6.8 : Impedance Combination

Consider a string of christmas lights, each bulb symbolizing an impedance element. In this series configuration, the flow of electric current remains uniform across every component. This behavior aligns with Kirchhoff's Voltage Law (KVL), which asserts that the total impedance in such a setup equals the sum of individual impedances—akin to resistors in series. It follows that the voltage from the power source is distributed proportionally among these components, adhering to the voltage division principle.

However, the drawback of this series connection is evident when a single bulb fails, causing an open circuit that interrupts the entire current flow. christmas lights are typically arranged in a parallel configuration to ensure a continuous and steady power supply. This setup guarantees a constant voltage across each bulb, as per Kirchhoff's Current Law (KCL), where the reciprocal of the equivalent impedance equates to the sum of the reciprocals of individual impedances—similar to resistors in parallel. It follows that the equivalent admittance is the sum of the individual admittances.

In this parallel arrangement, the source current divides inversely based on the impedances of the individual bulbs, exemplifying the current division principle. Notably, each bulb establishes an independent pathway to the power source, enabling isolated and uninterrupted current flow.

Furthermore, in more complex circuits with both series and parallel impedances, the delta-to-wye and wye-to-delta transformations can also be employed for impedance circuits, offering valuable circuit analysis and design tools. These transformations facilitate the conversion between different impedance configurations, enhancing the versatility of impedance-based circuits.