LC circuits: Sinusoidal Voltages and
Currents
Aims:
To appreciate:
•Similarities between oscillation in LC circuit and
mechanical pendulum.
•Role of energy loss mechanisms in damping.
•Why we study sinusoidal signals
•RMS Current and Voltage
To be able:
•To analyse some basic circuits.
Lecture 10
Lecture
9
1
L and C connected together
The capacitor is charged up to a voltage V
What happens when the switch is closed?
C
L
1.
2.
3.
C discharges through L
Current in L decays and
charges C with the reverse
e.m.f. (in the reverse polarity)
C discharges through L
and so on ….
Lecture 10
Lecture
9
2
1
LC oscillation
Pendulum
Energy
Energy
1
CV 2 +V
2
1 2
LI
2
C
L
mgh
h
C
1 2
mv
2
L
+I
v
1
CV 2
2
1 2
LI
2
-V
C
L
mgh
h
C
1 2
mv
2
L
-I
Lecture 10
Lecture
9
v
3
Form of oscillation
The voltage across the inductor is
but for the capacitor,
dI
dt
dV
I =C
dt
V = −L
so
This is the differential equation describing simple harmonic motion
The details depend on the initial
conditions.
The solution is
ω is the ANGULAR frequency (radians per second)
The true frequency
(oscillations per second or HERTZ) is
Lecture 10
Lecture
9
f =
ω
1
Hz
=
2π 2π LC
4
2
Form of oscillation
The voltage across the inductor is
but for the capacitor,
so
dI
dt
dV
I =C
dt
d 2I
I = − LC 2
dt
V = −L
This is the differential equation describing simple harmonic motion
The solution is
I = I 0 sin ωt
V = V0 cos ω t
The details depend on the initial
conditions.
ω is the ANGULAR frequency (radians per second) ω =
The true frequency
(oscillations per second or HERTZ) is
f =
Lecture 10
Lecture
9
1
rad s-1
LC
ω
1
Hz
=
2π 2π LC
5
Relationship between peak
current and voltage
Conservation of energy:
High capacitance – high current
High inductance – high voltage
Lecture 10
Lecture
9
6
3
Relationship between peak
current and voltage
Conservation of energy:
1
1
CV02 = LI 02
2
2
I0
C
=
V0
L
High capacitance – high current
High inductance – high voltage
Lecture 10
Lecture
9
7
Sinusoidal oscillations
Voltage
IP
Current
VP
phase difference φ
time
Period τ
Frequency = 1/τ
Sine waves are fundamental to electronic systems
• The natural form of oscillations in LC circuits (and sound and radio waves)
• The form of voltages generated by rotating dynamos
• Any complex waveform can be built up from superpositions of fundamental
sinusoidal waves (Fourier Series)
Lecture 10
Lecture
9
8
4
Damping
R
With ideal components, the oscillation will
continue indefinitely (no energy loss).
C
L
With real components, there is resistance and
power (I2R) is dissipated on each cycle.
dI
Q
+ iR + = 0 (KVL)
dt
C
d 2I
dI
LC 2 + RC + I = 0
dt
dt
L
This differential equation has a solution like
where γ is the damping coefficient γ =
I = − I 0 exp(−γ t ) sin(ωt + φ )
and ω =
Damping term
1
R2
− 2
LC 4 L
R
2L
Note a reduction
in the frequency
Lecture 10
Lecture
9
9
Damped oscillations
Time constant of
decay is
1 2L
τ=
ex p
γ
=
R
)
(-γt
The number of oscillations in
one time constant of the
decay is called the quality
factor, or Q of the circuit:
Q=
1γ
2π ω
When R is small, this is
given approximately by
Q=
1 L
R C
High Q means long ‘ringing’ time and high voltage (high L)
Lecture 10
Lecture
9
10
5
Fourier series
The first few HARMONICS
(multiples of the fundamental frequency)
f
3f
can be used to reconstruct any regular
waveform.
5f
The accuracy of the reconstruction
improves as you increase the number
of harmonics.
time
This is important because it means
that we can predict the behaviour of
an electronic circuit with any
complex waveform by studying the
effect on pure sine waves of
different frequencies
Time domain
Lecture 10
Lecture
9
11
Jean Baptiste Joseph Fourier
1768 – 1830
The son of a tailor in Burgundy he
developed much of the mathematical basis
of heat transfer, which led to the Fourier
expansion. This was heavily criticised by
Laplace and Lagrange.
His life was much affected by turbulent
French politics. He narrowly escaped the
guillotine during the Terror, and under
Napoleon he was a senior administrator in
Egypt (where he wrote his Description of
Egypt) and Prefect of Grenoble.
Lecture 10
Lecture
9
12
6
Frequency ranges
Frequency
Period
0 Hz
Applications
Constant voltages, battery circuits
10 – 100 Hz 100 ms – Power generation / transmission; mains
sockets; TV frame rate
50 / 60 Hz
10 ms
a.k.a.
DC
LF
50 Hz - 20
kHz
20 ms –
50 μs
Audible frequencies (speech, music), RS232, phone modem, TV line rate
audio
AF
60 kHz –
100 MHz
15 μs –
10 ns
AM – SW- FM radio; computer data bus;
Ethernet; CD sampling rate
Radio
RF, VHF
100 MHz –
100 GHz
10 ns –
10 ps
Mobile phones, TV channels, satellite links,
radar, microwave ovens, PC clocks
UHF
Microwave
> 100 GHz
< 10 ps
??? medical imaging? communication?
high speed super highways?
Terahertz
remember: time constants must be less than this
Lecture 10
Lecture
9
14
Heinrich Rudolf Hertz
1857 – 1894
Born in Hamburg and became Professor
of Physics in Bonn (via Berlin and
Karlsruhe).
In 1885 he was the first person to
demonstrate experimentally the
electromagnetic waves that had been
predicted theoretically by Maxwell in the
previous year.
(Marconi did not begin his work on radio
until 1894).
Lecture 10
Lecture
9
15
7
Some general properties of sine waves
Average voltage over one cycle is ZERO
Since power depends on V2 we get an
effective value by taking the square
root of the average of the SQUARE of
the voltage or current over one cycle:
V2 =
1
T
T
∫
VP
time
φ
V = VP sin(ω t + φ )
VP2 sin 2 (ω t + φ ) dt
0
with T (the period)= 2π ω
VRMS =
This works out to
VP
= 0.707VP
2
This is the RMS - root-mean-square – value of the voltage
Lecture 10
Lecture
9
16
RMS values and power
For resistors (current and voltage in phase), RMS values of
voltage or current can be used to calculate power dissipation:
P=
2
VRMS
R
2
= I RMS
R
Example: Mains voltage has a peak voltage of 339.4 V
so: RMS voltage is 0.707 x 339.4 = 240 V
If we connect this to a heater with a resistance of 57.6 Ω
the total power disputed is P=240 x 240/57.6 = 1000 W = 1kW
We can’t use this when the current and voltage are not in phase….
Lecture 10
Lecture
9
17
8