[Phys-l] impedance matching (was: stereo speakers)

[John Denker <jsd@av8n.com>]

On 03/09/2009 05:17 PM, Anthony Lapinski wrote:
> I was discussing Ohm's law today, and a few students were asking about
> stereo speakers and their resistance. Textbooks do few problems involving
> speakers. From my understanding, a typical speaker is 8 ohms. This will
> get a certain amount of current and voltage, providing a power (P = IV) to
> the speakers. If one replaces the speaker with one that has a lower
> resistance, does this mean that the current (and power) will increase for
> a given voltage? Or is it that for a given power (volume), the current
> will increase while the voltage will decrease (since R is now less)? And
> the implication is that when the volume is turned up (P increases), an
> amplifier must have a better power rating to supply the sound if the
> speaker resistance is lower.

It depends.  See below.

Rule #1:  Draw the circuit diagram. 

 Here's my diagram:
   http://www.av8n.com/physics/img48/impedance-match.png

We are treating the source as a black box with terminals A and 
B.  The voltage at the terminals of the black box is VAB. The 
source impedances is RS.  The load impedance is RL.  All circuit
elements are linear.

Now we can consider some limiting cases:

A) In the case of a low-impedance source, i.e. RS=small compared 
 to the load, then lowering the load impedance increases the 
 current with little if any change in VAB.  Power (i.e. power 
 delivered to the load) goes up.

B) In the case of a high-impedance source, i.e. RS=large compared
 to the source, then lowering the load impedance decreases the
 terminal voltage with little if any change in the current. 
 Power goes down.

C) In the case where the load resistance is equal to the source
 resistance, then to first order changing the load resistance
 makes no change in the power.  Larger changes in the load
 resistance -- in either direction -- will decrease the power.

 This is a well-known and very useful result:  given a linear
 circuit with fixed source impedance, if you want to maximize
 the amount of power you get out of it, the best strategy is
 _impedance matching_.

Homework:  Prove this.

Extra credit:  Consider the case where the source impedance
is somewhat reactive (i.e. capacitive or inductive) not just
purely resistive.

Extra credit:  Derive the Norton-equivalent result, i.e. all
circuit elements in parallel, as opposed to all in series.