Analog Circuits and Systems Prof. K Radhakrishna Rao

Lecture 32: LC Oscillator – Effect of Non-idealities

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Review

  Harmonic Oscillator   Second order differential equation   Tank Circuit   Negative resistance LC oscillator   Self starting and amplitude stabilization

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Review (contd.,)

  AGC/AVC - used in almost all communication receivers at the front end (RF)

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Coil in a Tank Circuit

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{ }( )

s2 2 2

s s

s s2 2 2 2 2 2s s

2 2s c c

c 2c s

R j L1R j L R L

R R j LR L R L

1 1 1 1R j L1 Q 1 1 Q

1 1High Qj LQ R

− ω=

+ ω − ω− ω

= −+ ω + ω

⎡ ⎤⎢ ⎥= +⎢ ⎥ω+ +⎣ ⎦

+ω

;

2p c s

cs

s

R Q R

LQ of the coil QR

R j L

=

ω=

+ ω

Non-ideal negative resistance (n-type)

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Non-ideal negative resistance (s-type)

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Non-ideal negative resistance (contd.,)

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{ }( ) ′+=

⎡ ⎤⎢ ⎥′⎢ ⎥

⎛ ⎞⎢ − −⎜ ⎟⎢ ⎝ ⎠⎣ ⎦

p

p

Consider a non-ideal op.amp being used for obtaining the gain stage (of 2). The op.amp finite GB makes

2gain of 2 because and R simulates at the 1 2s GB

1input admittance

R2s1 2 1GB

−= = +′ ′⎡ ⎤

′⎢ ⎥⎢ ⎥

⎥ ⎢ ⎥− +⎣ ⎦⎥

p pp

1 1 4sR GBR

R4s1GB

Non-ideal negative resistance (contd.,)

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′′

ω =⎛ ⎞

+⎜ ⎟⎜ ⎟′⎝ ⎠

= =ω⎡ ⎤ ⎡ ⎤ω ++⎢ ⎥ ⎢ ⎥′ ⎣ ⎦ω⎢ ⎥⎣ ⎦

′ω = = ω

pp

n

p

nn

n p

n n p

4Negative resistance R shunted by a capacitance GBR

1which changes the frequency to 4L C

GBR

1 122 LC 1LC 1 GBQGBR C

1where ; Q R CLC

Non-ideal negative resistance (contd.,)

  Higher the Q of the tank circuit better is the frequency of stability, means it is less sensitive to active device parameter .

  Crystals with very high Q (5000-50000) therefore are ideal elements for oscillators with stable frequency.

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φ −=ω ωω − φ∴ =ω

VVV V0 0

0

0

2Q

2Q

Negative resistance simulator

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3 1ii

i 2

3

1i

2

Z ZV ZI Z

Z Z

ZZ ZZ

= = −

= −

=

Negative Impedance Inverter (NII).

If

simulated using one more NII, then

is a Positive Impedance Inverter (PII).

Positive impedance inverter

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1 3 5i

2 4

2 4

2i

Z Z ZZZ Z

Z Z

Z sCR

=

=

or as capacitors

and rest resistors

can be simulated which is called a gyrator.

Nullator-Norator concept

  Application in synthesis

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Nullator-Norator concept (contd.,)

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  Application in synthesis - Other topologies

Nullator-Norator concept (contd.,)

  Application in synthesis - Additional topologies

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Two gyrators in one topology

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Gyrator

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It can simulate a grounded inductor using capacitor.

RC Oscillator using Gyrator

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p

n 2

p p

ZR

1 1RCRC C

R R

′

ω = =⋅

′=

For inductor simulator and gain of

for negative resistance

Another view point of oscillator

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o

2i

0 0

1 2 sLV 2 R R

2 1 2sLV sC s LC 1R sL R

1 .LC

×= =

+ + + +′ ′

ω = ω

If now a non-inverting amplifier of gain 2 is used,

loop to be formed thus has a gain of 1 at resonance,

loop if closed can sustain oscillations at

Another view point of oscillator (contd.,)

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( )2

1Q2

sCR1 3sCR sCR

⎛ ⎞

Another view point of oscillator (contd.,)

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  This is used with a non-inverting amplifier of gain 3, also acts as in an oscillator if output is connected to its input forming a loop.

Wien Bridge Oscillator

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01RC

ω =

Phase Shift Oscillators

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Quadrature oscillator/double integrator oscillator

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( )2 2

2o oo o 0 02 2 2

d V d V 1 1KV 0 or KV ; KRCdt dt RC

+ = = − ω = = ω =

If a second order differential equation is simulated it results

in the double integrator loop or harmonic oscillator.

that is

Amplitude stabilized quadrature oscillator

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p refV 10V=

Simulation 1

  Vref=0.4; Vp=2

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Simulation 2

  Vref=0.1; Vp=1

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Ring oscillator using opamp inverters

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( )( ) ( )

( ) ( )

( )

− −= = = ω − ω =+ − ω

ω = ω =

333 3

i 3 22 20

22 20

2 8g 1 ; 3 C 2R C 2R 0 1 2sCR 1 3 C 2R

3and C 2R 3;2RC

Conclusion

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