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RL and RC series and parallelImpedance index.Resonance.

RESISTANCE (R) - REACTANCE (X) - IMPEDANCE (Z)
A MORE ACCURATE DESCRIPTION

BASIC PROBLEMS

Reactance has been so far referred to as being either a positive or negative value. This is still true but it is not a direct translation to impedance.

That is, even when Ø = ±Π/2, Z≠X. On the other hand, provided Ø = ±Π/2 then |Z|=|X|. (Z=impedance & X=reactance)

As has been stated before, reactance is only one facet of impedance. The other is resistance. Impedance is a combination of both resistive and reactive components even if one of the quantities is zero or infinite.

Said another way, resistance is resistance and consumes power. Reactance is reactance and consumes no power. Impedance is a combination of both and can't be expressed in direct terms of either.

The question might arise, why not just treat them separately. A good answer can be found by looking at the resonance of a tank circuit. This is simply a capacitor and inductor in parallel. In this circuit, if a voltage is applied, current will flow through both. When the reactance of both is the same (see resonance later), the amount of energy stored in each is the same. Zero current needs to flow from the source to maintain the voltage because each transfers its stored energy in equal amounts to the other during each half cycle. A voltage that produces little or no current must be working into a high or infinite impedance load.

Simple tank circuit

This assumes perfect components BUT THERE IS NO SUCH THING. There will always be a resistance associated with the wire forming the coil and the circuit will look more like that shown at right. This will now change the amount of current flowing through the inductor. The inductor will not be able to store the same amount of charge and this will change the "driven resonant frequency" although the "natural resonant frequency" will remain the same.

Tank circuit with R in series with L.

RL REVISITED

At right are series RL and Parallel RL considered earlier. Both circuits will result in an impdeance containing both resistive and reactive components. This impedance will have a phase angle (or probably more accurately the resulting current will). For any series combination, as shown in the tank circuit above, there will be some combination of parallel components that will result in the same impedance.

Left - Series RL and Parallel RL.

As a simple example, suppose at some frequency, the inductor has a reactance of 1Ω. Suppose also we place a 1Ω resistor in series with it. The resulting impedance will be 1.414Ω (√2). The phase angle of the current will be Π/4 (45°).

At the same frequency, suppose we have inductor with a reactance of 2Ω and place a resistor of 2Ω in parallel with it. The resulting impdeance will be 1.414Ω (√2). The phase angle of the current will be Π/4 (45°).

This is only a very simple example but now means there is an equivalent to the tank circuit shown above with a resistor in parallel to both components like the one shown on the right. The effective amount of inductance, and therefore reactance, will have changed. Don't forget the inductive reactance of the equivalent circuit is not the same as the original reactance of the resistor in series with the inductor.

Left - Series RL and Parallel RL.

The driven resonant frequency will change if the resistor in series with the inductor is changed. The driven resonant frequency is that where the impedance is at a maximum or mimimum depending on whether it is parallel or series respectively. At this point it will be purely resistive. The current will be in phase with the voltage. (MORE ABOUT THIS IN RESONANCE LATER).

The natural resonant frequency is the frequency the circuit will continue to oscillate at after the applied voltage is removed, until of course, the energy is absorbed by the resistance.

Driven and natural resonant frequencies are only exactly the same for perfect components (which don't exist anywhere I know).

REPRESENTATION OF RESISTANCE, REACTANCE AND IMPEDANCE

All of these quantities are measured in ohms (symbol Ω). That doesn't mean they are the same. Just because both work and heat are measured in joules doesn't mean work and heat are the same. Just as an interesting point while on the topic. In classical physics, energy is divided up into only work and heat. Mass can also be measured in joules. 1kg = 8.9 x 1016J or better 1kg=89PJ (PetaJoules).

For resistance the value is simply given as a positive number in Ohms and the letter used in equations simply R. Conductance, the reciprocal of Ohms, normally uses the symbol σ. Since the equations that use it can be written as 1/Ω, I will avoid it to keep things simple. If you want to know more try Wikipedia.

Reactance is also measured in ohms and the value also quoted using the same symbol Ω. For equations the standard letter used is "X". Recatance can be positive for inductors or negative for capacitors. All of the same equations can be used for both. Normally, for an inductor, the subscript XL is used and for a capacitor XC but in complex circuits, at changing frequencies, whether the reactance is inductive or capacitive, may not be known so X by itself is sufficient.

Impedance is also measured in ohms using the symbol Ω. The standard letter used in equations is Z. Rather than just being a member of real numbers, impedance is a complex number. Impedance can be represented in a number of ways. One way is to simply state the magnitude and phase angle. Another is to represent the resistive part as a positive real number and also state the reactive part as a member of the complex number plane.

Both involve two values, that is:-

R ± j X ≡ ( √( R2 + |X|2 ) , ± Atan ( X / R ) )

Quoting an impedance as a complex number ( R ± j X ) is equivalent to quoting it as a magnitude ( √( R2 + |X| 2 ) ) and a phase angle ( ± Atan ( X / R ) ).

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