The difference between impedance and resistance

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 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