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 + (X_L - X_C)^2}\$

For parallel RLC circuit, the result is:

\$Z = 1/\sqrt{(1/R)^2 + (1/X_L - 1/X_C)^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. To determine the resonance frequency, just equate X_C and X_L using the first two equations then solve for f.

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

It might help us help you if we knew what is it you find confusing about RLC circuits ?

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.

[TimFox]

My favorite anecdote from graduate school (physics) in the 1970s.
Prof. Ugo Fano was lecturing on a quantum calculation of the dielectric constant in material.
He started a perturbation calculation for the polarization as a function of the applied electric field E, writing the Hamiltonian.
He then did a Fourier decomposition of E into frequency components E(w)exp(iwt).
A theory-freak in the front row objected, since the resulting Hamiltonian was not Hermitian (the energy, as written, was not real-valued).
Prof. Fano replaced the "i" with "j" and announced that the Hamiltonian was now Hermitian.

Also note than when writing the complex impedance, in the imaginary part jX, the term X is real-valued, positive for inductive reactance and negative for capacitive reactance.
When writing admittance, the inverse of impedance, in its imaginary part jB, B is also real-valued, but (due to complex algebra), inductive susceptance is negative and capacitive susceptance is positive.

[MrAl]

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

[Note: added a little more information]

Hi,

Probably the simplest way to look at impedance (Z) is to think of it as a variable resistance which varies with AC frequency. It usually carries some stored energy too that's a little different than just a regular resistance.

Example:
We might have a circuit where the magnitude of Z=1 at a frequency of 100Hz, but as the frequency goes to 200Hz the magnitude Z may go up to 1.5, or it may go down to 0.5 as it all depends on how the R, L and C are connected and their values.  There is series and parallel, and some other combinations.

To go deeper you should really study complex numbers first that's the only way to understand this in its entirety.  The three elements R, L and C have impedances all by themselves (w=2*pi*f with f the frequency in Hertz):
zC=1/(j*w*C)
zL=j*w*L
zR=R (no change with frequency)

You can then combine them just like you do with resistances.  For L and C in parallel:
Ztotal=1/(1/zC+1/zL)
for L and C in series:
Ztotal=1/zC+1/zL
and for all three in series:
Ztotal=1/zC+1/zL+R
You then go on to resolve these into just one real part and one imaginary part so you can calculate the magnitude and phase if needed.
The magnitude is the norm of the complex impedance (Ztotal above), and the phase is atan(imag/real) but that requires adjusting depending on the quadrant so better is atan2(imag,real), and that function is available in a lot of math software.