Lesson 2. Resistors in Series and Parallel
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Note that current is defined as the flow of positive charges.
v = Ri is Ohm's Law. It's an important equation; commit it to memory.
Click here (FloridaState University) to find out how to use the resistor color code.
Mega (106), kilo (103), milli (10-3), micro(10-6), nano(10-9), and pico(10-12) are themost commonly used multipliers. Commit them to memory.
Resistors in Series
Two elements are in series if they are connected together at one end with noother connection at that end. Use this definition, rather than your intuition,to determine if elements are in series. The following elements are inseries:
These elements are not in series:
For resistors in series, the net resistance is just the sum of theindividual resistances.
REQ = R1 + R2 + ..... Rn
Resistors in Parallel
Two elements are in parallel if both ends of each element are connectedtogether. These elements are in parallel:
These elements are not in parallel:
A parallel combination of resistors is found by the equation
1/REQ = 1/R1 + 1/R2 + 1/R3 +..... 1/Rn
A useful special case is for exactly two resistors in parallel:
REQ = R1R2/(R1 + R2)
The equations for elements in series and parallel are easily derivable fromKirchhoff's Laws.
Before going on, you should complete Tutorial 2 on resistorsin series and parallel.
Resistors not in Series or Parallel
Some circuits, such as the one shown below, cannot be simplified bycombining elements in series and parallel. When this happens, you just have togrit your teeth and apply Kirchhoff's Laws, or use the delta-wye transformation(discussed in a later section).
Short Circuits and Open Circuits
An open circuit is a place in a circuit where nodes are not connected, oropen. Zero amps flows between nodes that are not connected, meaning zero ampsflows in an open circuit. The resistance across an open circuit is equal toinfinity. Open circuits are represented as a broken wire. For calculating anequivalent resistance, a resistor connected to the circuit at only one node isopen. An open resistor (1) makes zero Ohms of contribution to the equivalentresistance and (2) can be removed from the circuit when calculating theequivalent resistance.
Open circuit = 0A of current
Open circuit = ∞Ω of resistance
An element (e.g., resistor, voltage source, etc.) is shorted if both of itsends are connected to the same one node. Short circuits are represented as awire. A wire is considered to have a negligible amount of voltage, or zerovolts, meaning the voltage is zero for a short circuit. The resistance of awire in electrical circuits is considered to be negligible, or 0Ω.Therefore, the resistance across a short circuit is negligible, and consideredequal to zero. For calculating an equivalent resistance, a shorted resistor isone whose both ends are connected to the same one node. A shorted resistor (1)makes zero Ohms of contribution to the equivalent resistance and (2) can beremoved from the circuit when calculating the equivalent resistance.
Short circuit = 0V of voltage
Short circuit = 0Ω of resistance
Voltage Divider
The voltage divider equation will be very useful to you. Consider the figurebelow.
It is easily derivable from Kirchhoff's Laws that
V2 = VSR2/(R1 + R2)
Current Divider
The current divider equation may be occasionally useful. Consider the figurebelow.
Similar to the voltage divider equation, Kirchhoff's Laws can be used tofind that
I2 = ISR1/(R1 + R2)
Notice that the numerator term uses the resistor that the currentdoesn't go through.
Before going on to the homework, you should complete Tutorial 2A on voltageand current dividers.
Homework Problems
Please note: It is not necessary to use delta-wye transformation in any ofthese problems.
- R1 = 11 Ω, R2 = 15 Ω, R3 = 30 Ω, R4 = 2 Ω. Find the resistance between X and Y. The answer is an integer.
- R1 = 42 Ω, R2 = 80 Ω, R3 = 120 Ω, R4 = 45 Ω. Find the resistance between E and F. The answer is an integer.
- R1 = 6 Ω, R2 = 9 Ω, R3 = 15 Ω, R4 = 14 Ω, R5 = 10 Ω, R6 = 30 Ω, R7 = 2 Ω. Find the resistance between A and B. The answer is an integer.
- Using exactly six 10 Ω resistors and no other resistors, design and sketch a circuit with a resistance of exactly 22 Ω. All six resistors must be significant parts of the circuit.
- a) Design and sketch a circuit with a resistance of exactly 1.4 MΩ between nodes A and B. Use only the following list of resistor values in your design: 100 kΩ, 620 kΩ, 2.4 MΩ, and 3.3 MΩ . Use as many of the listed resistor values as you choose for your design. Use no other resistor values. All resistors used must be a significant part of the circuit.
b) Go to digi-key.com. Search for resistors to build your design from part a). Use search filters for resistors with mounting type “Through Hole,” a power rating of “1/4W (.25W),” a material composition of “Carbon Film,” a resistance value tolerance of “±5% (directly above ‘Jumper’ listing in Tolerance),” from the manufacturer “Yageo,” and in packaging of type “Bulk.”
c) What would it cost (Subtotal) to build 10 versions of your above design? (Click on each ‘Digi-Key Part Number’ and add specific Quantities to your order.)
d) What would it cost (Subtotal) to build 10,000 versions of your above design?
e) Consider the point of view of a resistor manufacturer. Why would a manufacturer not stock a 1.4 MΩ resistor but would stock 1.3MO and 100 kΩ resistors? - Sketch a voltage divider circuit that uses a battery of your choice, a 47 kΩ resistor, and a 22 kΩ resistor to produce an output voltage of approximately 6.13 V.
- The 6.8 kΩ resistor has a tolerance of 10%. The 3.3 kΩ resistor has a tolerance of 20%. What is the maximum possible voltage for Vout? What is the minimum possible voltage for Vout?
- A student uses a voltage divider in hopes of converting 9 V to 1 V, using the circuit shown. When the student connects a cheap volt-ohm meter (VOM) to Vout, she gets just 0.67 V. Explain why the voltage is lower than expected. Note: This problem has nothing to do with tolerance; the resistor values are accurate.
Bonus (no partial credit). All resistors are 1 Ω. Find anexpression for RGH that can be expanded to as many decimal places asdesired, e.g.
RGH = (π - 3)/7 (Not the right answer). 
