Resistors in Circuits

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© 1998-2008 by Glenn Elert -- A Work in Progress
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Discussion

introduction

series

IT	=	I1	=	I2	=	I3	=	…
VT	=	V1	+	V2	+	V3	+	…
ITRT	=	I1R1	+	I2R2	+	I3VR3	+	…
RT	=	R1	+	R2	+	R3	+	…

parallel

VT	=	V1	=	V2	=	V3	=	…
IT	=	I1	+	I2	+	I3	+	…
VT	=	V1	+	V2	+	V3	+	…
RT		R1		R2		R3
1	=	1	+	1	+	1	+	…
RT		R1		R2		R3

Summary

• bullet

Problems

practice

1. Determine the following quantities for each of the two circuits shown on the right …

   the equivalent resistance, the total current from the power supply, the current through each resistor, the voltage drop across each resistor, and the power dissipated in each resistor.	a.
   	b.

   Solutions …
   1. Follow the rules for series circuits.
      1. Resistances in series add up.

         RT =	R1	+	R2	+	R3
         RT =	20 Ω	+	30 Ω	+	50 Ω
         RT =	100 Ω

      2. Total current is determined by the voltage of the power supply and the equivalent resistance of the circuit.

         IT = VT/RT = 125 V/100 Ω = 1.25 A

      3. Current is constant through resistors in series.

         IT = I1 = I2 = I3 = 1.25 A

      4. The voltage drops can be found using Ohm's law.

         V1 = I1R1 = (1.25 A)(20 Ω) = 25.0 V

         V2 = I2R2 = (1.25 A)(30 Ω) = 37.5 V

         V3 = I3R3 = (1.25 A)(50 Ω) = 62.5 V

         Verify your calculations by adding the voltage drops. On a series circuit they should equal the voltage increase of the power supply.

         VT	=	V1	+	V2	+	V3
         125 V	=	25.0 V	+	37.5 V	+	62.5 V
         125 V	=	125 V

         We're good, so let's finish.
      5. There are three formulas for determining power. Since we have three resistors, let's apply a different formula to each as an exercise.

         P1	=	V1 I1	=	(25.0 V)(1.25 A)	= 31.250 W
         P2	=	I22R2	=	(1.25 A)2(30 Ω)	= 46.875 W
         P3	=	V32/R3	=	(62.5 V)2/(50 Ω)	= 78.125 W

         In a series circuit, the element with the greatest resistance consumes the most power.
   2. Follow the rules for parallel circuits.
      1. Resistances in parallel combine according to the sum-of-inverses rule.

         1	=	1	+	1	+	1
         RT		R1		R2		R3
         1	=	1	+	1	+	1
         RT		20 Ω		100 Ω		50 Ω
         1	=	5	+	1	+	2	=	8
         RT		100 Ω		100 Ω		100 Ω		100 Ω
         RT	=	100 Ω	=	12.5 Ω
         		8

      2. Total current is determined by the voltage of the power supply and the equivalent resistance of the circuit.

         IT = VT/RT = 125 V/12.5 Ω = 10 A

      4. (Note: we'll answer part iv before part iii.) On a parallel circuit, each branch experiences the same voltage drop.

         VT = V1 = V2 = V3 = 125 V

      3. The current in each branch can be found using Ohm's law.

         I1 = V1/R1	=	(125 V)/(20 Ω)	= 6.25 A
         I2 = V2/R2	=	(125 V)/(100 Ω)	= 1.25 A
         I3 = V3/R3	=	(125 V)/(50 Ω)	= 2.50 A

         Verify your calculations by adding the currents. On a parallel circuit they should add up to the current from the power supply.

         IT	=	I1	+	I2	+	I3
         10 A	=	6.25 A	+	1.25 A	+	2.50 A
         10 A	=	10 A

         Good, it works.
      5. Again as an exercise, use a different formula to determine the electric power of each resistor.

         P1	=	V1I1	=	(125 V)(6.25 A)	= 781.25 W
         P2	=	I22R2	=	(1.25 A)2(100 Ω)	= 156.25 W
         P3	=	V32/R3	=	(125 V)2/(50 Ω)	= 312.50 W

         In a parallel circuit, the element with the least resistance consumes the most power.
2. A kitchen in North America has three appliances connected to a 120 V circuit with a 15 A circuit breaker: an 850 W coffee maker, a 1200 W microwave oven, and a 900 W toaster.
   1. Draw a schematic diagram of this circuit.
   2. Which of these appliances can be operated simultaneously without tripping the circuit breaker?
   Solutions …
   1. Outlets are wired in parallel so that the appliances on a circuit are independent of one another. Turning the coffee maker off will not result in the toaster turning off (assuming both were on at the same time). Each appliance will also get the same regulated voltage, which simplifies the design of electrical devices. The downside to this scheme is that the parallel currents can add up to dangerously high levels. A circuit breaker in series before the parallel branches can prevent overloads by automatically opening the circuit.
   2. A 15 A circuit operating at 120 V consumes 1800 W of total power.

      P = VI = (120 V)(15 A) = 1800 W

      Total power in a parallel circuit is the sum of the power consumed on the individual branches.

      coffee maker	+	microwave oven	=	850 W	+	1200 W	=	2050 W
      microwave oven	+	toaster	=	1200 W	+	900 W	=	2100 W
      toaster	+	coffee maker	=	900 W	+	850 W	=	1750 W

      On this circuit, only the coffee maker and toaster can be operated simultaneously. All other combinations exceed the safe capacity.
3. The diagram below shows a circuit with one battery and 10 resistors; 5 on the left and 5 on the right. Determine …
   1. the current through,
   2. the voltage drop across, and
   3. the power dissipated by each resistor.
   The way to solve a complex problem is to break it down into a series of simpler problems. Be careful not to lose sight of your goal among all the bits and pieces, however. Before beginning plot your course. In this case we'll start by finding the effective resistance of the entire circuit and the total current from the battery. This sets us up to get the current in all the different segments of the circuit. (The current divides and divides again in an effort to follow the path of least resistance.) After that, it's a simple matter to calculate the voltage drops in each resistor using V = IR and the power dissipated using P = VI. No part of this problem is difficult by itself, but since the circuit is so complex we'll be quite busy for a little while.
   1. Let's begin the process by combining resistors. There are four series pairs in this circuit.

      Rs = 3 Ω + 1 Ω	Rs = 4 Ω + 2 Ω	Rs = 2 Ω + 3 Ω	Rs = 1 Ω + 4 Ω
      Rs = 4 Ω	Rs = 6 Ω	Rs = 5 Ω	Rs = 5 Ω
      These pairs form two parallel circuits, one on the left and one on the right.
      1/Rp = 1/(4 Ω) + 1/(6 Ω) = 5/(12 Ω)		1/Rp = 1/(5 Ω) + 1/(5 Ω) = 2/(5 Ω)
      Rp = (12/5) Ω = 2.4 Ω		Rp = (5/2) Ω = 2.5 Ω
      Each gang of four resistors is in series with another.
      Rs = 2.4 Ω + 0.6 Ω		Rs = 2.5 Ω + 0.5 Ω
      Rs = 3 Ω		Rs = 3 Ω
      The left and right halves of the circuit are parallel to each other and to the battery.
      1/Rp = 1/(3 Ω) + 1/(3 Ω) = 2/(3 Ω)
      Rp = (3/2) Ω = 1.5 Ω
      Now that we have the effective resistance of the entire circuit, let's determine the total current from the power supply using Ohm's law.
      Itotal = Vtotal/Rtotal = (24 V)/(1.5 Ω) = 16 A
      Now walk through the circuit (not literally of course). At each junction the current will divide with more taking the path with less resistance and less taking the path with more resistance. Since charge doesn't leak out anywhere on a complete circuit, the current will be the same for all those elements in series with one another.
      The left and right halves of the circuit are identical in overall resistance, which means the current will divide evenly between them.
      8 A for the 0.6 Ω resistor.		8 A for the 0.5 Ω resistor.
      On each side the current divides again into two parallel branches.
      The branches on the left have resistances in the ratio R1&3/R2&4 = (4 Ω)/(6 Ω) = 2/3, which means the currents will divide in the ratio I1&3/I2&4 = 3/2.		The branches on the right are identical, so the current splits into two equal halves.
      (3/5)(4 A) = 2.4 A	(2/5)(4 A) = 1.6 A	(8 A)/2 = 4 A	(8 A)/2 = 4 A
      for the 1 Ω and 3 Ω resistors on the left.	for the 2 Ω and 4 Ω resistors on the left.	for the 2 Ω and 3 Ω resistors on the right.	for the 1 Ω and 4 Ω resistors on the right.

   2. Use V = IR over and over and over again to determine the voltage drops.
   3. Use P = VI (or P = I²R or P = V²/R) over and over again to determine the power dissipated.
      
      These last two tasks are so tedious you should use the spreadsheet function on a graphing calculator. Enter the resistance values given and the current values we just calculated into columns and instruct the calculator to multiply the appropriate columns. Something like this …

      resistance (Ω)	current (A)	voltage (V)	power (W)
      0.6	8	4.8	38.4
      1	2.4	2.4	5.76
      2	1.6	3.2	5.12
      3	2.4	7.2	17.28
      4	1.6	6.4	10.24
      0.5	8	4	32
      1	4	4	16
      2	4	8	32
      3	4	12	48
      4	4	16	64

4. Here's a circuit [magnify] …

   Calculate the equivalent resistance of the circuit. Calculate the current through the battery. Graph voltage as a function of location on the circuit assuming that Va = 0 V at the negative terminal of the battery. Graph current as a function of location on the circuit.

   Here are the solutions …
   1. The total resistance in a series circuit is the sum of the individual resistances …

      RT = R1 + R2 + R3 = 3 Ω + 9 Ω + 6 Ω = 18 Ω

   2. The total current can be found from Ohm's law …

      IT = VT/RT = (12 V)/(18 Ω) = ⅔ A = 0.667 A

   3. The voltage in a circuit rises in a battery and drops in a resistor (when we follow the flow of conventional current). The rise in the battery is given as 12 V and the drops in each resistor can be found through repeated use of Ohm's law …

      V1 = I1R1 = (⅔ A)(3 Ω) = 2 V

      V2 = I2R2 = (⅔ A)(9 Ω) = 6 V

      V3 = I3R3 = (⅔ A)(6 Ω) = 4 V

      Starting at zero volts on the negative terminal of the battery, the voltage goes up 12 V then drops 2 V, 6 V, and 4 V, which brings us back to zero. (We are assuming that the battery and wires have negligible resistance.) Here's how it looks when graphed [magnify]. Here's how it looks when the graph is superimposed on the circuit [magnify].
   4. Current is everywhere the same in a series circuit. We've already determined it's 0.667 A. All that remains is to draw a horizontal line at two-thirds of an amp [magnify]. Here's how it looks when the graph is superimposed on the circuit [magnify].

conceptual
A power source and four identical, numbered light bulbs are connected together as shown in the circuit to the right. Rank the light bulbs in order of decreasing brightness. Briefly explain your reasoning. Describe the effect that removing any one of the light bulbs has (if any) on the brightness of the other three. Compile your results into a table like the one shown below. Briefly explain your reasoning.
R1 R2 R3 R4 ×         ×         ×         ×
numerical
Given the circuit to the right, find … the potential difference across the 4 Ω resistor, the potential difference across the 5 Ω resistor, the current through the 2 Ω resistor, and the current through the 7 Ω resistor.
Given the circuit to the right, determine … the equivalent resistance of the entire circuit, the total current from the power supply, the current through each resistor, and the voltage drop across each resistor.
The circuit on the right consists of eleven identical 1 Ω resistors connected to a 15 V power supply. Determine … the total resistance of the entire circuit, the total current from the power supply, the current through resistor R1, the current through resistor R2, and the current through resistor R3.
Twelve identical 1 Ω resistors are connected into a circuit that consists of two square loops as show in the diagram to the right. Determine the resistance between the following pairs of points … b & g d & g f & g a & g a & h d & h g & h f & h

Resources

• general
  • The Mechanical Universe and Beyond (video on demand, login required)
    • Electric Circuits, The work of Wheatstone, Ohm, and Kirchhoff leads to the design and analysis of how current flows.

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