7.5. BUILDING SERIES-PARALLEL RESISTOR CIRCUITS221voltage and resistance have decreased, but without knowing how mucheach one has changed,we can’t use the I=E/R formula to qualitatively determine the resulting change in current.However, we can still apply the rules of series and parallel circuits horizontally. We know thatthe current through the R1//R2 parallel combination has increased, and we also know that thecurrent through R1 has decreased. One of the rules of parallel circuits is that total current isequal to the sum of the individual branch currents. In this case, the current through R1//R2 isequal to the current through R1 added to the current through R2. If current through R1//R2 hasincreased while current through R1 has decreased, current through R2 musthave increased:EIRVoltsAmpsOhmsR1R2R3TotalR4R1 // R2R3 // R4samesamesamesamesameAnd with that, our table of qualitative values stands completed. This particular exercisemay look laborious due to all the detailed commentary, but the actual process can be performedvery quickly with some practice. An important thing to realize here is that the general proce-dure is little different from quantitative analysis: start with the known values, then proceedto determining total resistance, then total current, then transfer ﬁgures of voltage and currentas allowed by the rules of series and parallel circuits to the appropriate columns.A few general rules can be memorized to assist and/or to check your progress when proceed-ing with such an analysis:• For any singlecomponent failure (open or shorted), the total resistance will always changein the same direction (either increase or decrease) as the resistance change of the failedcomponent.• When a component fails shorted, its resistance always decreases. Also, the currentthrough it will increase, and the voltage across it maydrop. I say ”may” because insome cases it will remain the same (case in point: a simple parallel circuit with an idealpower source).• When a component fails open, its resistance always increases. The current through thatcomponent will decrease to zero, because it is an incomplete electrical path (no continu-ity). This mayresult in an increase of voltage across it. The same exception stated aboveapplies here as well: in a simple parallel circuit with an ideal voltage source, the voltageacross an open-failed component will remain unchanged.7.5Building series-parallel resistor circuitsOnce again, when building battery/resistor circuits, the student or hobbyist is faced with sev-eral different modes of construction. Perhaps the most popular is the solderless breadboard: aplatform for constructing temporary circuits by plugging components and wires into a grid ofinterconnected points. A breadboard appears to be nothing but a plastic frame with hundredsof small holes in it. Underneath each hole, though, is a spring clip which connects to otherspring clips beneath other holes. The connection pattern between holes is simple and uniform: