Linear Algebra

70 Chapter One. Linear SystemsComponents with this property, that the voltage-amperage response curve is aline through the origin, are resistors. For example, if a resistor measures 2 ohmsthen wiring it to a 12 volt battery results in a flow of 6 amperes. Conversely, ifelectrical current of 2 amperes flows through that resistor then there must bea 4 volt potential difference between it’s ends. This is the voltage drop acrossthe resistor. One way to think of the electrical circuits that we consider here isthat the battery provides a voltage rise while the other components are voltagedrops.The two facts that we need about networks are Kirchhoff’s Laws.Current Law. For any point in a network, the flow in equals the flow out.Voltage Law. Around any circuit the total drop equals the total rise.We start with the network below. It has a battery that provides the potentialto flow and three resistors, drawn as zig-zags. When components are wired oneafter another, as here, they are in series.20 voltpotential2 ohmresistance3 ohmresistance5 ohmresistanceBy Kirchhoff’s Voltage Law, because the voltage rise is 20 volts, the total voltagedrop must also be 20 volts. Since the resistance from start to finish is 10 ohms(the resistance of the wire connecting the components is negligible), the currentis (20/10) = 2 amperes. Now, by Kirchhoff’s Current Law, there are 2 amperesthrough each resistor. Therefore the voltage drops are: 4 volts across the 2 oh mresistor, 10 volts across the 5 ohm resistor, and 6 volts across the 3 ohm resistor.The prior network is simple enough that we didn’t use a linear system butthe next one is more complicated. Here the resistors are in parallel.20 volt 12 ohm 8 ohmWe begin by labeling the branches as below. Let the current through the leftbranch of the parallel portion be i 1 and that through the right branch be i 2 ,and also let the current through the battery be i 0 . Note that we don’t need toknow the actual direction of flow — if current flows in the direction opposite toour arrow then we will get a negative number in the solution.