WEEK 7 and 8

Resistive circuits may be analyzed by combining networks of parallel and series resistances into a single equivalent resistance, then using Ohm's law to find the current or voltage across that equivalent resistance. Once this is known, it is possible to work backward and use Ohm's law to calculate the voltage and current across any resistance in the network.

The equations necessary to perform the analysis are briefly introduced along with worked examples. References are cited or linked, but enough information is presented here to apply the concepts without needing to refer elsewhere. The step-by-step style is used only in sections where there is more than one step.

All intentional resistances will be shown as resistors (schematically, a zig-zag line). Connections shown as lines are assumed to be of zero resistance (at least approximately, relative to the resistors shown).

In summary, the basic steps are shown below.

1.If there is more than 1 resistor in the circuit, find the equivalent resistance "R" of the entire network as illustrated in "Combination of Resistances in Series and in Parallel" below.

2.Apply Ohm's law to this value of "R" as illustrated in the "Ohm's law" section below.

3.If there is more than 1 resistor in the circuit, the value of voltage or current calculated in the preceding step may be used in Ohm's law to find the voltage across or current through any other resistor in the network

Ohm's law may be written ^[1] in 3 equivalent forms depending on what is being solved for:

(1) V = IR

(2) I = V / R

(3) R = V / I

"V" is the voltage across the resistance (the "potential difference"), "I" is the current through the resistance, and "R" is the value of the resistance. If the resistance is a resistor (a component having a calibrated value of resistance) it is usually labelled with "R" followed by a number, such as "R1", "R105", etc.

Form (1) is easily converted into forms (2) or (3) by algebraic manipulation. In some cases the letter "E" is used in place of "V" (for example, E = IR), where "E" stands for EMF or "electromotive force" which is another name for voltage.

Form (1) is used when the current is known through a resistor of known value.

Form (2) is used when the voltage is known across a resistor of known value.

Form (3) is used when the resistor value is unknown, but the voltage across it and current through it are known, allowing the resistance to be computed.

The standard SI units of each parameter in Ohm's law are:

• Voltage drop across the resistor "V" is in volts, abbreviated "V". The abbreviation "V" for "volts" is not to be confused with the voltage "V" in Ohm's law.
• Current "I" is in amperes, often shortened to "amps", abbreviated "A".
  
• Resistance "R" is in ohms, often represented by the Greek symbol capital omega (Ω). Letter "K" or "k" signifies a multiplier of a "thousand" ohms, "M" or "MEG" signifies multiplier of a "million" ohms. Often the Ω symbol is not written after a multiplier, for example a 10,000 Ω resistor is usually labelled "10K" rather than "10 K Ω".

Ohm's law applies to any circuit containing only resistive elements (such as resistor components, or the resistance of conductors such as wires or PC board runners). If there are reactive elements (inductors or capacitors) it does not directly apply in the form shown above (the above equation only contains "R", which does not encompass inductance or capacitance). Ohm's law can be used on resistive circuits whether the applied voltage or current is DC (direct current), AC (alternating current), or some randomly time-varying signal examined at any instant of time. If the driving voltage or current is sinusoidal AC (such as from a 60 Hz household electric outlet), the units of voltage and current are usually volts or amps RMS.

Resistances in Series

Resistors connected in series.

The total end-to-end resistance of a string of resistors connected in "series" (see figure) is simply the sum of all the resistances. For "n" resistors labeled R1, R2, ..., Rn.

R_total = R1 + R2 + ... + Rn

Example: Resistors in Series

Suppose there are 3 resistors connected in series:
R1 = 10 Ohms
R2 = 22 Ohms
R3 = 0.5 Ohm

The total end-to-end resistance is:

R_total = R1 + R2 + R3 = 10 + 22 + 0.5 = 32.5 Ω

Resistances in Parallel

Resistors connected in parallel.

The total resistance across a set of resistors connected in parallel (see diagram at right) is given by:

Common notation meaning "in parallel with" is to write two parallel slashes ("//"). For example, R1 in parallel with R2 may be denoted as "R1//R2". Note that R1//R2 = R2//R1. A set of 3 resistors R1, R2, and R3 all in parallel could be denoted "R1//R2//R3".

Example: Resistors in Parallel

For 2 resistors in parallel, R1 = 10 Ω and R2 = 10 Ω (both the same value), we have:

Combination of Resistances in Series and in Parallel

Networks of combination of series and parallel resistors may be analyzed by combining them into a single "equivalent" or "total" resistance.

Steps

1. 1
   Generally, combine all parallel resistors into equivalent parallel resistances using "Resistances in Parallel" above. Note that if there are parallel branches that also contain series elements, these must first be combined by adding the resistances in that branch.
2. 2
   Combine series resistances by adding them, to obtain the total resistance of the network, R_total.
3. 3
   Use Ohm's law to find the total current into the network for a given applied voltage, or the total voltage across the network for a given applied current.
4. 4
   The total voltage or current computed in the previous step is used to compute voltages and currents in the network using Ohm's law.
5. 5
   Apply this current or voltage to Ohm's law to find the voltage across or current through any other resistor in the network.

Example: Series/Parallel Network

For the network shown at right, first the parallel resistances will be combined to find R1//R2, then the total resistance of the network (across the terminals) is found from:

R_total = R3 + R1//R2

Suppose R3 = 2 Ω, R2 = 10 Ω, R1 = 15 Ω, and a 12 V battery is applied across the network so V_total = 12 volts. Solving using the above steps we have:

The voltage across R3 (denoted V_R3) could now be calculated from Ohm's law, since the current through it is known to be 1.5 amperes:

V_R3 = (I_total)(R3) = 1.5 A x 2 Ω = 3 volts

The voltage across R2 (which is the same as the voltage across R1) could be calculated using Ohm's law by multiplying the current I = 1.5 amps times the equivalent parallel resistance R1//R2 = 6 Ω, giving 1.5 x 6 = 9 volts, or could be calculated by subtracting the voltage across R3 (V_R3, just calculated above) from the applied voltage of 12 volts, that is, 12 volts - 3 volts = 9 volts. Once this is known the current through R2 (denoted I_R2) may be calculated from Ohm's law (where the voltage across R2 is denoted by "V_R2"):

I_R2 = (V_R2) / R2 = (9 volts) / (10 Ω) = 0.9 amp

The current through R1 could similarly be found using Ohm's law by dividing the voltage across it (9 volts) by its resistance (15 Ω), giving 0.6 amps through R1. Notice that the current through R2 (0.9 amps) plus the current through R1 (0.6 amps) equals the total current into the terminals of 1.5 amps.