Resistance / Frequency relationship

[DruidArmy]

Hi,

How are the resistance of a conductor and frequency of an AC current related? If frequency goes up, does resistance go down or vice versa? Is this relationship constant for all types of material? Are some materials more frequency sensitive than others?

For instance in an incandescent light, if we change the frequency of AC can we make it brighter or dimmer?

Thanks.

 

[Redbelly98]

Resistance and frequency are not related.

 

[chroot]

That's not true, Redbelly98.

The relationships are extremely complex, but the resistance of even something as simple as a length of wire varies with frequency.

The term 'impedance' is usually used in this context, rather than 'resistance.' The impedance seen by a line driver driving a wire at 1 MHz is different from the impedance seen by a line driver at 1 GHz, for example. The reason? Real wires have some capacitance (and some inductance). A thorough understanding of a real wire also has to include some pretty complicated effects, like the skin effect.

The bottom line, in general, is that resistance of a length of wire goes up with frequency. In other words, a length of wire will attenuate high frequencies much more than it will low frequencies.

- Warren

 

[dx]

Impedance and resistance are different things, no? Isn't resistance (resistivity to be precise) a property of the material?

 

[chroot]

Impedance is a generalized term for resistance. Impedance includes reactance (capacitance and inductance), whereas resistance does not.

Capacitance and inductance are represented with complex numbers; resistances are always real, while impedances can be complex. When a specific frequency is given, however, a complex impedance can be evaluated at that frequency, producing a real number. That real number is the resistance of the system as seen by a signal of that frequency.

- Warren

 

[Bob S]

At higher frequencies (above about 1 MHz) an ac current is confined to a depth, called the skin depth, on the surface of conductors. The depth varies inversely as the square root of frequency, so the resistance increases as the square root of frequency. See
http://en.wikipedia.org/wiki/Skin_effect
Also see Jackson Classical Electrodynamics 2nd Ed. Section 8.1.
Litz wire, made up of many small conductors, is sometimes used at high frequencies, because using many small conductors increases the amount of surface area for a given total cross section of copper wire, and lowers the effective wire resistance.

 

[Gnosis]

Although the OP presented his question per the “resistance” of a conductor, once an AC voltage is introduced, a circuit must be viewed from the standpoint of it being an “impedance” (essentially an “equivalent resistance”).

Since he specifically referred to a “conductor”, the inductor will posses an inductance (L) value of some magnitude (in “henrys”), in which case, it will oppose changes in current per its inductance and the applied frequency (f), unlike that of a simple carbon resistor.

If it were a simple carbon resistor, the current could be determined simply by dividing the AC’s rms voltage by the resistance value. Voltage rms / resistance = current

However, per applying an AC voltage across a lone inductor, first the capacitive reactance (equivalent resistance) must be calculated,

2pi f L = inductive reactance (in ohms)

where,

f = frequency (in Hz)
L = inductance (in henrys)

Examples:

2pi * (60 Hz) * (2 henrys) = 753.9822369 ohms of inductive reactance

12 VAC / 753.9822369 ohms = .015915494 amperes



2pi * (120 Hz) * (2 henrys) = 1507.964474 ohms of inductive reactance

12 VAC / 1507.964474 ohms = .007957747 amperes


Per the inductive reactance equation, it’s evident that when increasing the frequency (f) of the applied AC voltage, or increasing the inductor’s inductance (L), or both; the inductive reactance increases thereby producing a higher impedance that further limits current flow.

Increasing f and/or L therefore decreases current flow.

 

[Redbelly98]

Warren, I'm aware of the whole complex impedance issue. But the OP asked about resistance, and

... resistances are always real,

... and therefore I wasn't considering capacitive or inductive effects. Or rather I didn't think the OP was asking about these effects. On the other hand, Bob S makes a good point about skin effect.

For instance in an incandescent light, if we change the frequency of AC can we make it brighter or dimmer?

If the frequency is to be anywhere near typical line frequencies of 50 or 60 Hz then no, changing the frequency will not make the light noticeably brighter or dimmer.

If the frequency is high enough that either skin effects or the inductance of the coiled wire plays a significant role in the impedance, then yes. For inductive effects, we'd have to be in the 100 MHz range for this to be noticeable, using the 0.2 μH that this guy measured for a 60W bulb:
https://www.physicsforums.com/showpost.php?p=1928977&postcount=10

Moreover, there are much easier ways to vary the brightness of a light bulb. Building a MHz, variable-frequency source is an impractical way to do this.