MEL TECH
CIRCTRONICS
Review of R, X and Z
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 90oout 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 (Figure below) possess resistance, but not reactance. Perfect inductors and perfect capacitors (Figure below) 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.
Series R, L and C
Let's take the following example circuit and analyze it:
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:
Example series R, L, and C circuit.
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 +90degree), 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 0degree):
Example series R, L and C circuit with component values replaced by impedances.
-
SIGNIFICANCES:
-
Impedances of any kind add in series: ZTotal = Z1 + Z2 + . . . Zn
-
Although impedances add in series, the total impedance for a circuit containing both inductance and capacitance may be less than one or more of the individual impedances, because series inductive and capacitive impedances tend to cancel each other out. This may lead to voltage drops across components exceeding the supply voltage!
-
All rules and laws of DC circuits apply to AC circuits, so long as values are expressed in complex form rather than scalar. The only exception to this principle is the calculation of power, which is very different for AC.
Prallel 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:
Example R, L, and C parallel 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:
Example series R, L and C circuit with component values replaced by impedances.