Transcript EGN 3373 Week 4 – Caps and Ind Explained - Help-A-Bull

EGN 3373 Introduction
to Electrical Systems I
A Systems Approach to Electrical Engineering
Graphics Adapted from “Physical, Earth, and Space Science”, Tom Hsu, cpoScience.
Section 3.1
RECALL: Whereas resistors convert electrical energy into heat,
inductors and capacitors are energy storage elements. Capacitors
and inductors do not generate energy. Resistors, capacitors, and
inductors are all passive elements.
q  Cv
In an ideal capacitor, the stored charge, q, is
proportional to the voltage between the plates.
Section 3.1
Example
q(t) = C v(t) = 1 x 10-6 v(t)
Given this v(t) –
notice units
Recal dV/dt =
slope of the line =
[v2 – v1]/[t2 – t1]
Section 3.1
Given the Current through a Capacitor.
Find the voltage.
q  Cv
t
Voltage in Terms of Current
q(t)   i(t )dt  q(t 0)
0
Thus, the voltage expression for the
circuit symbol for capacitance is
t
q(t ) 1
v (t ) 
  i(t )dt v(t 0)
C
C0
Section 3.1
Given this i(t) in radians; q(0) = 0
Plot i(t) =>
sin wave formation
Example
t
q(t)   i(t )dt  q(t 0)
0
Then V(t) = q(t)/C
Section 3.1
Example
t
p(t )  v(t )i (t ) w (t )   p(t )dt  12 Cv2 (t )
t0
Given a voltage waveform applied to a 10 μF capacitance. Plot current,
power delivered, and energy stored from 0 to 5s.
Section 3.1
Exercise
Given the square wave current through a 0.1 μF capacitor. Find and
plot the voltage, power, and stored energy.
Section 3.1
Exercise
Given the square wave current through a 0.1 μF capacitor.
Find and plot the voltage, power, and stored energy.
Section 3.1
Exercise
Given the square wave current through a 0.1 μF capacitor.
Find and plot the voltage, power, and stored energy.
Section 3.2 Capacitance in Series and Parallel
Capacitors in parallel
Capacitors in a parallel configuration each have the same
applied voltage. Their capacitances are added. Charge is
apportioned among them by size.
Section 3.2 Capacitance in Series and Parallel
Capacitors in series
The capacitors each store instantaneous charge build-up equal to
that of every other capacitor in the series. The total voltage
difference from end to end is apportioned to each capacitor
according to the inverse of its capacitance. The entire series acts as
a capacitor smaller than any of its components.
Capacitors are combined in series to achieve a higher working voltage, for
example for smoothing a high voltage power supply. For example, in a
pacemaker.
Exercise
a) Ceq =
b) Ceq =
Section 3.4 Inductance
RECALL Inductors
Bar-Coil
Surface Mount
Thin Film
Toroid Type
An inductor's ability to store magnetic energy is measured
by its inductance, in units of henries. Typically an inductor is
a conducting wire shaped as a coil, the loops helping to
create a strong magnetic field inside the coil due to
Faraday's Law of Induction.
Section 3.4 Inductance
Current in Terms of Voltage
1
di  v(t )dt
L
t
1
i (t )   v(t )dt  i(t 0)
L t0
Stored Energy
p(t )  v(t )i (t )
di
p(t )  v(t )i (t )  Li (t )
dt
t
t
di
w (t )   p(t )dt   Li dt 
dt
t0
t0
i (t )
2
1
Lidi

Li
(t )
2

0
Section 3.4 Inductance
We Can Present the Same Type of Example for an Inductor
Note: As the current magnitude increases, power is + and stored
energy accumulates; when the current is constant, the voltage is
0, thye power is 0, and the stored energy is constant; when the
current magnitude falls toward 0, the power is -, showing that the
energy is being returned to the other parts of the circuit.
Section 3.4 Inductance
Example
t
t
1
1
i (t )   v(t )dt  i(t 0)   10dt  5t
L t0
20
Section 3.5 Inductors in Series and Parallel
Exercise
Using KCL, write
v(t) = v1(t) + v2(t) + v3(t)
v(t) = L1 di/dt + L2 di/dt + L3 di/dt
then if we define Leq = L1 + L2 + L3 (Add like R in series)
v(t) = Leq di/dt
Exercise
Leq = ?
Leq = ?
Exercise