Multiple Voltage Sources

Multiple Voltage Sources

There are two voltage sources when a battery charger is used. Voltage sources connected in series are relatively simple. When voltage sources are in series, their internal resistances add and their emfs add algebraically. (See this figure.) Series connections of voltage sources are common—for example, in flashlights, toys, and other appliances. Usually, the cells are in series in order to produce a larger total emf.

But if the cells oppose one another, such as when one is put into an appliance backward, the total emf is less, since it is the algebraic sum of the individual emfs.

A battery is a multiple connection of voltaic cells, as shown in this figure. The disadvantage of series connections of cells is that their internal resistances add. One of the authors once owned a 1957 MGA that had two 6-V batteries in series, rather than a single 12-V battery. This arrangement produced a large internal resistance that caused him many problems in starting the engine.

A series connection of two voltage sources. The emfs (each labeled with a script E) and internal resistances add, giving a total emf of \({\text{emf}}_{1}+{\text{emf}}_{2}\) and a total internal resistance of \({r}_{1}+{r}_{2}\).

Batteries are multiple connections of individual cells, as shown in this modern rendition of an old print. Single cells, such as AA or C cells, are commonly called batteries, although this is technically incorrect.

If the series connection of two voltage sources is made into a complete circuit with the emfs in opposition, then a current of magnitude \(I=\cfrac{({\text{emf}}_{1}–{\text{emf}}_{2})}{{r}_{1}+\phantom{\rule{0.25em}{0ex}}{r}_{2}}\) flows. See this figure, for example, which shows a circuit exactly analogous to the battery charger discussed above. If two voltage sources in series with emfs in the same sense are connected to a load \({R}_{\text{load}}\), as in this figure, then \(I=\cfrac{({\text{emf}}_{1}+\phantom{\rule{0.25em}{0ex}}{\text{emf}}_{2})}{{r}_{1}+\phantom{\rule{0.25em}{0ex}}{r}_{2}+{R}_{\text{load}}}\) flows.

These two voltage sources are connected in series with their emfs in opposition. Current flows in the direction of the greater emf and is limited to \(I=\cfrac{({\text{emf}}_{1}-\phantom{\rule{0.25em}{0ex}}{\text{emf}}_{2})}{{r}_{1}+\phantom{\rule{0.25em}{0ex}}{r}_{2}}\) by the sum of the internal resistances. (Note that each emf is represented by script E in the figure.) A battery charger connected to a battery is an example of such a connection. The charger must have a larger emf than the battery to reverse current through it.

This schematic represents a flashlight with two cells (voltage sources) and a single bulb (load resistance) in series. The current that flows is \(I=\cfrac{({\text{emf}}_{1}+\phantom{\rule{0.25em}{0ex}}{\text{emf}}_{2})}{{r}_{1}+\phantom{\rule{0.25em}{0ex}}{r}_{2}+\phantom{\rule{0.25em}{0ex}}{R}_{\text{load}}}\). (Note that each emf is represented by script E in the figure.)

This figure shows two voltage sources with identical emfs in parallel and connected to a load resistance. In this simple case, the total emf is the same as the individual emfs. But the total internal resistance is reduced, since the internal resistances are in parallel. The parallel connection thus can produce a larger current.

Here, \(I=\cfrac{\text{emf}}{({r}_{\text{tot}}\phantom{\rule{0.25em}{0ex}}+\phantom{\rule{0.25em}{0ex}}{R}_{\text{load}})}\) flows through the load, and \({r}_{\text{tot}}\) is less than those of the individual batteries. For example, some diesel-powered cars use two 12-V batteries in parallel; they produce a total emf of 12 V but can deliver the larger current needed to start a diesel engine.

Two voltage sources with identical emfs (each labeled by script E) connected in parallel produce the same emf but have a smaller total internal resistance than the individual sources. Parallel combinations are often used to deliver more current. Here \(I=\cfrac{\text{emf}}{({r}_{\text{tot}}\phantom{\rule{0.25em}{0ex}}+\phantom{\rule{0.25em}{0ex}}{R}_{\text{load}})}\) flows through the load.