Before analysing the (possible) effects of bi-wiring we need to establish a suitable simplified model of the loudspeaker. One of the simplest possible arrangements is illustrated in Figure 4.

In reality, any practical speaker will have much more complicated properties. However since we are only concerned with seeing if it is

For convenience we can assume that , where is a conveniently chosen standard resistance value. Each of the crossover/unit combinations will have a turn-over frequency set by the chosen values. This will determine the frequency range within which each part of the loudspeaker is responsible for ensuring signals are audible. Again, for simplicity and convenience we can assume these have been set to both equal the same value, . Hence we can say that

In this specific, simplified, case it turns out that when we calculate it equals at all frequencies. This result will, of course, not be true in general, so we must interpret any results based upon this particular speaker model with care.

In terms of acoustic output we may say that the pressure radiated by each speaker unit will vary in proportion with the current at each instant through its radiation resistance. We may represent the current through the tweeter as and that through the woofer as . Since the radiation will add coherently we may say that the total sound pressure created will be proportional to the vector total current, . The mean power will therefore be proportional to , and any phase changes will depend upon the relative phase of the driving signal and the total current.

When using the loudspeaker modelled here we can now represent the situation when employing a single cable (i.e. not bi-wired) by the circuit shown in Figure 5.

When the amplifier asserts an output (a.c.) voltage of this will result in a voltage, , appearing at the loudspeaker terminals. In this simple case we know our loudspeaker has an input impedance (tweeter and woofer sections linked in parallel) of . Hence we can say that

The circuit is a fairly simple one and has only one current node. This means we can immediately say that

The situation when using a bi-wiring arrangement may be represented by the circuit shown in Figure 6. For simplicity it is assumed that the two cables are identical.

As with the single-cable arrangement, the system illustrated in Figure 6 has a single node. Hence we can say that the power efficiency will simply depend upon how may vary with signal frequency. We may therefore look to see if the total load impedance now varies with frequency as if it does, this implies that the above bi-wired arrangement may be expected have a frequency response that differs from that produced with a single cable.