The circuits we have looked at so far involve components such as resistances, capacitors, and inductors. Until tensions and currents remain continuous, in other words, have a constant amplitude level, resistance, capacity and inductance values keep constant. However, when these circuits are fed with tensions and alternate currents (fixed-frequency sinusoids, or signals, such a sound signals, which contain various and extended bands of frequencies), the components' values vary when frequencies vary. This means that a circuit will respond differently to different frequencies. Limiting ourselves to our three components R, L, C, we can begin to introduce Ohm's Law on a general level, with the following formula:
Equation 4.11. Generalized Ohm's Law
This equation is saying that all involved quantities depend upon frequency. In particular the Z(f) value, measures impedance, in other words, the resistance quantity and overall reactances within the entire circuit. These quantities being variable, they can't be described by a simple constant value, but rather, they shall be represented on a graph which will show the value of all the frequency values of the signals involved in the circuit. Actually, all these quantities are described in two graphs, one pertaining to amplitude (indicated by the letter A), and one relating to phase (indicated by the letter Fi). We will now give an example which illustrates on a practical level all the concepts we have been exploring.
Let's consider a high-pass filter, which, as we have seen, requires the use of a capacitor. Seeing that a loudspeaker can be seen from the circuit's point of view as a resistance (to be more precise it is experienced as an impedance, but in this case we'll exclude the reactance part). So, the high-pass filter will have the following diagram:
The impedance of this circuit shall be give by the following formula:
Equation 4.12. Impedance of a high-pass filter
Where Rc is the resistance part of the capacitor. Through calculation (which we won't show here, in that they require mathematical knowledge of imaginary numbers) we can calculate the amplitude and phase of quantity Z when frequency varies. More than the calculation itself what we're interested in is the rate of the two curves and their meaning. A general filter could have the following curves for amplitude and phase:
Diagrams of the amplitude and phase of a high-pass filter
Amplitude diagram: seeing that in a high-pass filter all minor frequencies of the cut frequency (in our case 440 Hz) get eliminated by the signal, this means that the impedance at such frequencies is too high and this blocks a signal from passing. Above 440 Hz we have a gain of 0dB, in other words, no impedance which means that above the cut frequency all amplitudes remain unaltered.
Phase diagram: This diagram shows the phase displacement between the two quantities bound by the impedance. In our case the tension V(f) of the circuit and the current I(f) that flow through the components.
Phase is a very important factor, yet often neglected, in sound engineering, in that it can bring along rather evident undesired effects. Generally speaking we'd desire a flat-phase 0 degrees diagram- all quantities are in phase, and we have no problems whatsoever. However this is impossible in that the circuit's components each introduce a different displacement to each different frequency. There are however, highly sophisticated mathematical means to project circuits with amplitude rates and desired phase.
Related topics on Wikipedia
High-pass filter