The difference between impedance and resistance


My Dad would be ashamed for me for writing this thread (he had a background in electrical engineering) but can someone explain the differnce between impedance and resistance (other than the former is for ac and the latter dc) as it pertains to audio circuitry?
russellrcncom
"Impedance" is a broader term than "resistance," and reflects the opposition to current flow that can be imposed by resistance, capacitance, or inductance.

The impedance of an ideal resistor (in practice there is no such thing, as all resistors will also have some amount of inductance and capacitance) is the same as its resistance and is the same at all frequencies. Resistance is usually denoted as R and is measured in ohms.

The impedance of an ideal capacitor (again, there is no such thing) decreases as frequency increases, and is equal to:

Xc = 1/(2 x pi x f x C)

where Xc denotes what is referred to as capacitive reactance (one form of impedance), and is also measured in ohms;
f = frequency in Hertz
C = capacitance in Farads

The impedance of an ideal inductor (again, there is no such thing) increases as frequency increases, and is equal to:

Xl (that's a small "L", not the number "1" or a capital "i") = 2 x pi x f x L

where Xl denotes what is referred to as inductive reactance (one form of impedance), and is also measured in ohms;
f = frequency in Hertz
L = inductance in Henries

Depending on the frequency and the circuit application, it is often (but certainly not always) possible to treat a practical resistor as being a close enough approximation to an ideal resistor to neglect its stray capacitance and inductance, as well as other non-ideal effects it may have. Likewise for practical capacitors and inductors.

The combined impedance of some amount of inductance, capacitance, and resistance is NOT calculated by directly combining the number of ohms of each, because the three parameters have different effects on the phase relation between voltage and current.

In the case of a pure resistance, voltage and current are in phase with each other. In the case of a pure inductance, voltage leads current by 90 degrees (1/4 cycle of the signal frequency). In the case of a pure capacitance, voltage lags current by 90 degrees.

Given that, the magnitude of the impedance of a circuit element combining all three types of impedance is calculated, for a specific frequency, by subtracting Xc from Xl, and then taking the square root of the sum of the squares of that difference and R. The phase angle between voltage and current at a specific frequency, corresponding to that combined impedance, is equal to the (arctangent of ((Xl-Xc)/R)), among other ways that it can be calculated.

Best regards,
-- Al
Holy shit.
Great answer.
I always wondered about this.
Thank you very much, Al!
Sincerely,
Truman
Al, Could you please add your take on Power Factor?
The Wiki article says that when the phase angle is 90degrees NO power is delivered.
My 'pet' idea is that reactive speaker loads are what really defines a 'bad load', not merely low impedance.
Hi Magfan,

Yes, your statements are correct, and the posts you have made emphasizing the importance of phase angle and how resistive vs. reactive a speaker load is are good.

A pure inductance or capacitance (with a phase angle of plus or minus 90 degrees) cannot dissipate (consume) any power at all.

Consider a sine wave at some frequency, with a phase angle of 90 degrees between voltage and current. When the sine wave reaches its maximum voltage, current will be zero. When current reaches its maximum, voltage will be zero. Since the power dissipated (consumed) at any instant of time is the product (multiplication) of voltage and current at that time, power at those instants will be zero.

In between those times, the product of voltage and current will alternate each quarter-cycle between being positive and being negative. That can be seen by drawing out two sine waves, with one delayed by 90 degrees from the other.

The alternating polarities correspond to the fact that incoming energy is stored during one quarter-cycle, and then discharged back to the source during the next. So the net power dissipation in the load is zero, and the energy that the amplifier tries to send to the speaker winds up being dissipated in the amplifier as heat.

In a purely resistive load, all incoming energy is consumed by the load. Voltage and current are always in phase, and their product is always positive. During positive-going quarter cycles the power is the product of two positive numbers, which is positive, and during negative-going quarter-cycles it is the product of two negative numbers, which is also positive.

Power factor, as defined here, reflects the degree to which the load is resistive vs. reactive, with a value of 1 being purely resistive, and 0 being purely reactive (capacitive or inductive). A value of 1 will correspond to a phase angle of 0 degrees, and a value of 0 will correspond to a phase angle of plus or minus 90 degrees.

Best regards,
Al
Hi Al,

Can you shed some theoretical light on the 2 * Pi constant in the equations for Xc and Xl?

Also, can you explain the physics as to why voltage and current are out of phase with reactive devices?

Many thanks,
Bob
Hi Bob,

The 2pi factor converts the units of frequency from cycles per second (Hertz) to radians per second, one radian equaling 360 degrees/2pi = approximately 57.3 degrees.

The need for using radians per second is inherent in the calculus (differentiation and integration) that I'll get into below, but those mathematical subjects were not exactly my forte and I can't explain that aspect of it any further :-)

The 90 degree phase shifts basically stem from the relations between voltage and current for inductors and capacitors. An inductor resists abrupt changes in current, and the voltage across an inductor and the current through it are related by the equation:

V = L(di/dt)

Where di/dt is the rate of change of current, or the derivative of current to use the calculus term. V is voltage, of course, and L is inductance.

That's the basis of the theory behind spark coils, btw. If the rate of change of current is near instantaneous, as would happen if some part of the path of the current flowing through the inductor were to suddenly be opened, the resulting voltage will be extremely large.

Capacitors resist abrupt changes in voltage. The relation between voltage and current is:

i = C(dV/dt)

Where dV/dt is the rate of change of voltage, or the derivative of voltage, in calculus terminology. C is capacitance, of course.

Based on standard calculus and differential equations:

The derivative of a sine wave is a cosine wave, which is the same as a sine wave except that the sine wave lags by 90 degrees.

The integral of a sine wave is an inverted cosine wave plus some constant. A sine wave leads an inverted cosine wave by 90 degrees.

For an inductor, rearranging the equation V = L(di/dt) to (di/dt) = V/L, and taking the integral of both sides of the equal sign, we have:

i = integral(V/L)

So if V is a sine wave, V will lead i by 90 degrees

For a capacitor we have:

i = C(dV/dt), so if V is a sine wave, per the above relationship that the derivative of a sine wave is a cosine wave, V will lag i by 90 degrees.

Whew! And all we want to do is to listen to well reproduced music :-)

Best regards,
-- Al
Al covered lots of good details but I would like to summarize it for those that might have gotten lost.

The fundamental difference has to do with frequency, as in frequencies of the recorded music, for example. There are innate qualities of capacitance and/or inductance in certain components (like capacitors and inductors) but they are not called into action until there is a signal composed of frequencies (such as music) interacting with them.

As the frequency of the signal increases, the way the signal sees every electrical part in the system changes dramatically. The higher the frequency, the more inductance and capacitance, quite literally, appear. Eventually you get to a point where even a resistor acts like a capacitor and an inductor! And then after that, even a wire exhibits inductive and capacitive effects! It is a crazy world - and it's no wonder cable and component break-in occurs.

Since music is not a DC signal (frequency of zero), technically speaking there is no such thing as a pure resistance in a stereo system. Everything has some sort of reactance associated with it, be it capacitive or inductive, which requires the use of the term "impedance." This is ever more accurate as frequencies go above human hearing, but there is no mathematical lower limit for this effect. This is why it pertains to our stereos.

As far as a "bad load" is concerned, it is tougher to drive a capacitive load than an inductive or resistive one. Capacitive loads can be seen in impedance plots as negative phase angles, meaning the current is ahead of the voltage. Not only does the efficiency of the circuit decrease if the load is capacitive, but the current demands during transients can be significantly more than looking at resistance alone. That double whammy is hard on an amplifier.

Arthur

Aball

As the frequency of the signal increases, the way the signal sees every electrical part in the system changes dramatically. The higher the frequency, the more inductance and capacitance, quite literally, appear. Eventually you get to a point where even a resistor acts like a capacitor and an inductor! And then after that, even a wire exhibits inductive and capacitive effects! It is a crazy world - and it's no wonder cable and component break-in occurs.

I don't see what any of that has to do with break-in.
Hi Al,

The 2pi factor converts the units of frequency from cycles per second (Hertz) to radians per second, one radian equaling 360 degrees/2pi = approximately 57.3 degrees.

I remember that 2Pi rads = 360 degs, but I don't understand why a cycle is assumed to be in degrees instead of radians.

1 Hz = 1 cycle/s = 360 deg/s * 2Pi rad/360 deg = 2Pi rad/s

Oh, I see it now. There's no implicit units of degrees.

1 Hz = 1 cycle/s * 2Pi rad/cycle = 2Pi rad/s

It's been a few decades since I studied any of this, but I guess calculations are done in radians for historical reasons. Formulas involving constants were derived using radians and have become part and parcel of the scientific literature. I assume there's no reason that a parallel literature could not be derived based on degrees.

Thanks Al. I'll need some more time to look over the calculus explanation. Again, it's been a few decades.
Hi Al,

I understand that sine and cosine are out of phase by pi/2, but why do you say that one leads or lags the other?

What's the point of reference?
Hi Bob,

Re the use of radians vs. degrees, see the section on this page headed "Advantages of measuring in radians," which begins by stating:
In calculus and most other branches of mathematics beyond practical geometry, angles are universally measured in radians. This is because radians have a mathematical "naturalness" that leads to a more elegant formulation of a number of important results.

Most notably, results in analysis involving trigonometric functions are simple and elegant when the functions' arguments are expressed in radians.
I'm not certain, but I believe that the results of differentiating and integrating a sine or cosine function would be much less "clean" and easy to work with if degrees or cycles were used instead of radians.
I understand that sine and cosine are out of phase by pi/2, but why do you say that one leads or lags the other?

What's the point of reference?
Just take any voltage level on one of the waveforms, such as a zero crossing or a peak level, and find the corresponding point on the other waveform. The two points will be pi/2 radians or 90 degrees apart, with one occurring first. Of course, for continuous waveforms a 90 degree lead is the same as a 270 degree lag, but characterizing it that way would not seem sensible physically. That would say that for a capacitor a voltage change precedes the corresponding current change, and for an inductor a current change precedes the corresponding voltage change. Given that capacitors resist instantaneous changes in voltage, and inductors resist instantaneous changes in current, viewing a 90 degree lead as a 270 degree lag would not work for a transient or non-continuous waveform.

Best regards,
-- Al
Hey we go again.. kudos to Al. As methodical and detail as ever. Way to go Prof Al.
The reason radians are used is because the derivative of sin(x) is directly cos(x) when using radians. This is why the formulations using radians come out cleaner.

I would like to add too that you very rarely get 90 degrees in a circuit. The parasitic effects will cause the angle to be all over the map, positive and negative, the extent of which is dependent on the frequency of the signal.

Simply-q: Break-in effects are largely due to parasitic capacitance effects through the air, insulation, cable sheaths, PC boards, etc., surrounding the signal-carrying wire(s). The capacitance is formed between the wire and "ground," which can be just about anything at a lower potential, and everything in between ground and the signal is the dielectric. These effects are part of what I was talking about. The fact the ground is indeterminate is what leads to the mystery of break-in. As time goes on, the more-dominant capacitances charge to a "neutral" level and finally break-in "ends." This is in addition to any physical material changes due to heat or an applied field.

There has been extensive research done by the French government (and the Germans to a certain extent) on this effect as it relates to high-power transmission lines. It is easier to witness when you're dealing with MV instead of mV. (When you stand under a power line, you are quite literally standing inside a capacitor.) They have shown that just the heavy ions in air can have a big impact on the capacitive resonance effects that affect a signal. But they are far from nailing down all the variables in all instances, of course.

Arthur
One way of dealing with any signal is via Fourier analysis. That analysis, along with a host of other signal analysis techniques, benefit greatly from Euler's identity, without which the analysis would be unnecessarily cumbersome. There is simply no good reason to express signals in any other way when engaged in any such analysis.
Arthur has a decent point.
I associate many speaker break-in effects to capacitors forming.
A 'virgin' cap doesn't know anything about electricity until it is in effect shown.

What got me started on this was many years ago buying a high power on-camera flash. Instructions say to fire several full power flashes when first turned on........than your good to go.

As a further aside, when i first turned on my new speakers.....some Magnepans, the image floated around quite a bit for a couple hours. Sometimes quite rapidly...flicking back and forth. Than it settled down and has been fine ever since.
Hi Al,

I'm not certain, but I believe that the results of differentiating and integrating a sine or cosine function would be much less "clean" and easy to work with if degrees or cycles were used instead of radians.
Maybe numerically, but only because the function values based on radians are readily available.

Arthur,

The reason radians are used is because the derivative of sin(x) is directly cos(x) when using radians.
The derivative of sin(x) is cos(x) regardless of the units of x. The derivation based on limits doesn't indicate any relevance to the units of x.

Radians is one of those unitless units like percent. Sine and cosine are defined in rectangular coordinates as the ratio of the lengths of two vectors on a unit circle. The ratio has no units yet they are noted as radians.
Hi Al,
Just take any voltage level on one of the waveforms, such as a zero crossing or a peak level, and find the corresponding point on the other waveform. The two points will be pi/2 radians or 90 degrees apart, with one occurring first.
Where occurring first means x1 is less than x2 where cos(x1) = sin(x2). I'm on the right page now. Thanks.

Ok, so the math works out. Statements from your earlier post...
An inductor resists abrupt changes in current...
Capacitors resist abrupt changes in voltage
pretty much leads to the "voltage leading current" for an inductor and "voltage lagging current" for a capacitor.

Physically, what's the explanation for this? That's what I was curious about initially.
Bob: It goes back to basic em fields. As to inductance the motion of a charge (which is current) causes a magnetic field. If the current is varying with time the magnetic field will vary wtih time. A time varying magnetic field causes a voltage in any conductor that is linked by that field. "Inductance" is by definition the circuit paramater that relates the voltage just described with the current that caused the voltage. There is a somewhat analogous explanation for capacitance, with the difference being that field is not a magnetic field but rather an electric field, the cause of which is a separation of charge. Remember that a "circuit" can consist of a wire. We construct specific elements to control the relationship which is inductance or capacitance.
Hi Bob,

In the case of a capacitor, the voltage between the two plates or conductors is proportional to the charge that has accumulated on them, the charges being equal but of opposite polarity on the two plates. Charge accumulates as a result of current flow between each plate and the circuit the capacitor is connected to. Therefore the charge and hence the voltage across the capacitor are proportional to the integral of current. Therefore (assuming an idealized capacitor model) an infinitely fast step increase in current will result in a gradual linear ramp-up of voltage.

In the case of an inductor, a change in current results in a change of magnetic flux, which results in a voltage being induced that opposes the change in current. See these writeups on Faraday's Law and Lenz's Law. Therefore, voltage is proportional to the rate of change (the derivative) of current; therefore current is proportional to the integral of voltage, and so a step increase in applied voltage will result in a gradual linear ramp-up of current.

Best regards,
-- Al
Musicnoise and Al, thanks for the posts. I'll read the links. I recall some of the aspects of inductance and parallel plate capacitors, but not the phase difference between voltage and current and why it physically occurs. My sophomore physics and calculus texts were given to the local library long ago.
Bob: While reading the links keep in mind that the 'phase' difference is a way of describing the special case of the transfer function for a system when the input a sine wave and where the transients have died out (steady state).
Bob - the difference between the derivative in radians and degress is the scalar multiplier of k*pi. The key thing to note in my description is the word "directly". I found this description that should go into enough detail for you to understand what I was saying:

http://mathforum.org/library/drmath/view/54181.html

As for lead and lag, you can do a search for "ELI the ICE man" and the first few links that pop up offer good explanations.

Arthur
As to inductance the motion of a charge (which is current) causes a magnetic field.

Agreed. And the magnetic field can be shown to be a relativistic effect (Einstein - special relativity) of the pure electrical field. It all depends on your frame of reference.
To dumb it down.

Resistance(R) => DC circuits design.
Impedance(Z) => AC circuits design.

Impedance includes resistance(R) + capacitance(C) OR - inductance(L). (BUT you cant have C and L at the same time)
BUT you cant have C and L at the same time
Actually, that's not correct. A real-world (physical) capacitor will have some small amount of stray inductance, and a real world inductor will have some small amount of stray capacitance. Both will also have some non-zero, finite amount of dc resistance (the resistance of an ideal inductor is zero; the resistance of an ideal capacitor is infinite).

The magnitude and phase angle that define the overall impedance of a combination of resistance, inductance, and capacitance are calculated for a specific frequency as described near the end of my first post in this thread.

Best regards,
-- Al