Generation of electrical oscillations with lc circuit

1. Generation of Electrical
oscillations with LC circuit
Dr.R.Hepzi Pramila Devamani,
Assistant Professor of Physics,
V.V.Vanniaperumal College for Women,
Virudhunagar

2. Generation of Electrical oscillations
with LC circuit
• Consider a circuit, made up of an inductance (L)
connected parallel to a capacitance C.
• Let the capacitor be charged by connecting A and
B across a battery.
• The battery is now removed. The capacitor
discharges through the inductance.
• The capacitor gets charged with opposite polarity,
due to the action of the inductance, which
provides path for electric charges to flow from
one plate to the other plate of the capacitor
externally.

3. Generation of Electrical oscillations
with LC circuit

4. Generation of Electrical oscillations
with LC circuit
• The inductance makes use of the magnetic energy
established around it for this.
• Again, the capacitor discharges by sending electric
charges in the opposite direction through the inductance
and recharges.
• The process continues indefinitely, each time the
magnetic energy around the coil and the electrostatic
energy in the capacitor getting interchanged.
• The process by which the voltage (VAB) across A and B
varies sinusoidally is known as electromagnetic
oscillation. The frequency of oscillation is given by f =
1/2π 𝐿𝐶.

5. Generation of Electrical oscillations
with LC circuit

6. Generation of Electrical oscillations
with LC circuit
• In Practice, the inductance coil may possess some
ohmic resistance and due to this, there is loss of
power in each cycle of oscillations.
• The result is that the amplitude goes on
decreasing at each cycle i.e., the oscillations are
damped.
• In order to get oscillations with constant
amplitude (sustained oscillations), the energy loss
must be compensated from an external source
such as battery.

7. Generation of Electrical oscillations
with LC circuit

8. The oscillator
• An oscillator is an amplifier with sufficient
gain and positive feedback.
• It provides a periodically varying voltage
waveform as its output V0 even without any
external input.
• The oscillator actually draws electric energy
for this purpose from the d.c power supply
connected to the oscillator.

9. The oscillator

10. Principle of the oscillator and
Barkhausen criterion for oscillations
• The basic elements of a feedback oscillator are
shown in the figure.199.
• When the oscillator circuit is switched on, random
noises (small voltages of wide range of
frequencies) are produced by the resistors or even
from the power supply switching on.
• This is supposed to be equivalent to applying a
small (noise) voltage at the input terminal y.
• This is amplified by the inverting amplifier,
producing phase change of 1800at the output.

11. Principle of the oscillator and
Barkhausen criterion for oscillations

12. Principle of the oscillator and
Barkhausen criterion for oscillations
• The amplified output signal is passed on to the
input side through a feedback circuit (LC or RC
network).
• The feedback circuit has a resonant frequency (fc)
• It allows the signal of this frequency (fc) to pass
through it.
• The feedback circuit also produces 1800 phase
shift and permits only a fraction of the output to
appear at the input side (x) of the amplifier.

13. Principle of the oscillator and
Barkhausen criterion for oscillations
• The phase shift of the signal around the closed
path containing the amplifier and feedback path is
1800+1800 = 3600 or 00.
• The amplifier amplifies this signal of frequency
(fc).
• It is fed back again and again and the amplitude
of the output (Vo) goes on increasing.
• This is a result of the initial condition that closed
loop gain Aβ is greater than 1. This is due to the
positive feedback.

14. Principle of the oscillator and
Barkhausen criterion for oscillations
• However, due to the non-linear behaviour of
the amplifier device, the closed loop gain of
the amplifier reduces to 1 and the amplitude of
the output remains the at a steady value.
• Thus we get sustained electrical oscillations of
the output (Vo) at a fixed frequency (fo).
• The frequency (fo) is determined by the circuit
parameters of the feedback path.

15. Principle of the oscillator and
Barkhausen criterion for oscillations
• The above principle is summarized and given as
Barkhausen criterion for oscillations as follows.
• The net phase shift around the feedback loop must
be zero. (i.e., the feedback voltage must be in
phase with the input signal to the amplifier).
• The closed loop voltage gain (A β ) of the circuit
must be equal to unity. i.e., A β = 1. (Here A is
the gain of the amplifier and β is the feedback
ratio).
• A= 1/ β .

17. Essential sections of an oscillator
• An amplifier of sufficient gain, with non-linear
active device like transistor, to fix the amplitude
of oscillations.
• A network produce oscillations at the desired
frequencies. This part is known as tank circuit.
• A feedback circuit to supply a part of output
power to the tank circuit in correct phase to aid
the oscillations, i.e., it provides positive feedback.
• A d.c power supply.

18. Hartley Oscillator with transistor
• Hartley oscillator is a type of oscillator circuit
which generates sinusoidal output signals i.e., the
output voltage varies with time as a sine curve.
• Hartley oscillator circuit using npn transistor in
the common emitter mode is shown in the
diagram. It is an amplifier with a positive
feedback arrangement.
• The transistor amplifier contains Rc as the load
resistance, RE as the emitter resistance and CE the
by-pass capacitor to provide the a.c path.

19. Hartley Oscillator with transistor
• Vcc is the power supply to the transistor.
• The resistance R1 and R2 form the potential divider
arrangement.
• The voltage across R2 gives the necessary base bias to
the transistor.
• The input to the amplifier appears to the base through
the capacitor C1.
• The output is available at the collector through the
capacitor C2 with respect to the ground.
• The value of the components Rc and RE are chosen
such that the stage gain of the amplifier is greater than
unity.(A>1).

20. Hartley Oscillator with transistor

21. Hartley Oscillator with transistor
• The feedback circuit is a tank circuit
containing the inductances L1 and L2 and a
variable capacitance C.
• L1 and L2 are inductively coupled i.e., any
change of flux in L1 occurs at L2 also.
• Because of earth connections at G, the
potential at Q and P are 1800 out of phase.
• If Q is positive going, P will be negative going
in potential.

22. Working
• When the switch is closed, the collector current starts
increasing.
• The potential at the collector Vc starts decreasing,
making the end Q negative going in potential.
• Hence the potential at P is positive going.
• This makes the transistor to conduct more and more
due to positive feedback action.
• The collector voltage continues to decrease till
• Vc = 0i.e., the transistor is in saturation conducting the
maximum current. Now the rate of variation
• (dIc/dt) = 0

23. • With no current in the feedback path, the
magnetic field in L1 collapses and discharges
the capacitor.
• The potential at P and hence that at the base
now starts decreasing and the collector current
starts decreasing.
• This causes Vc to increase and the end Q
becomes more and more positive and the end P
goes more and more negative.

24. • This trend goes on till the collector current is zero
and Vc = Vcc (the maximum).
• The transistor is now in cut-off state. Again
(dIc/dT) = 0.With no current in the feedback
circuit, the magnetic field in L1 collapses in the
opposite direction and the whole process repeats.
• Thus output goes on decreasing, increasing and
again decreasing and so on, producing sinusoidal
voltage waves at the output.

25. • The coil L2 couples the collector circuit energy back to
the tank circuit by means of the mutual inductance
between L1 and L2.
• In this way, energy is continously supplied to the tank
circuit to overcome the energy lossess in the tank
circuit.
• The LC tank , which is the feedback path, introduces a
phase shift of 1800 (π radian) at its resonant frequency
and the transistor in the common emitter mode
produces a phase shift of another 1800 (π radian)
between the input and the output.

26. • Thus the net phase shift around the loop = 2π
radian, and there is strong positive feedback.
• As these two conditions are met with, the circuit
readily functions as an oscillator and the electrical
oscillations are maintained.
• The frequency of oscillations is given by
• F = 1/2 𝐿𝐶 where L = L1 + L2 + 2M.
• Here M denotes mutual inductance between the
two coils of self inductance L1 and L2. C is the
capacitance in the tank circuit .

27. • Sustained oscillations will be produced if
• hfe ≥ (L2 +M) / (L1 +M)
• Where hfe is the forward current gain of the
transistor.
• The oscillator can be set into working by
adjusting the value of L1 and L2 initially.
• The frequency of oscillation can be changed
by varying the capacity of the capacitor C in
the tank circuit.

28. • The Hartley oscillator is easy to tune.
• It can be used to produce electrical oscillations
of a wide range of frequencies.
• Hartley oscillator is commonly used as high
frequency local oscillator in radio receives and
for production of ultrasonic waves using piezo-
electric crystal in the tank circuit.

29. Condition for oscillation
• For an oscillator to start functioning, the positive
feedback must be strong. i.e., A β > 1, A>1/ β
where A is the open loop gain and β is the
feedback ratio.
• For a common emitter amplifier circuit used in
this oscillator, the voltage gain A – Rc/re
• Where Rc is the load resistance and re is the
dynamic resistance of the base of the base-emitter
junction.
• Rc/re >1/ β (1)

30. Condition for oscillation
• The feedback ratio β = voltage fed to the base/
output voltage at the collector = Vf/Vo
• The output voltage is across L2 and the
feedback voltage is across L1
• β = (i L1ω) / (i L2ω)
• Where I is the circulating current at any instant
in the tank circuit
• β = (L1 / L2)

31. Condition for oscillation
• Using the value in equation (1),
• Rc/re > (L2 / L1)
• In terms of hfe of the transistor, since the gain
A = hfe, we get the condition
• Hfe > (L2 / L1) (since A = Rc/re )
• This is the condition for oscillation.

32. Expression for frequency
• The phase difference of 1800 by the feedback
network required for the oscillator is available
only at the resonant frequency of the network.
Hence this resonant frequency is the oscillator
frequency.
• At resonance, reactance of inductance branch
= reactance of capacitance branch
• XL = XC
• (L1 + L2 + 2M)ω = 1/C ω

33. Expression for frequency
• ω2 = 1/ (L1 + L2 + 2M)C
• ω2 = 1/ LC where L = L1 + L2 + 2M
• ω = 1/ 𝐿𝐶 i.e.,2πf = 1/ 𝐿𝐶
• f= 1/2π 𝐿𝐶
• This gives the frequency of the Hartley
oscillator.