ECE 220 Network Analysis I

Lesson 2. Resistors in Series and Parallel


Browser note: Your Web browser should be capable of displaying symbols. Here is the Greek letter omega: Ω. This looks like a W, instead of an omega, on some primitive browsers.

Printer note: Even if the document looks ok on your browser, it may still print bogus characters on your printer.  If this happens, see if you can get an update on your printer driver.


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 (Florida State 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 the most commonly used multipliers.  Commit them to memory.

Resistors in Series

Two elements are in series if they are connected together at one end with no other connection at that end. Use this definition, rather than your intuition, to determine if elements are in series. The following elements are in series:

You should see a picture here.

These elements are not in series:

You should see a picture here.

For resistors in series, the net resistance is just the sum of the individual resistances.

REQ = R1 + R2 + ..... Rn

Resistors in Parallel

Two elements are in parallel if both ends of each element are connected together. These elements are in parallel:

You should see a picture here.

These elements are not in parallel:

You should see a picture here.

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 from Kirchhoff's Laws.


Before going on, you should complete Tutorial 2 on resistors in series and parallel.


Resistors not in Series or Parallel

Some circuits, such as the one shown below, cannot be simplified by combining elements in series and parallel. When this happens, you just have to grit your teeth and apply Kirchhoff's Laws, or use the delta-wye transformation (discussed in a later section).

You should see a picture here.

Short Circuits and Open Circuits

An open circuit is a place in a circuit where nodes are not connected, or open. Zero amps flows between nodes that are not connected, meaning zero amps flows in an open circuit. The resistance across an open circuit is equal to infinity. Open circuits are represented as a broken wire. For calculating an equivalent resistance, a resistor connected to the circuit at only one node is open. An open resistor (1) makes zero Ohms of contribution to the equivalent resistance and (2) can be removed from the circuit when calculating the equivalent resistance.

Open circuit = 0A of current

Open circuit = ∞Ω of resistance

An element (e.g., resistor, voltage source, etc.) is shorted if both of its ends are connected to the same one node. Short circuits are represented as a wire. A wire is considered to have a negligible amount of voltage, or zero volts, meaning the voltage is zero for a short circuit. The resistance of a wire in electrical circuits is considered to be negligible, or 0Ω. Therefore, the resistance across a short circuit is negligible, and considered equal to zero. For calculating an equivalent resistance, a shorted resistor is one 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 be removed 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 figure below.

You should see a picture here.

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 figure below.

You should see a picture here.

Similar to the voltage divider equation, Kirchhoff's Laws can be used to find that

I2 = ISR1/(R1 + R2)

Notice that the numerator term uses the resistor that the current doesn't go through.


Before going on to the homework, you should complete Tutorial 2A on voltage and current dividers.


Homework Problems

Please note:  It is not necessary to use delta-wye transformation in any of these problems.

  1. R1 = 11 Ω, R2 = 15 Ω, R3 = 30 Ω, R4 = 2 Ω.  Find the resistance between X and Y.  The answer is an integer.
    You should see a picture here.
  2. R1 = 42 Ω, R2 = 80 Ω, R3 = 120 Ω, R4 = 45 Ω.  Find the resistance between E and F.  The answer is an integer.
    You should see a picture here.
  3. 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.
    You should see a picture here.
  4. 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.
  5. 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?
  6. 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.
  7. 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?
    You should see a picture here.
  8. 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.
    You should see a picture here.

Bonus (no partial credit). All resistors are 1 Ω. Find an expression for RGH that can be expanded to as many decimal places as desired, e.g.
RGH = (π - 3)/7 (Not the right answer).
You should see a picture here.