends to +90°.

High frequencies• the upper cut‐off frequency reduces the

magnitude of the gain and produces a phase lag.

• For f the gain falls towards zero and the phase angle tends to −90°.

f=fLow

f=fHigh


• if the locus of P does not enter the unit circle centred on (–1, 0), the circuit is stable and has negative feedback

• if the locus of P enters the unit circle, the feedback is positive within that region• if the locus of P encircles the point (–1, 0), the amplifier will oscillate


• 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


• 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

AX YY AB X

• 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

Im 0

Re 1

AB

AB

1

0

AB

phase AB


AX Y

• In order to realize an oscillator at frequency f0, the following two conditions must be fulfilled1. The phase shift along the loop must be zero2. The loop gain must be (theoretically*) equal to 1

* 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

• 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 2reactive elements (eventually including parasitic elements of the amplifier)


• 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


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

20

1 2

2

1 2 1

2

1 2 1 1

A

AB

A

A BV I I

A B

ZV V

Z ZZ Z

VZ Z Z Z

Z Z ZA V V

Z Z Z Z Z Z

Z1

v1 v0v1Avv1 Z2 Z2 Z2

Z1 Z1-

- --

+ ++

AB

2 1 2

2 1

/ /

/ /A

B A

Z Z Z Z

Z Z Z Z


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

3 2

1 1 1

2 2 2

1 1

5 6 1VA

Z Z ZZ Z Z

Z1


Z1 Z1-

- --

+ ++

AB

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

3

1 1

2 2

6 0Z ZZ Z

2

1

2

6ZZ


1

2

1Zj C

Z R

Z1


Z1 Z1-

- --

+ ++

AB

• In both cases

01

6 RC

1

21

Z R

Zj C

06

RC

2

1

2

1 1

5 1VA

ZZ

1 1

5 6 1VA

29VA


• The complete oscillator could be

01

6 RC


• Consider the following network (Wien‐bridge)

2 20

1 1 2 2

/ // /I

R CV V

R C R C

2

2 20

21

1 2 2

11

1

I

Rj R C

V VRR

j C j R C

01

3IV V

j j RCRC

• If R1=R2=R and C1=C2=C, it follows

• The network produces a phase‐shift of 0° at a single frequency 01

RC

• The gain is 0 13I

VV


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

10

1

0

1 3

13

RV V V

R

V V


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

Z2Z1

Z3

Gm-Z2Z1

Z3

gm vv+

-

21

1 2 3m

Zv g v Z

Z Z Z

1 2 1 2 3mg Z Z Z Z Z

• 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


R1 j X1

j X3

j X2

V

-

+

gmV

1 1 1

2 3

//Z R jXZ Z jX

1 1 1 12 2 3

1 1 1 1m

jX R X Rg jX j jX jXR jX R jX

1 2 1 1 2 3 1 1 2 3mg X X R j X X X R X X X


1 2 3

1 2 1 1 2 3

0

mg X X RX X

X XX

X

1

12

mXg RX


1 3

2 2 2/ /Z Z jXZ R jX


1 2 3

1 2 2 2 1 3

0

mg X X RX X

X XX

X

2

21

mXg RX

+

V

-

j X1

j X3

j X2R2gmV

2 2 2 21 1 3

2 2 2 2m

jX R X Rg jX jX j jXR jX R jX

1 2 2 1 2 3 2 2 1 3mg X X R j X X X R X X X


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

L

C1

C2 C1C2

L

rdVgs

+

-

gm Vgs

00 1 0 2

1 1 0j Lj C j C

01LC

1 2

1 2

C CCC C

1

2m d

Cg rC


C

L1

L2

L1 L2

C

rbb’

rb’e

rb’c

rce

vb’e

gmvb’e

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

0 1 0 20

1 0j L j Lj C

01LC

1 2L L L

1'

2m b e

Lg rL


• 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

C

R

L

C’

X Reactance

0

(inductive)

(capacitive)

ω ωp ωs

Useful bandwidth

(a) (b) (c)

prof. paolo colantonio a.a. 2011 12...• the requirements for oscillation are described by the...

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