The mathematical development of the Virtual Impedance concept starts with the well-known formulas for the series and parallel interconnection of two loads with complex impedances Z1 and Z2, respectively:
In the idealized case where the two impedances are perfectly identical:
For this idealized case, substitution of 3) into 1) and 2) yields, respectively:
For this idealized case, it therefore follows that:
From 4)-6) we see that, for identical load impedances, the impedances of the loads, their aggregate series interconnection, and their aggregate parallel interconnection are all simple scalar multiples of one another. The implication is that there is no essential difference in the dynamic responses of the loads, their parallel interconnection or their series interconnection, i.e. the frequency response phase curves are all identical, and the frequency response magnitude curves are identical in shape, differing only by a fixed scalar multiple which is constant across all frequencies.
For this reason, the decision to use multiple loads, and whether to connect them in series or parallel with the source, has historically been somewhat arbitrary as regards considerations of frequency response and dynamical behavior, given that, for identical loads, there is no essential difference in the frequency responses or dynamical behaviors. Specifically, the decision to use multiple speakers in guitar amplifiers and speaker cabinets used with guitar amplifiers is typically based on power handling and volume considerations. Further, the speakers are typically “hard-wired” somewhat arbitrarily in either fixed series or parallel interconnections. As the speakers are typically the same brand and model (i.e. apparently identical), there has been no reason to believe that any of these choices impact dynamical behavior or frequency response. Further, even when different brands and/or models of speakers are used, there has been no reason to prefer one of series or parallel interconnection based on considerations of frequency response and dynamical behavior.
The above notwithstanding, we have discovered that even apparently identical loads have impedances that necessarily differ from one another due to manufacturing tolerance variations in the component manufacturing processes and the final assembly process. In addition, different rates of in-service deterioration and aging can also cause differences in the load impedances. This discovery is universal, applying across all types of apparently identical loads. In particular, in guitar amplifiers and speaker cabinets used with guitar amplifiers, even in the case of identical make/model speakers, the frequency response characteristics of the speakers will differ, corresponding to differences in dynamical behavior and impedance.
Of course, the impedances of the loads may also differ intentionally in some applications. In particular, in guitar amplifiers and speaker cabinets used with guitar amplifiers, sometimes different make/model speakers are used for certain reasons, such as having the characteristics of one speaker compliment those of the other in a “hard-wired” arrangement; for example, to achieve a “choir” like effect. Further, in this example, one of the loads may correspond to a “dummy load”, or power resistor, used to dissipate power to reduce volume, or may also correspond to a headphone set or recording equipment, while the other load may correspond to a loudspeaker, corresponding to dramatically different load impedances.
Whether unintentional or intentional, the difference in load impedances can be exploited through a switching device to obtain great variety in frequency response and dynamical behavior. To fully develop this idea, we now return to the mathematics to consider the case of non-identical load impedances. To analyze the case where the load impedances are not identical we first define the difference in the two load impedances:
Solving equations 1) and 7) for Z1 and Z2 yields:
Substituting 8) and 9) into 2) yields the following expression:
Rearranging 10) yields:
Defining now the virtual impedance as:
We can recast 11) as:
Note the striking similarity of 13) to 6). Note also that 13) yields 6) only when Zv = 0, which is only achieved when Z1 = Z2 , corresponding to the idealized, practically unachievable case of perfectly identical loads. When the loads are not identical, Zv is not zero, and the aggregate parallel and series impedances are no longer simple scalar multiples of either each other or the individual load impedances. Hence, in the case of non-identical loads, the individual loads, their parallel interconnection and their series interconnection exhibit distinctly different frequency responses and dynamical behaviors.
Referring to 1), we see that the left-hand side of 13) can be interpreted as the aggregate impedance of the series interconnection of the virtual impedance with the aggregate series impedance of the two physical loads. Accordingly, we can represent the left-hand side of 13) by the following diagram:
Now, the right-hand side of 13) involves the parallel impedance of the two physical loads, which can be depicted as follows:
Recognizing that the fixed scalar multiple of four in 13) doesn’t alter the essential character of the frequency response or dynamical behavior of the right-hand side of 13), we can interpret 13) as saying that the frequency responses and dynamical behaviors of the aggregate impedances represented in the two diagrams above are essentially identical.
In effect, the parallel interconnection of the two physical loads is equivalent to adding a third “virtual” load – in series - to the series interconnection of the two physical loads, and this third “virtual” load is zero only in the practically unachievable case where the two physical loads are identical. Thus, the virtual impedance makes clear the essential difference in character between the frequency responses and dynamical behaviors of the individual physical loads, their parallel interconnection, and their series interconnection, in the practically ever-present case of non-identical physical loads.
The discovery and development of the virtual impedance makes clear that switching the source between driving one load, the other (necessarily different) load, both loads in series, or both loads in parallel, or some subset thereof, will result in different system frequency responses and dynamical behaviors. A system augmented with such a switching device can thus be selectively switched (e.g., by its user) between frequency responses and dynamical behaviors to tailor its operation to a given situation, providing much greater flexibility, utility and/or capability to the system. Clearly, a switching device that switches only between even any two of the four modes described above will provide greater flexibility, utility and/or capability to the system. It is also clear to one skilled in the art that this analysis and approach can be readily extended to encompass the situation of more than two physical loads (possibly also involving multiple sources), since the interconnection of the multiple physical loads can be viewed as a sequence of nested parallel and/or series interconnections of two (aggregate) loads.
The discovery and characterization of the impedance differences inherent in apparently identical loads, the related discovery and characterization of virtual impedance, and the related discovery of the utility of switching devices can be exploited in many different ways across many classes of physical systems in a wide range of applications.
A preferred embodiment of the present invention enables switching between all four of the modes previously described (load 1 alone, load 2 alone, both loads in parallel, both loads in series). Other embodiments switch between only a subset of these four modes; specifically, between only two and also three of the modes. However, the invention is not limited to the embodiments thus described, as other related embodiments will be obvious to one skilled in the art.
Extensions to the case of more than two (aggregate) loads and switching between them and or different combinations of nested parallel and/or series interconnections of them will be obvious to one skilled in the art.
The loads under consideration may be apparently identical or purposefully different. In the case of purposefully different loads, the desired level and character of the differences between the load impedances can be specified by design to accentuate the variations in the resulting frequency response characteristics and dynamical behaviors selectively enabled by the switching device of the present invention. For example, in the case of guitar amplifiers and speaker cabinets, one of the loads may correspond to a “dummy load”, or power resistor, used to dissipate power to reduce volume, or may also correspond to a headphone set or recording equipment, while the other load may correspond to a loudspeaker, corresponding to dramatically different load impedances. Further, it is recognized that the load impedances considered may each be the aggregate impedance of arbitrarily complex networks of interconnected impedances of similar or vastly different types, providing further utility in applying the invention.
