Impedance

[blackfin29]

Hello all: 

I am trying to understand what impedance is in an RLC circuit.  Can you please give me a simple definition?  I have searched on Google and read some books and still seems to be a bit confusing to me.   
Thank You,
Sal

[Andy Chee]

Think of impedance as "AC resistance" rather than "DC resistance".

For ideal resistors, AC resistance and DC resistance will be exactly the same.

For capacitors and inductors, AC resistance called "reactance" will depend on frequency.

[boB]

Impedance has a real part and an imaginary part for a RLC circuit.   Real part is the  AC resistance which is the in-phase  part of voltage across the RLC and current through the RLC.  i.e.  Voltage and current you would measure at any one frequency will be in phase with each other like a resistor at DC would have.  But because of skin effect and all that, AC resistance changes with frequency usually going higher with frequency for a component.

The imaginary part comes from the reactance of the capacitor and inductor and is the amount of 90 degree phase shifted current vs. voltage of the RLC network.

A single (absolute?)  impedance number (real) can be obtained but  Z = Sqrt(R^2 + X^2)

AC resistance is not the same as DC resistance except for a pure ideal resistance.  It changes with frequency and is related to the in-phase only part of voltage and current through you RLC circuit.  An RLC circuit will be purely resistive at resonance and in phase VI. 

Not important to the question but real Rs will have some self inductance and capacitance. Capacitors will have some self AC resistance and some inductance.  Inductors will have some capacitance across turns and of course AC and DC resistance so those may not be important depending on how accurate you need to know.   Because the parts are not perfectly, exactly R, L or C, each will have its own self resonant frequency.

Hopefully I didn't screw that up too much but maybe it will help.  Someone else can explain further and/or correct me.  I may have some stray reactance in my brain today.  i.e. I am old

[TimFox]

Complex numbers, such as Z = R + jX, where the complex impedance Z has a "real part" R (resistance) and "imaginary part" X (reactance), can be confusing if your mathematical training does not include them.
Another way is to consider impedance as an "in-phase" part I and a "quadrature-phase" component Q, both of which are functions of frequency.
If the input is proportional to cos(wt), then the in-phase part of the response is proportional to cos(wt) and the quadrature-phase is proportional to sin(wt), where the angular frequency w = (2 pi)xf

[Laval]

The answer to that question can be simple or complicated depending on how many things you are ready to accept. Impedance is the opposition to alternating current in a circuit. Of course, resistance will oppose both DC and AC in the same way but capacitors and inductors will "add" some opposition only to AC and this is called reactance. The capacitive reactance, symbol X_C decreases as frequency increases and the inductive reactance, symbol X_L, increases as frequency increases. These quantities are given by the following formulae:

\$\boldsymbol{X_C} = \frac{1}{2 \pi fC}\$

\$\boldsymbol{X_L} = 2 \pi fL\$

In a RLC circuit, pure resistance, capacitive reactance and inductive reactance are combined together. Reactance and resistance can be represented by vectors or complex numbers (they are not the same but both can be used). Capacitive reactance and inductive reactance oppose each other and are along the y (or imaginary) axis while resistance is along the x axis. It means that, when combining them, you cannot just add them together. They need to be added as vectors or complex numbers. This leads to the following formula for series RLC circuit:

\$Z = sqrt[R^2 + (XL - XC)^2]\$

For parallel RLC circuit, the result is:

\$Z = 1/sqrt[(1/R)^2 + (1/XL - 1/XC)^2]\$

The fact that the reactance are subtracted is just a consequence of the fact that they oppose each other.

In any RLC circuit, for a given capacitance and inductance, there is a frequency for which the capacitive reactance and the inductive reactance are equal and cancel each other. This frequency is called the resonance frequency and at that frequency, the impedance of the circuit is reduced to only the resistance.

All this probably didn't make thing much clearer. All these equations can be derived from the definition of capacitator and inductor as well as the representation of alternating current using sine functions with calculus and basic differential equations. Like I said at the beginning, it comes to how much things you are ready to accept.

I hope this helped a bit.

Laval

[Andy Chee]

Complex numbers, such as Z = R + jX

A side trivia note, when one encounters complex numbers in mathematics the imaginary component is typically denoted with small letter i.  Contrast with electronics engineering, the imaginary component is denoted with small letter j.

The use of "j" is because in electronics "i" is already allocated to symbolize current.