Analog Circuits and Systems · 2017. 8. 4. · Gyrator 16 It can simulate a grounded inductor using...
Transcript of Analog Circuits and Systems · 2017. 8. 4. · Gyrator 16 It can simulate a grounded inductor using...
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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
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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
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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
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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
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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).
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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.
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Nullator-Norator concept
Application in synthesis
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Nullator-Norator concept (contd.,)
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Application in synthesis - Other topologies
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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.
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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
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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
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Another view point of oscillator (contd.,)
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( )2
1Q2
sCR1 3sCR sCR
⎛ ⎞
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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.
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Wien Bridge Oscillator
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01RC
ω =
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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
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Amplitude stabilized quadrature oscillator
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p refV 10V=
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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
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Conclusion
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