Download electromagnetic oscillations

ELECTROMAGNETIC OSCILLATIONS
LC OSCILLATIONS
When a circuit is
comprised
only of
inductor and
capacitor
(resistance not
present)
• Initially the capacitor is fully charged and no
current flows in the inductor, as capacitor
begins to discharge, current I flows through
the inductor which increases to maximum
when the capacitor has fully transferred to the
magnetic field of the inductor. Energy now
flows from the inductor back to the capacitor
and when fully charged the capacitor begins
to discharge again except that it is in reverse
direction.
• Energy stored in the electric field of the
capacitor at time t is
• Energy stored in the magnetic field of inductor
at any time
• At any time in the circuit the total energy
;
LC Oscillator
• Since the circuit has no resistance, energy
transferred to thermal energy is zero & U is
constant with time
But
• Compare this with
with solution
;
(LC
oscillation eqn)
• The solution to LC eqn is thus
(charge oscillation)
• Q is the amplitude of charge oscillation and is
the angular frequency.
• Differenciating to get i, we have
• Differentiating current we have
• Sub for q and
we have
angular frequency.
OSCILLATIONS IN RLC CIRCUIT
• When a resistor is added then we have RLC
circuit, the total energy no longer remain the
same because the resistance convert part of it
into thermal energy i.e energy decreases at a
rate
thus
• Dividing through by i and rearranging we have
Damped oscillation in an RLC circuit
• The solution is
• The total energy of the electric field in the
capacitor as a function of time is given as
• Thus the energy of the electric field oscillates
according to a cosine-squared term, and the
magnitude of the oscillation decreases
exponentially with time.
ALTERNATING CURRENT
• To make up for damping in the RLC circuit, an
external emf device is made to supply an
alternating current or simply ac. The
oscillating current or emf vary sinusoidally
with time, reversing directions. The emf is
given as
= max (amplitude) is the driving frequency
• The current
; is a
convention adopted instead of
= phase
constant since i may not be in phase with
.
• The driving frequency can be got from
Three simple circuits
A resistive load
• from the loop rule
vR also has an amplitude,
VR=εm.
Or
(Resistor)
A capacitive load
• The p.d across the capacitor is
Also we have
to find the current we differentiate q
The voltage amplitude
Vc and current
amplitude Ic are
related thus
We then define a quantity called
capacitive reactance
And also noting that
hence,
AN INDUCTIVE LOAD
The potential drop across
the inductor
• Recall that
hence p.d across
inductor with current
changing
• Combining we have
And replacing -cos(ωdt)
with sin(ωdt-90)
for inductor
• To get i, we integrate
t
The series RLC circuit
• The alternating emf
is now
applied to the circuit containing R, L and C
which are in series this implies that same
current
passes through all.
(vC and vL) are opposite in
direction
The denominator is called the impedance Z
The denominator is called the impedance Z
thus
If we sub for XL and XC
;
• If XL > Xc then the circuit is said to be more
inductive than capacitive
• If XC>XL then the circuit is said to be more
capacitive than inductive ( is negative)
• If XC=XL the circuit is said to be in resonance,
RESONANCE
• Consider the eqn
• I is max when
recall that this is same as , the
natural frequency of the circuit
without external emf.
resonance; I is maximum
POWER IN AC CIRCUITS
Power P is given as
• Average power
• The quantity
• Average power
• We can also write
And
(root mean square current)
also
But from the geometry of the system
THE TRANSFORMER
• When an ac circuit has only a resistive load, the
power factor, cos =1 ;
• (I & V are by convention the root mean square
values measured by meters at home).
• In power transmission, one seeks to transmit at the
highest possible voltage & lowest possible current to
reduce energy dissipation as a result of the
resistance of the conducting wire.
• At the same time the end users need low voltage
and high current hence the need to be able to
raise(for transmission) and lower (for use) the ac
voltage in a circuit keeping the power constant. The
device for this is called transformer.
• The ideal transformer consists of two coils, with
different numbers of turns, wound around an iron
core. The coils are insulated from the core. The
primary winding is connected to an alternating
current generator. The relationship between the
voltage and current in the two coils and their
number of turns is given by
- transformaton of voltage
If
- it is a step up transformer and
if
it is a step down transformer
• But power is constant in both the primary and
secondary coils thus,
- transformation of current.
Sample problem
• Solution
In the above figure, R=200
ohms C= 13 microfarad and
L=230 mili Henry fd =60 Hz
Q-1. if L=0 find a),
impedanze Z; b), phase
angle and current I
Sample problem/exercise
Q-2. in the problem
figure;
If C=0; find a). Z, b).
Phase angle and c).
The current I
Q-3 in the problem figure;
find a). Z, b). Phase angle
and c). The current I for
the RLC circuit.