LC oscillations.pdf

Lic oscillations

1. Welcome to
Classes
BYJU’S
Alternating Current
What you already know
What you will learn
S9: LC Oscillations
1.Pure AC circuits
2.RC and LR AC circuits
3.Impedance
4.Power in AC circuits
5.Resonance


1.Radio tuning
2.Bandwidth
3.Sharpness
4.Quality factor
5.Everything about LC
oscillations

2. CASE 3
If ??????
??????=??????
??????,??????=0
????????????????????????=
??????
??????−??????
??????
??????

CASE 1
If ??????
??????>??????
??????,??????≠0
Circuit is predominantly
capacitive
Current leads the source
voltage
CASE 2
If ??????
??????<??????
??????,??????≠0
Circuit is predominantly
inductive
Current lags the source
voltage

3. Resonance
If ?????? is varied, then at a particular frequency (??????
0), ??????
??????=??????
??????
Impedance is minimum (&#3627408461;=&#3627408453;
2
+0
2
=&#3627408453;) and purely resistive circuit
Current is maximum (??????
&#3627408474;=&#3627408457;
&#3627408474;/&#3627408453;)

For resonance condition, ??????
??????=??????
??????
1
??????
0&#3627408438;
=??????
0?????? ⇒??????
0=
1
??????&#3627408438;
⇒??????
0=
1
2????????????&#3627408438;

Resonant frequency

4. Resonance (Graphical representation)

5. Resonance (Graphical representation)

6. |
BIG FM
(92.7)
RADIO MIRCHI
RADIO MANGO
To hear one particular radio station,
radio tuning is required.

7. &#3627408453;
??????=??????
&#3627408474;sin(??????&#3627408481;)
&#3627408438; ??????
|
To hear a particular radio station, tuning of radio is necessary i.e., vary the capacitance of
a capacitor in tuning circuit of the radio such that the resonant frequency of the circuit
becomes nearly equal to the frequency to that of radio signal received. At this condition,
the amplitude of the current with the frequency of the signal of the particular radio
station in the circuit is maximum.

8. |
??????
&#3627408474;

??????(&#3627408479;&#3627408462;??????/&#3627408480;) ??????
0
2∆??????
??????
&#3627408474;&#3627408474;????????????
??????
&#3627408474;&#3627408474;????????????
2

??????
1 ??????
2
Bandwidth (2∆??????)

9. |
??????
&#3627408474;

??????(&#3627408479;&#3627408462;??????/&#3627408480;) ??????
0
2∆??????
??????
&#3627408474;&#3627408474;????????????
??????
&#3627408474;&#3627408474;????????????
2

??????
1 ??????
2
&#3627408451;
&#3627408474;????????????=(??????
&#3627408474;????????????)
2
&#3627408453;/2
At resonance, Power will be maximum.
&#3627408451;=
??????
&#3627408474;????????????
2
2
&#3627408453;
2
=
&#3627408451;
&#3627408474;????????????
2


??????
1 and ??????
2 are called as
half-power points.
•The current will be 1/2 times the maximum value of current at ??????
1 and ??????
2.

10. |
??????
&#3627408474;

??????(&#3627408479;&#3627408462;??????/&#3627408480;) ??????
0
2∆??????
The quantity (??????
0/2∆??????) is regarded as
a measure of sharpness of resonance.
GOOD TUNNING
HIGH SHARPNESS
LOW BANDWIDTH
SMALL ∆??????
∆??????=
&#3627408453;
2??????

11. |
⇒
??????
0
2∆??????
=
??????
0??????
&#3627408453;
Sharpness of resonance
Quality Factor, &#3627408452;
∆??????=
&#3627408453;
2??????

??????
&#3627408474;

??????(&#3627408479;&#3627408462;??????/&#3627408480;) ??????
0
2∆??????
The quantity (??????
0/2∆??????) is regarded as
a measure of sharpness of resonance.
At resonance, &#3627408452; can also be defined as,
&#3627408452;=
&#3627408457;
??????
&#3627408457;
??????
=
&#3627408457;
??????
&#3627408457;
??????

&#3627408452;=
??????
0??????
0??????
??????
0&#3627408453;
=
??????
0
??????
0&#3627408438;??????
0&#3627408453;

&#3627408452;=
??????
0??????
&#3627408453;
=
1
??????
0&#3627408453;&#3627408438;
=
1
&#3627408453;
??????
&#3627408438;

12. |
??????
&#3627408474;

??????(&#3627408479;&#3627408462;??????/&#3627408480;) ??????
0
2∆??????
&#3627408452;=
??????
0
2∆??????

&#3627408452;=
??????
0??????
&#3627408453;

&#3627408452;=
1
&#3627408453;
??????
&#3627408438;

&#3627408452;=
1
??????
0&#3627408453;&#3627408438;

13. Which of the following combination should be selected for better tuning of an L-C-R
circuit used for communication?
a b c
&#3627408453;=25 Ω,
??????=1.5 ??????,
&#3627408438;=45 ????????????
&#3627408453;=20 Ω,
??????=1.5 ??????,
&#3627408438;=35 ????????????
&#3627408453;=25 Ω,
??????=2.5 ??????,
&#3627408438;=45 ????????????
&#3627408453;=15 Ω,
??????=3.5 ??????,
&#3627408438;=30 ????????????
d

14. ?????? should be high.
For better tuning, the quality factor of the circuit should be high.
&#3627408452;=
1
&#3627408453;

??????
&#3627408438;

Value of R and C should be minimum
&#3627408452;=
1
&#3627408453;

??????
&#3627408438;

a b c d
&#3627408453;=25 Ω,
??????=1.5 ??????,
&#3627408438;=45 ????????????
&#3627408453;=25 Ω,
??????=1.5 ??????,
&#3627408438;=35 ????????????
&#3627408453;=25 Ω,
??????=2.5 ??????,
&#3627408438;=45 ????????????
&#3627408453;=15 Ω,
??????=3.5 ??????,
&#3627408438;=30 ????????????

15. ??????
&#3627408438; +&#3627408478;
−&#3627408478;
−&#3627408478;
+&#3627408478;
)

16. ??????
&#3627408438;
−&#3627408478;
?????? +&#3627408478;
??????
+
+
+

??????
&#3627408437;=0
Step 1
??????
&#3627408438;
&#3627408437;=&#3627408437;
?????? ??????=??????
??????
Step 2
??????
&#3627408438;
&#3627408437;=&#3627408437;
??????
Step 3
??????=??????
??????
Now the same oscillation
and steps will occur.

17. ??????
&#3627408438;
−&#3627408478; +&#3627408478;
At &#3627408481;=0,
Current, ??????=0
Charge on capacitor, &#3627408478;=&#3627408478;
&#3627408474;
Induced emf across the inductor, ??????
&#3627408465;??????
&#3627408465;??????
=0
( ))
At time, &#3627408481;
Current =??????
Charge on capacitor =&#3627408478;
Potential difference =
??????
&#3627408464;

Induced emf across the inductor, ??????
&#3627408465;??????
&#3627408465;??????
=0
??????
&#3627408438;
−&#3627408478; +&#3627408478;

18. Applying Kirchhoff's Law
&#3627408478;
&#3627408438;
−??????
????????????
??????&#3627408481;
=0…(1) ??????=−
??????&#3627408478;
??????&#3627408481;

&#3627408478;
&#3627408438;
−??????
??????−
??????&#3627408478;
??????&#3627408481;
??????&#3627408481;
=0 ⇒
&#3627408478;
??????&#3627408438;
+
??????
2
&#3627408478;
??????&#3627408481;
2
=0
??????
2
&#3627408478;
??????&#3627408481;
2
+
&#3627408478;
??????&#3627408438;
=0
??????
&#3627408438;
−&#3627408478; +&#3627408478;
+ −
( ))

19. For a particle in SHM
??????
2
??????
??????&#3627408481;
2
+??????
??????
2
??????=0…(3) ??????=
??????
??????

??????
2
=
1
??????&#3627408438;
⇒??????=
1
??????&#3627408438;

??????
2
&#3627408478;
??????&#3627408481;
2
+
&#3627408478;
??????&#3627408438;
=0…(2)
??????
&#3627408438;
−&#3627408478; +&#3627408478;
( ))

20. ??????
2
??????
??????&#3627408481;
2
+??????
??????
2
??????=0…(3) ??????=
1
??????&#3627408438;

??????=&#3627408436;cos(??????
??????&#3627408481;+??????)
&#3627408478;=&#3627408478;
&#3627408474;cos??????
??????&#3627408481;+??????…(4)
At &#3627408481;=0,&#3627408478;=&#3627408478;
&#3627408474;
⇒&#3627408478;
&#3627408474;=&#3627408478;
&#3627408474;cos??????
⇒cos??????=1⇒??????=0
&#3627408478;=&#3627408478;
&#3627408474;cos??????
??????&#3627408481;
??????
&#3627408438;
−&#3627408478; +&#3627408478;
( ))

21. ??????=
1
??????&#3627408438;
&#3627408478;=&#3627408478;
&#3627408474;cos??????
??????&#3627408481;…(5)
??????=−
??????&#3627408478;
??????&#3627408481;

⇒??????=−
??????
??????&#3627408481;
[&#3627408478;
&#3627408474;cos??????
??????&#3627408481;]
⇒??????=??????
??????&#3627408478;
&#3627408474;sin??????
??????&#3627408481;
??????
&#3627408474;=??????
??????&#3627408478;
&#3627408474;
??????=??????
&#3627408474;sin??????
??????&#3627408481;
??????
&#3627408438;
−&#3627408478; +&#3627408478;
( ))

22. ??????=
1
??????&#3627408438;
&#3627408478;=&#3627408478;
&#3627408474;cos??????
??????&#3627408481;…(5) ??????
&#3627408474;=??????
??????&#3627408478;
&#3627408474;
??????=??????
&#3627408474;sin??????
??????&#3627408481;…(6)
&#3627408456;=
&#3627408478;
2
2&#3627408438;
, &#3627408456;
′
=
1
2
????????????
2
??????=
1
??????
??????
2
&#3627408438;

&#3627408456;
????????????????????????&#3627408473;=&#3627408456;+&#3627408456;′
&#3627408456;
????????????????????????&#3627408473;=
&#3627408478;
2
2&#3627408438;
+
1
2
????????????
2

??????
&#3627408438;
−&#3627408478; +&#3627408478;
( ))

23. ??????=
1
??????&#3627408438;
&#3627408478;=&#3627408478;
&#3627408474;cos??????
??????&#3627408481;…(5) ??????
&#3627408474;=??????
??????&#3627408478;
&#3627408474;
??????=??????
&#3627408474;sin??????
??????&#3627408481;…(6)
&#3627408456;
????????????????????????&#3627408473;=
&#3627408478;
2
2&#3627408438;
+
1
2
????????????
2

&#3627408456;
????????????????????????&#3627408473;=
[&#3627408478;
&#3627408474;cos??????
??????&#3627408481;]
2
2&#3627408438;
+
1
2
1
??????
??????
2
&#3627408438;
[??????
&#3627408474;sin??????
??????&#3627408481;]
2

??????=
1
??????
??????
2
&#3627408438;

&#3627408456;
????????????????????????&#3627408473;=
&#3627408478;
&#3627408474;
2
2&#3627408438;
[cos
2
??????
??????&#3627408481;)+(sin
2
??????
??????&#3627408481;)
&#3627408456;
????????????????????????&#3627408473;=
&#3627408478;
&#3627408474;
2
2&#3627408438;

??????
&#3627408438;
−&#3627408478; +&#3627408478;
( ))