In this chapter we have introduced and examined parallel circuits using either one or more current sources or a single voltage source, along with two or more resistors. The hallmark of a parallel configuration is that all components are connected to just two nodes. This means that all elements in a parallel configuration see the same voltage. If multiple current sources are present, they may be combined into a single equivalent current source by adding their values, taking care to watch for the directions of source currents; like directions add while opposing directions subtract, in much the same manner as series-connected voltage sources. Voltage sources generally are not placed in parallel as it would be a practical impossibility to establish different voltages across the same two nodes. The major exception to this rule is if the sources use the same voltage and the goal is to extend battery life. Dissimilar parallel voltage sources will most likely cause excessive heat and an unstable voltage.

The equivalent resistance of a group of parallel resistors will always be smaller than the smallest resistor in the group because each resistor adds another path for current flow, thus enhancing conductivity and reducing resistance. In general, the effective resistance is found by summing the individual conductances to find the total conductance of the group and then taking the reciprocal of this value. Two shortcut techniques when just two resistors are involved are the product-sum rule and the ratio rule (which is particularly handy when one resistor is the integer multiple of the other).

Kirchhoff's current law (KCL) states that the sum of currents entering a node must equal the sum of currents leaving that node. Another way of stating this is that the sum of all currents entering and exiting a node must be zero (obeying polarities). The current divider rule (CDR) is useful for determining the division of current between two parallel resistors. It states that the current through one resistor must equal the current entering the pair times the ratio of the other resistance value to the summed resistance of the pair.

To determine individual branch currents in a circuit driven by a voltage source, Ohm's law may be used by dividing the source voltage by the individual resistor values. These branch currents must sum to the total current delivered by the voltage source due to KCL. If the parallel network is driven by current sources, the individual branch currents can be found by determining the effective parallel resistance and then using Ohm's law to find the system voltage. Once the voltage is known, Ohm's law is used again on each resistor to find the associated branch current. Alternately, the resistors can be simplified into successive pairings and then CDR can be applied repeatedly.

Real world ammeters and voltmeters exhibit internal resistances that can load down circuit elements and cause errors in measurement. It is best if voltmeters have an internal resistance that is lat east an order of magnitude (preferably two) larger than the resistors they are placed across in order to limit undesired current divider effects. Similarly, ammeters should have an internal resistance at least an order of magnitude (preferably two) smaller than the resistors with which they are placed in-line to limit undesired Ohm's law effects.

Fuses and circuit breakers are used to limit potentially damaging current in a system. Both devices will activate by opening the circuit if the current passing through them exceeds their rated value for a specified period of time. This results in zero current flow. Fuses are one-shot devices that must be replaced after being activated. In contrast, circuit breakers can be activated numerous times and are simply reset via a switch or button after being tripped.

## Review Questions

1. Define the term parallel connection.

2. How is the equivalent resistance for a group of parallel connected resistors computed?

3. How is the equivalent value for parallel connected current sources computed?

4. Why are voltage sources generally not placed in parallel?

5. Define the product-sum rule and its use.

6. Define Kirchhoff's current law.

7. Define the current divider rule.

8. Explain the basic operation and usage of fuses and circuit breakers.