The circuits we have looked at so far involve components such as resistances, capacitors and inductors. As long as voltages and currents remain continuous, in other words have a constant amplitude level, resistance, capacity and inductance values stay constant. However, when these circuits are fed with alternate voltages and currents (fixed frequency sinusoids, or signals, such a sound signals, which contain various frequencies), the components' values vary when frequency varies. This means that a circuit will respond differently to different frequencies. Keeping to our three components R, L, C, we can introduce the, so called, generalized Ohm's law with the following formula:
Equation 4.11. Generalized Ohm's law
This equation states 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. Seeing that these quantities are variable, they cannot be described by a simple constant value so we will need to use a mathematical function instead. So, we can plot on a graph the impedance function, namely the impedance values at all the frequencies of the applied signals.
In actual fact, all these quantities require two graphs to be fully described, one pertaining to amplitude (indicated by the letter A), and one relating to phase (indicated by the letter Φ). We will now give an example which illustrates on a practical level all the concepts we have just explored.
Let's consider a high-pass filter, which, as we have seen, requires a capacitor. Since, from a circuit's point of view, a loudspeaker can be seen as a resistance (to be more precise it is experienced as an impedance, but in this case we'll exclude the reactance part), the high-pass filter will be described by the circuit illustrated in the following diagram:
The impedance of this circuit shall be given 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 undertake here, because it would require a mathematical knowledge of imaginary numbers) we can calculate the amplitude diagram and the phase diagram related to the impedance, Z(f). More than the calculation itself, what we're interested in is the rate of the two diagrams and their meaning.
A generic high-pass filter circuit could have the following diagrams for amplitude and phase:
Amplitude diagram: seeing that in a high-pass filter all frequencies below the cut-off frequency (in our case 440 Hz) get removed from the signal, this means that the impedance at such frequencies is so high that it blocks the signal from passing. Above 440 Hz we have a gain of 0 dB, in other words no impedance, which means that above the cut-off frequency all amplitudes remain unaltered.
Phase diagram: this diagram shows the phase displacement between the two quantities bound by the impedance, which in our case are: the voltage V(f) of the circuit and the current I(f) that flows through the components.
Phase is a very important, yet often neglected factor in sound engineering, in that it can generate rather evident undesired effects. Generally speaking we'd want a diagram with a flat phase at 0 degrees. This would mean that quantities are in phase, and we'd have no problems whatsoever. However, this is impossible because the circuit's components each introduce a different displacement at each different frequency. However, highly sophisticated systems are employed to design circuits with the desired amplitude and phase rates.