Chapter – 10
More on Basic Electronics
Capacitance is also a measure of the amount of electric charge
stored (or separated) for a given electric potential. A common form of
charge storage device is a two-plate capacitor. If the charges on the
plates are +Q and −Q, and V give the voltage between the plates,
then the capacitance is given by
C=Q/V
The SI unit of capacitance is the farad; 1 farad = 1 coulomb per volt.
The energy (measured in joules) stored in a capacitor is equal
to the work done to charge it. Consider a capacitance C, holding a
charge +q on one plate and -q on the other. Moving a small element
of charge dq from one plate to the other against the potential
difference V = q/C requires the work dW:
dW= q/C dq
Where W is the work measured in joules, q is the charge
measured in coulombs and C is the capacitance, measured in farads.
We can find the energy stored in a capacitance by integrating
this equation. Starting with an uncharged capacitance (q=0) and
moving charge from one plate to the other until the plates have
charge +Q and -Q requires the work W.
Inductance
Inductance is the property in an electrical circuit where a
change in the current flowing through that circuit induces an
electromotive force (EMF) that opposes the change in current.
Definition of Inductance
The quantitative definition of the (self-) inductance of a wire
loop in SI units is.
Where Φ denotes the magnetic flux through the area spanned
by the loop, and N is the number of wire turns. The flux linkage λ =
NΦ thus is.
There may, however, be contributions from other circuits.
Consider for example two circuits C1, C2, carrying the currents i1, i2.
The flux linkages of C1 and C2 are given by.
According to the above definition, L11 and L22 are the selfinductances of C1 and C2, respectively. It can be shown (see below)
that the other two coefficients are equal: L12 = L21 = M, where M is
called the mutual inductance of the pair of circuits.
The number of turns N1 and N2 occur somewhat
asymmetrically in the definition above. But actually Lmn always is
proportional to the product NmNn, and thus the total currents Nmim
contribute to the flux.
Self and mutual inductances also occur in the expression.
For the energy of the magnetic field generated by K electrical
circuits where in is the current in the nth circuit. This equation is an
alternative definition of inductance that also applies when the currents
are not confined to thin wires so that it is not immediately clear what
area is encompassed by the circuit nor how the magnetic flux through
the circuit is to be defined.
The definition L = NΦ / i, in contrast, is more direct and more
intuitive. It may be shown that the two definitions are equivalent by
equating the time derivative of W and the electric power transferred to
the system.
Properties of Inductance
Taking the time derivative of both sides of the equation NΦ = Li
yields:
In most physical cases, the inductance is constant with time and so
By Faraday's Law of Induction we have:
Where the Electromotive force (emf) and v is is the induced
voltage. Note that the emf is opposite to the induced voltage thus:
OR
These equations together state that, for a steady applied
voltage v, the current changes in a linear manner, at a rate
proportional to the applied voltage, but inversely proportional to the
inductance. Conversely, if the current through the inductor is
changing at a constant rate, the induced voltage is constant.
The effect of inductance can be understood using a single loop
of wire as an example. If a voltage is suddenly applied between the
ends of the loop of wire, the current must change from zero to nonzero. However, a non-zero current induces a magnetic field by
Ampère's law. This change in the magnetic field induces an emf that
is in the opposite direction of the change in current. The strength of
this emf is proportional to the change in current and the inductance.
When these opposing forces are in balance, the result is a current
that increases linearly with time where the rate of this change is
determined by the applied voltage and the inductance.
Multiplying the equation for di / dt above with Li leads to
Reactance
Reactance is a circuit element's opposition to an alternating
current, caused by the build up of electric or magnetic fields in the
element due to the current. Both fields act to produce counter EMF
that is proportional to either the rate of change (time derivative), or
accumulation (time integral) of the current.
In vector analysis, reactance is the imaginary part of electrical
impedance, used to compute amplitude and phase changes of
sinusoidal alternating current going through the circuit element. It is
denoted by the symbol . The SI unit of reactance is the ohm both
reactance and resistance re required to calculate the impedance
although in some circuits one of these may dominate: an
approximate knowledge of the minor component is useful to
determine if it may be neglected.
Both the magnitude
and the phase
of the impedance
depend on both the resistance and the reactance.
The magnitude is the ratio of the voltage and current
amplitudes, while the phase is the voltage–current phase difference.
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•
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If
If
If
the reactance is said to be inductive
then the impedance is purely resistive
the reactance is said to be capacitive.
Resonance
Resonance is the tendency of a system to oscillate at its
maximum amplitude, associated with specific frequencies known as
the system's resonance frequencies (or resonant frequencies). At
these frequencies, even small periodic driving forces can produce
large amplitude vibrations, because the system stores vibrational
energy.
When damping is small, the resonance frequency is
approximately equal to the natural frequency of the system, which is
the frequency of free vibrations. Resonant phenomena occur with all
types of vibrations or waves: there is mechanical resonance, acoustic
resonance, electromagnetic resonance, NMR, ESR and resonance of
quantum wave functions. Resonant systems can be used to generate
vibrations of a specific frequency, or pick out specific frequencies
from a complex vibration containing many frequencies.
Theory
For a linear oscillator with a resonance frequency Ω, the
intensity of oscillations I when the system is driven with a driving
frequency ω is given by:
The intensity is defined as the square of the amplitude of the
oscillations. This is a Lorentzian function, and this response is found
in many physical situations involving resonant systems. Γ is a
parameter dependent on the damping of the oscillator, and is known
as the line width of the resonance. Heavily damped oscillators tend to
have broad line widths, and respond to a wider range of driving
frequencies around the resonance frequency. The line width is
inversely proportional to the Q factor, which is a measure of the
sharpness of the resonance.