CONVENTIONS - VECTORS
HOW VALUES ARE QUOTED ON THIS SITE
SPECIAL NOTE: This entire page involves impedances NOT capacitance or inductance. Reactances are determined always by frequency. Everything here relies on the frequency remaining the same throughout. Changing frequencies will come later.
This page might be better placed after a description of RESISTANCE, CAPACITANCE, INDUCTANCE, REACTANCE and IMPEDANCE. It has been placed at the top of the list in case you don't quite understand something and want to refer back to it. If you are unfamiliar with any of the concepts listed, it would be better to skip this page.
Impedance is that quality of a circuit that impedes or restricts current through it as a result of an applied voltage. With DC voltage, like the lights in a car, the impedance can be considered as always purely resistive. Reactive components like coils and capacitors will affect a DC current for the first few milli-seconds only. Alternating current (including Radio frequency) presents a different situation entirely. The way reactive components impede current is different to a normal resistance. In the case of a resistor, the current is in phase with the voltage. With reactive components there is a phase angle of ± π/2 between them. (SEE CAPACITANCE AND INDUCTANCE LATER.)
A resistor consumes power. Neither an inductance or a capacitor do even though there is still a current through them. An IMPEDANCE contains both RESISTIVE and REACTIVE parts even though one of them might be zero or infinite (although neither condition will ever occur). It is only the resistive parts that consume power even thought there may be both present.
Reactance is only a measure of impedance that consumes no power. It can be inductive or capacitive. For the moment we are only talking about reactance.
Many (most) sources give the formula for capacitive reactance as Xc = 1 / ω C. This is a short way of writing |Xc| = 1 / ω C. On this site (and a few others) it will stated as Xc = -1 / ω C. This is more convenient (and more correct) because complex impedances will have both a magnitude ( |Z| ) and a phase angle which can be either +ve in the case of inductors and -ve in the case of capacitors.
1) Resistance is always a +ve value (unless 0)
2) Reactance is a +ve value for inductance and a -ve value for capacitance
3) Impedance contains elements of 1) and 2) even though one might be 0
Note, the real formula for the impedance (not reactance) of a capacitor is neither of the above. It is ZC = -j / ω C where ω is the frequency in rad/sec, C is the capacitance in Farads and j = √-1. For an inductor it is ZL = j ω L, where j and ω are the same but inductance is in Henries.
It should be noted here that, just like conventional current versus actual electron flow, the whole thing works just as well upside down. We can just as easily consider inductive -ve and capacitive +ve OR phase angle of current to voltage instead of voltage to current. The choice of +ve for inductive impedances is entirely arbitrary just like the column order DEBIT and CREDIT in double entry accounting. Some countries, like the Middle east where this accounting method was developed, use the opposite order to Western Countries. There is no correct way around. It just helps if everybody uses the same convention to avoid misinterpretation.
On this site, the capacitive portion of an impedance will be -ve while the inductive portion will be +ve. This way, unknown impedances can be combined using the same formula whether capacitive or inductive.
A good example is if two reactances are in series. If the two reactances are inductive the formula is ZL = ZL1 + ZL2. The same formula can be used for capacitance but when one component is inductive and the other capacitive, the formula is often stated as Z = ZL - ZC.
It is a whole lot easier to simply state that, in all circumstances, reactances in series are all simply added thus,
Z = Z1 + Z2.
The fact that either one of them might be capacitive is taken into account by the fact that a capacitive reactance is -ve.
Many values, especially impedance, are often complex values. These are called imaginary numbers. Many people find this concept difficult, especially those who didn't pay a lot of attention in high school. Counting numbers are easy for most people to imagine. 1 means one of something, 2 means two etc. If the numbers are imaginary, how can we visualize something imagined in someone else's mind. In actual fact, complex numbers as they apply to inductance and capacitance, are only imaginary when compared to resistance.
COMPLEX IMPEDANCES IN SERIES
One way to conceptualise imaginary numbers is to use vectors. There are several ways to represent imaginary numbers. Vectors is only one of them. At right are the vector diagrams for two complex impedances with reactive and resistive components. If these two impedances are added, the result is shown in the lower diagram. REACTANCE IS NOT THE SAME AS RESISTANCE IS NOT THE SAME AS IMPEDANCE. As mentioned elsewhere, reactance consumes no power. Resistance does. Impedance can consume power provided the reistive component is non-zero.
Instead of vectors, an impedance can be stated as a two dimensional array eg Z = (X,R) where Z is the impedance, X the reactive component and R the resistive component. This is really just the same thing. Two impedances can be added by adding the elements of the array, that is:-
Z = Z1 + Z2 = (X1,R1) + (X2,R2) = (X1+X2,R1+R2)
The current through the circuit is determined by the impedance. The power consumed is determined by the resistive component only.
In the above diagram, when the two impedances are added, the result is resistive only. In this case, the capacitive and inductive reactances cancel each other out. This works with the same formula whether the reactances are capacitive or inductive provided the capacitive reactance is always a -ve value.
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