Electric Current in Complex Circuits

Introduction

Outline

• Current, Resistors, and Kirchhoff's Laws
  • DC Circuits and Resistance
  • Series and Parallel Circuits
  • Kirchhoff's Rules
• Practice with the Concepts
• Discussion Points

Current, Resistors, and Kirchhoff's Laws

Charges moving through circuits chose the path of least resistance, so at a junction, more current will flow into the path with less resistance. We add total resistance for a section of circuit differently if the circuits are in parallel than if they are in series, and the same holds true for adding total capacitance.

DC Circuits and Resistance

Direct current (DC) circuits are circuits where the electron flow is always in one direction. This is the current supplied by batteries or other chemical cells. By convention, current in circuit diagrams is always shown as positive flow — that is, in the direction opposite the actual flow of electrons. We create circuit schematics using standard symbols to represent power sources, resistors (any load where the electrical current does work), capacitors, and other components such as diodes.

The simplest circuit contains a voltage source and a load (resistor):

The current source has a positive and negative terminals at locations A (postive) and B (negative). The difference in voltage between the two sources V_A-V_B is the ΔV or voltage V_AB available to drive the electrical current. In a real battery, the available potential difference produced as V_AB is the total output of the source less any internal resistance within the device. For historical reasons, this total ouput is called the electromotive force, although it is really an energy potential and not a force at all. We use the "emf" symbol ℰ for this potential, so:

V_A = ℰ - Ir_internal

where I is the current on the circuit between these two points. If the source potential is given as the emf ℰ, we need to consider the internal resistance of the battery. If the source potential is given as the net voltage V, we can assume that the internal resistance has already been accounted for in the representation of the source.

We can put a switch into the circuit; if the switch is open, it breaks the circuit, since the electrons cannot continue their movement across this gap.

Series and Parallel Circuits

We can arrange the components of a circuit in whatever we we like, depending on our goals for the circuit. A circuit (or part of a circuit) with a single continuous loop is a series circuit. A circuit which branches is a parallel circuit. Because the current must split up in a parallel circuit, the effects of loads on the circuit is different.

• In a series circuit, all of the components lie on the same loop, and the available current must pass through each component.
• In a parallel circuit, components lie on multiple loops, each of which carries only part of the original current output from the ℰ source.
• A complex circuit will involve multiple parallel loops that can be treated individually as series loops for analysis.

Kirchhoff's rules

The current through each part of the circuit depends the potencial difference across the load. In a series circuit, the current depends on the total resistance, as though only a single resistor existed in the circuit. For the series circuit above, I = V/R_total = V/(R₁ + R₂ + R₃) In a parallel circuit, the current must divide at the junctions, so that some current flows through each part of the circuit. The current going through each loop of the circuit depends on the load at that point: I₁ = V/R₁, I₂ = V/R₂, I₃ = V/R₃. The individual currents through each loop must add up to the initial current produced by the ℰ source: I₀ = I₁ + I₂ + I₃, since electrons cannot be created or destroyed. The total resistance is related to the individual resistances as 1/R_total = 1/R₁ + 1/R₂ + 1/R₃.

A complex circuit may involve resisters or other components in parallel for part of the circuit and in series for part of the circuit. To analyze a circuit, we use Kirchhoff's rules.

• Junctions: the sum of all the currents entering a junction must equal the sum of all the currents leaving the junction.
• Conservation of energy: the sum of changes in potential around a closed loop must be zero.

We can analyze the total resistance in the circuit by determining equivalent resistances for the parallel loops and the series loops. The actual order of loop resolution depends on the circuit.

Practice with the Concepts

Discussion Points

• Discuss the advantages and disadvantages of Christmastree lights connected in parallel versus those connected in series.
• With two identical lightbulbs and two identical batteries, explain how and why you would arrange the bulbs and bat- teries in a circuit to get the maximum possible total power to the lightbulbs. (Ignore internal resistance of batteries.)

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