Formula Sheets: Oscillators | Analog and Digital Electronics - Electrical Engineering (EE) PDF Download

 Page 1


Formula Sheet for Oscillators (Analog and Digital
Electronics) – GATE
1. Basic Concepts
• Oscillator: Circuit that generates a periodic signal without an external input.
• Types: Sinusoidal (harmonic) and non-sinusoidal (relaxation).
• Components: Active device (BJT, MOSFET, Op-Amp), feedback network (LC,
RC, crystal).
• Barkhausen Criteria:
– Loop gain: |Aß|= 1.
– Phase shift: Total loop phase = 0
?
or multiple of 360
?
.
2. LC Oscillators
2.1 Hartley Oscillator
• Frequency of Oscillation:
f =
1
2p
v
L
eq
C
where L
eq
=L
1
+L
2
+2M, M: Mutual inductance.
• Loop Gain Condition:
Aß =
g
m
R
1+
L
2
L
1
= 1
where g
m
: Transconductance, R: Tank circuit resistance.
2.2 Colpitts Oscillator
• Frequency of Oscillation:
f =
1
2p
v
LC
eq
where C
eq
=
C
1
C
2
C
1
+C
2
.
• Loop Gain Condition:
Aß =g
m
R·
C
2
C
1
= 1
2.3 Clapp Oscillator
• Frequency of Oscillation:
f =
1
2p
v
LC
eq
, C
eq
=
C
1
C
2
C
3
C
1
C
2
+C
2
C
3
+C
3
C
1
• Advantage: Improved frequency stability due to additional capacitor C
3
.
1
Page 2


Formula Sheet for Oscillators (Analog and Digital
Electronics) – GATE
1. Basic Concepts
• Oscillator: Circuit that generates a periodic signal without an external input.
• Types: Sinusoidal (harmonic) and non-sinusoidal (relaxation).
• Components: Active device (BJT, MOSFET, Op-Amp), feedback network (LC,
RC, crystal).
• Barkhausen Criteria:
– Loop gain: |Aß|= 1.
– Phase shift: Total loop phase = 0
?
or multiple of 360
?
.
2. LC Oscillators
2.1 Hartley Oscillator
• Frequency of Oscillation:
f =
1
2p
v
L
eq
C
where L
eq
=L
1
+L
2
+2M, M: Mutual inductance.
• Loop Gain Condition:
Aß =
g
m
R
1+
L
2
L
1
= 1
where g
m
: Transconductance, R: Tank circuit resistance.
2.2 Colpitts Oscillator
• Frequency of Oscillation:
f =
1
2p
v
LC
eq
where C
eq
=
C
1
C
2
C
1
+C
2
.
• Loop Gain Condition:
Aß =g
m
R·
C
2
C
1
= 1
2.3 Clapp Oscillator
• Frequency of Oscillation:
f =
1
2p
v
LC
eq
, C
eq
=
C
1
C
2
C
3
C
1
C
2
+C
2
C
3
+C
3
C
1
• Advantage: Improved frequency stability due to additional capacitor C
3
.
1
3. RC Oscillators
3.1 Phase-Shift Oscillator
• Frequency of Oscillation:
f =
1
2pRC
v
6
(for 3-stage RC network, each stage provides 60
?
phase shift).
• Loop Gain Condition:
A= 29
(minimum gain for oscillation).
3.2 Wien Bridge Oscillator
• Frequency of Oscillation:
f =
1
2pRC
• Loop Gain Condition:
A= 3
• Bridge Balance:
ß =
1
3
at ? =
1
RC
4. Crystal Oscillator
• Frequency of Oscillation:
f =
1
2p
v
LC
s
where L, C
s
: Equivalent inductance and capacitance of crystal.
• Quality Factor (Q): Very high (10
4
-10
6
), ensuring stable frequency.
• Equivalent Circuit: Series resonance (C
s
) and parallel resonance (C
p
).
5. Relaxation Oscillators
• Frequency (Astable Multivibrator):
f =
1
2RCln(1+2ß)
where ß: BJT current gain.
• Time Period:
T =T
on
+T
o?
= 2RCln
(
1+
2R
B
R
E
)
2
Page 3


Formula Sheet for Oscillators (Analog and Digital
Electronics) – GATE
1. Basic Concepts
• Oscillator: Circuit that generates a periodic signal without an external input.
• Types: Sinusoidal (harmonic) and non-sinusoidal (relaxation).
• Components: Active device (BJT, MOSFET, Op-Amp), feedback network (LC,
RC, crystal).
• Barkhausen Criteria:
– Loop gain: |Aß|= 1.
– Phase shift: Total loop phase = 0
?
or multiple of 360
?
.
2. LC Oscillators
2.1 Hartley Oscillator
• Frequency of Oscillation:
f =
1
2p
v
L
eq
C
where L
eq
=L
1
+L
2
+2M, M: Mutual inductance.
• Loop Gain Condition:
Aß =
g
m
R
1+
L
2
L
1
= 1
where g
m
: Transconductance, R: Tank circuit resistance.
2.2 Colpitts Oscillator
• Frequency of Oscillation:
f =
1
2p
v
LC
eq
where C
eq
=
C
1
C
2
C
1
+C
2
.
• Loop Gain Condition:
Aß =g
m
R·
C
2
C
1
= 1
2.3 Clapp Oscillator
• Frequency of Oscillation:
f =
1
2p
v
LC
eq
, C
eq
=
C
1
C
2
C
3
C
1
C
2
+C
2
C
3
+C
3
C
1
• Advantage: Improved frequency stability due to additional capacitor C
3
.
1
3. RC Oscillators
3.1 Phase-Shift Oscillator
• Frequency of Oscillation:
f =
1
2pRC
v
6
(for 3-stage RC network, each stage provides 60
?
phase shift).
• Loop Gain Condition:
A= 29
(minimum gain for oscillation).
3.2 Wien Bridge Oscillator
• Frequency of Oscillation:
f =
1
2pRC
• Loop Gain Condition:
A= 3
• Bridge Balance:
ß =
1
3
at ? =
1
RC
4. Crystal Oscillator
• Frequency of Oscillation:
f =
1
2p
v
LC
s
where L, C
s
: Equivalent inductance and capacitance of crystal.
• Quality Factor (Q): Very high (10
4
-10
6
), ensuring stable frequency.
• Equivalent Circuit: Series resonance (C
s
) and parallel resonance (C
p
).
5. Relaxation Oscillators
• Frequency (Astable Multivibrator):
f =
1
2RCln(1+2ß)
where ß: BJT current gain.
• Time Period:
T =T
on
+T
o?
= 2RCln
(
1+
2R
B
R
E
)
2
6. Ampli?er Parameters for Oscillators
• BJT Transconductance:
g
m
=
I
C
V
T
, V
T
˜ 25mV at 300K
• MOSFET Transconductance:
g
m
=k
n
(V
GS
-V
TH
), k
n
=µ n
C
ox
W
L
• Output Resistance (BJT):
r
o
=
V
A
I
C
• Output Resistance (MOSFET):
r
o
=
1
?I
D
7. Frequency Stability
• Stability Factor:
S
f
=
? f/f
? X/X
where X: Parameter a?ecting frequency (e.g., L, C).
• Crystal Oscillator: High stability due to high Q.
• Temperature Coe?cient: Minimize using temperature-compensated compo-
nents.
8. Start-Up Conditions
• Initial Loop Gain: |Aß|> 1 to start oscillations.
• Gain Reduction: Automatic gain control (AGC) or diode limiting to stabilize
amplitude.
9. Power Consumption
• BJT Oscillator:
P =V
CE
I
C
• MOSFET Oscillator:
P =V
DS
I
D
• E?ciency: Higher in tuned LC oscillators than RC oscillators.
3
Page 4


Formula Sheet for Oscillators (Analog and Digital
Electronics) – GATE
1. Basic Concepts
• Oscillator: Circuit that generates a periodic signal without an external input.
• Types: Sinusoidal (harmonic) and non-sinusoidal (relaxation).
• Components: Active device (BJT, MOSFET, Op-Amp), feedback network (LC,
RC, crystal).
• Barkhausen Criteria:
– Loop gain: |Aß|= 1.
– Phase shift: Total loop phase = 0
?
or multiple of 360
?
.
2. LC Oscillators
2.1 Hartley Oscillator
• Frequency of Oscillation:
f =
1
2p
v
L
eq
C
where L
eq
=L
1
+L
2
+2M, M: Mutual inductance.
• Loop Gain Condition:
Aß =
g
m
R
1+
L
2
L
1
= 1
where g
m
: Transconductance, R: Tank circuit resistance.
2.2 Colpitts Oscillator
• Frequency of Oscillation:
f =
1
2p
v
LC
eq
where C
eq
=
C
1
C
2
C
1
+C
2
.
• Loop Gain Condition:
Aß =g
m
R·
C
2
C
1
= 1
2.3 Clapp Oscillator
• Frequency of Oscillation:
f =
1
2p
v
LC
eq
, C
eq
=
C
1
C
2
C
3
C
1
C
2
+C
2
C
3
+C
3
C
1
• Advantage: Improved frequency stability due to additional capacitor C
3
.
1
3. RC Oscillators
3.1 Phase-Shift Oscillator
• Frequency of Oscillation:
f =
1
2pRC
v
6
(for 3-stage RC network, each stage provides 60
?
phase shift).
• Loop Gain Condition:
A= 29
(minimum gain for oscillation).
3.2 Wien Bridge Oscillator
• Frequency of Oscillation:
f =
1
2pRC
• Loop Gain Condition:
A= 3
• Bridge Balance:
ß =
1
3
at ? =
1
RC
4. Crystal Oscillator
• Frequency of Oscillation:
f =
1
2p
v
LC
s
where L, C
s
: Equivalent inductance and capacitance of crystal.
• Quality Factor (Q): Very high (10
4
-10
6
), ensuring stable frequency.
• Equivalent Circuit: Series resonance (C
s
) and parallel resonance (C
p
).
5. Relaxation Oscillators
• Frequency (Astable Multivibrator):
f =
1
2RCln(1+2ß)
where ß: BJT current gain.
• Time Period:
T =T
on
+T
o?
= 2RCln
(
1+
2R
B
R
E
)
2
6. Ampli?er Parameters for Oscillators
• BJT Transconductance:
g
m
=
I
C
V
T
, V
T
˜ 25mV at 300K
• MOSFET Transconductance:
g
m
=k
n
(V
GS
-V
TH
), k
n
=µ n
C
ox
W
L
• Output Resistance (BJT):
r
o
=
V
A
I
C
• Output Resistance (MOSFET):
r
o
=
1
?I
D
7. Frequency Stability
• Stability Factor:
S
f
=
? f/f
? X/X
where X: Parameter a?ecting frequency (e.g., L, C).
• Crystal Oscillator: High stability due to high Q.
• Temperature Coe?cient: Minimize using temperature-compensated compo-
nents.
8. Start-Up Conditions
• Initial Loop Gain: |Aß|> 1 to start oscillations.
• Gain Reduction: Automatic gain control (AGC) or diode limiting to stabilize
amplitude.
9. Power Consumption
• BJT Oscillator:
P =V
CE
I
C
• MOSFET Oscillator:
P =V
DS
I
D
• E?ciency: Higher in tuned LC oscillators than RC oscillators.
3
10. Design Considerations
• Frequency Selection: Choose L, C, or R, C for desired f.
• Transistor Biasing: Ensure active region (BJT: V
BE
˜ 0.7V; MOSFET: V
GS
>
V
TH
).
• Applications: Signal generators, clocks, RF circuits, microcontrollers.
• Phase Noise: Minimize by using high-Q components (e.g., crystal).
4