# Basic Circuits

It is common to start a discussion of electronics by comparing the flow of electricity (current) to the flow of water. In this scenario a battery or other power source can be considered a pump which raises the level of the water, and any devices which expend energy can be considered slopes along which the water descends. This is demonstrated in the figure above. Like all analogies it breaks down eventually, but imagining how current will flow through your circuit is important, so I believe it is a good place to start.

In circuits there are three physical parameters which have a good analogs in our model of flowing water. There is voltage (\(V\)), or electric potential, represented by the height of the water above some arbitrary point which we will call the ground. There is resistance (\(R\)), represented by the downward slope of the water channel. Finally, there is current (\(I\)) which is represented by the amount of water flowing along a point in the channel. In circuits current flows from higher potential to lower potential encountering resistance to flow along the way. The relationship between voltage, current, and resistance is expressed by the equation \[ V = IR. \] Voltage is measured in volts (V), resistance is measured in Ohms (Ω), and current is measured in amps (A). 1 volt is defined as 1 amp times 1 ohm. So for example if you have a 5 volt potential across a 100 Ω resistor, you can get the current that will flow across that resistor as \[ I = \frac{V}{R} = \frac{5~\mathrm{V}}{100~\mathrm{Ω}} = 50~\mathrm{mA}. \]

A circuit consists of one or more loops, called a complete circuit, which flows from a battery or other power source through the circuitry and then back to the power source. On any circuit loop, the current flowing through that loop is constant along the loop, unless the loop splits into multiple paths. We will cover how current splits into multiple paths later, but right now the current through a circuit at all points within the circuit is given by the above equation. It is important to know the approximate current running through a circuit as components and wiring have current limits which cannot be exceeded without sometimes explosive consequences. The limits of components can be found in their data sheets, but for the wiring it depends on the size of the wire (expressed as a gauge, with smaller numbers being larger diameter wires). A quick and very conservative guide to current limits for copper wire is shown below. Long wire runs and wires made of different metals may have more restrictive current limits.

AWG gauge | Practical Amp Limit |
---|---|

10 | 30 |

12 | 20 |

14 | 15 |

18 | 10 |

22 | 5 |

24 | 2 |

30 | 1 |

## Analyzing Circuits

When two resistive loads exist in series along a wire, as in the figure above, the total resistance is just the sum of the two resistances. If we consider the two resistances to be \(R_{1}\) and \(R_{2}\), then the total resistance is just \(R_{1}+R_{2}\), and the current which flows through the system is \[ I = \frac{V}{R_{1}+R_{2}}. \]

When two loads are in parallel the current splits into two directions, and figuring out the currents in each part of the circuit is more complex. The important principle to understand in analyzing these circuits is that the total current coming from the battery must be equal to the sum of the currents running through each path. In the figure below, the total current coming out of the battery (on the left with the + and - on each side), would be equal to the sum of the currents through \(R_{1}\) and \(R_{2}\).

The other principle to remember is all portions of a circuit which are tied together only by conductors (wires) have the same electric potential (voltage). So the potential across both loads is equal to the voltage of the battery. Given this we can find the current in each leg of the circuit to be: \[ I_{1} = \frac{V}{R_{1}}, \] and \[ I_{2} = \frac{V}{R_{2}}. \] Therefor the total current coming from the battery would be \[ I = I_{1}+I_{2} = V\left(\frac{1}{R_{1}} + \frac{1}{R_{2}}\right). \] Additional analysis of circuits involves using the equation \(V=RI\) along each path and load, and this holds however complex the circuit becomes.

## Power

Another quantity which is often used in circuits is the power, measured in watts. Power is a measure of energy produced or used per unit time, so watts is joules per second. The equation for power (\(P\)) is \[ P = VI, \] where \(P\) is watts, \(V\) is voltage difference, and \(I\) is current. Combining this with the previous equation, we can get the power dissipated in a resistor is \[ P = VI = (IR)I = I^{2}R. \] An important note is that in many passive components the energy is converted to heat, which means that a component like a resistor will get hotter as the power dissipated by it increases. Most components will indicate how much power they can take, but in some cases it may be important to provide heat sinks to help dissipate the heat and keep the component from getting too hot.