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Kirchhoff's Voltage and Current Laws

In Current and Resistance, we described the term ‘resistance’ and explained the basic design of a resistor. Basically, a resistor limits the flow of charge in a circuit and is an ohmic device where V=IR. Most circuits have more than one resistor. If several resistors are connected together and connected to a battery, the current supplied by the battery depends on the equivalent resistance of the circuit. The equivalent resistance of a combination of resistors depends on both their individual values and how they are connected. The simplest combinations of resistors are series and parallel connections (Figure 1011).

Figures (10)

Figure 10.11 (a) For a series connection of resistors, the current is the same in each resistor. (b) For a parallel connection of resistors, the voltage is the same across each resistor.
Figure 10.12 (a) Three resistors connected in series to a voltage source. (b) The original circuit is reduced to an equivalent resistance and a voltage source.
Figure 10.13 A simple series circuit with five resistors.
Figure 10.14 (a) Two resistors connected in parallel to a voltage source. (b) The original circuit is reduced to an equivalent resistance and a voltage source.
Figure 10.15 (a) The original circuit of four resistors. (b) Step 1: The resistors R3R3 and R4R4 are in series and the equivalent resistance is R34=10Ω.R34=10Ω. (c) Step 2: The reduced circuit shows resistors R2R2 and R34R34 are in parallel, with an equivalent resistance of R234=5Ω.R234=5Ω. (d) Step 3: The reduced circuit shows that R1R1 and R234R234 are in series with an equivalent resistance of R1234=12Ω,R1234=12Ω, which is the equivalent resistance Req.Req. (e) The reduced circuit with a voltage source of V=24VV=24V with an equivalent resistance of Req=12Ω.Req=12Ω. This results in a current of I=2AI=2A from the voltage source.
Figure 10.16 These three resistors are connected to a voltage source so that R2R2 and R3R3 are in parallel with one another and that combination is in series with R1.R1.
Figure 10.17 Why do lights dim when a large appliance is switched on? The answer is that the large current the appliance motor draws causes a significant IR drop in the wires and reduces the voltage across the light.
Figure 10.18 To find the unknown voltage, we must first find the equivalent resistance of the circuit.
Figure 21.2 (a) A series connection of resistors. (b) A parallel connection of resistors.
Figure 21.3 Three resistors connected in series to a battery (left) and the equivalent single or series resistance (right).

Key Points

  • A matrix version of Kirchhoff's current law is the basis of most circuit simulation software, such as SPICE.
  • The loop rule is stated in terms of potential V rather than potential energy, but the two are related since U=qV.
  • A junction, also known as a node, is a connection of three or more wires.
  • In a closed loop, whatever energy is supplied by a voltage source, the energy must be transferred into other forms by the devices in the loop, since there are no other ways in which energy can be transferred into or out of the circuit.
  • So the voltage drop across R1 is V1=IR1 , that across R2 is V2=IR2 , and that across R3 is V3=IR3 .
  • The current through the circuit is the same for each resistor in a series circuit and is equal to the applied voltage divided by the equivalent resistance: I=VRS=9V90Ω=0.1A.

Terms

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