The difference between impedance and resistance

"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

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

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

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

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