Before we begin to explore the effects of resistors, inductors, and
capacitors connected together in the same AC circuits, let's briefly
review some basic terms and facts.
Resistance is essentially friction against the motion of electrons. It is present in all conductors to some extent (except superconductors!),
most notably in resistors. When alternating current goes through a
resistance, a voltage drop is produced that is in-phase with the
current. Resistance is mathematically symbolized by the letter “R” and
is measured in the unit of ohms (Ω).
Reactance is essentially inertia against the motion of
electrons. It is present anywhere electric or magnetic fields are
developed in proportion to applied voltage or current, respectively; but
most notably in capacitors and inductors. When alternating current goes
through a pure reactance, a voltage drop is produced that is 90o
out of phase with the current. Reactance is mathematically symbolized
by the letter “X” and is measured in the unit of ohms (Ω).
Impedance is a comprehensive expression of any and all forms of
opposition to electron flow, including both resistance and reactance. It
is present in all circuits, and in all components. When alternating
current goes through an impedance, a voltage drop is produced that is
somewhere between 0o and 90o out of phase with the
current. Impedance is mathematically symbolized by the letter “Z” and
is measured in the unit of ohms (Ω), in complex form.
Perfect resistors possess resistance, but not reactance.
Perfect inductors and perfect capacitors
possess reactance but no resistance. All components possess impedance,
and because of this universal quality, it makes sense to translate all
component values (resistance, inductance, capacitance) into common terms
of impedance as the first step in analyzing an AC circuit.
Perfect resistor, inductor, and capacitor.
The impedance phase angle for any component is the phase shift between
voltage across that component and current through that component. For a
perfect resistor, the voltage drop and current are always in phase with each other, and so the impedance angle of a resistor is said to be 0o. For an perfect inductor, voltage drop always leads current by 90o, and so an inductor's impedance phase angle is said to be +90o. For a perfect capacitor, voltage drop always lags current by 90o, and so a capacitor's impedance phase angle is said to be -90o.
Impedances in AC behave analogously to resistances in DC circuits: they
add in series, and they diminish in parallel. A revised version of Ohm's
Law, based on impedance rather than resistance, looks like this:
Kirchhoff's Laws and all network analysis methods and theorems are true
for AC circuits as well, so long as quantities are represented in
complex rather than scalar form. While this qualified equivalence may be
arithmetically challenging, it is conceptually simple and elegant. The
only real difference between DC and AC circuit calculations is in regard
to power. Because reactance doesn't dissipate power as
resistance does, the concept of power in AC circuits is radically
different from that of DC circuits.
Series R, L, and C
Let's take the following example circuit and analyze it:
The first step is to determine the reactances (in ohms) for the inductor and the capacitor.
The next step is to express all resistances and reactances in a mathematically common form: impedance. Remember that an inductive reactance translates into a positive imaginary impedance (or an impedance at +90o), while a capacitive reactance translates into a negative imaginary impedance (impedance at -90o). Resistance, of course, is still regarded as a purely “real” impedance (polar angle of 0o):
Parallel R, L, and C
We can take the same components from the series circuit and rearrange them into a parallel configuration for an easy example circuit:
The fact that these components are connected in parallel instead of series now has absolutely no effect on their individual impedances. So long as the power supply is the same frequency as before, the inductive and capacitive reactances will not have changed at all:
No comments:
Post a Comment