Circuit Theory/Resistive Circuit Analysis

Resistive Circuit Analysis

There are certain established rules that can be used to examine a circuit that is comprised entirely of resistors and sources. Among these are tools for combining resistors in certain configurations into a single conceptual resistor that has an equivalent resistance to the two that have been replaced. In this manner, very complicated circuits can be reduced to be a very simple circuit with few components.

The Resistor

The resistor is an electrical component that limits the flow of electrical current. The physical characteristics of a resistor are described in the Electronics Wikibooks and information on resistors as found in real life are in Practical Electronics.

The v-i characteristic of a resitor is simply a straight line, passing through the origin:

We call the gradient of this line the resistance of the element. It is defined as the voltage developed across the element per ampere of current through it. An ideal resistor therefore obeys Ohm's Law:

$v=iR$

The resistance is given the symbol "R" and is the unit of ohm which is denoted by a capital omega, Ω. It is named after Georg Ohm so the word "ohm" is always lowercase (unless it begins the sentence) - this is the standard for SI units. The ohm is given in base units by:

$\mathrm{\Omega }=\frac{\text{V}}{\text{A}}=\frac{{\text{m}}^2\cdot \text{kg}}{{\text{s}}^3\cdot {\text{A}}^2}$

In real life, resistors are generally made of carbon or metal films or ceramic, depending on the application and desired accuracy. Resistance is always positive for such devices. Negative resistances are possible, but they are not found in resistors.

Power in a Resistor

The power, P, dissipated in a resistor is given by

$P=vi$

And from Ohm's law it can be seen that, since v=iR,

$P=i^{2}R$

This means that the power is the product of a square and a resistance, both of which must be positive. Therefore, the power dissipated in a resistor is always positive.

Conductance

The resistor is an invertible component. This means that we can rearrange the v-i characteristic into an i-v characteristic:

$v=iR$

$i=\frac{1}{R}v$

The constant of proportionality (1/R) is called the conductance and is generally given the symbol "G":

$i=Gv$

Conductance has the unit siemens, named after the German scientist Ernst Werner von Siemens. The symbol for siemens is S. Conductance is expressed in base units as:

$S=\frac{\text{A}}{\text{V}}=\frac{{\text{s}}^3\cdot {\text{A}}^2}{{\text{m}}^2\cdot \text{kg}}$

A siemens is therefore the current in the resistor per unit voltage through it. As with all SI units named after people, the word "siemens" is lowercase, but the symbol is uppercase (S).

Real Resistors

Resistors tend to heat up as they pass more current, due to an increase in dissipated power. Since usually the resistance of a material changes with temperature, this produces a distorted v-i graph:

Non-ideal resistor value with nonlinear resistance characteristic

This generally has a very complicated v-i characteristic, dependent of the thermal coefficient (how the resistance changes with temperature), the ability of the resistor to dissipate heat into its surroundings (which is itself dependent on many things) as well as the nominal resistance and the applied voltage or current.

Equivalent Resistances

An unknown circuit element can be modeled as a single resistance if it has a directly proportional v-i relationship. This resistance is called the equivalent resistance, and is often written as R_eq.

In order to determine the value of the equivalent resistance, either a voltage can be applied to the terminals of the unknown element, and the resulting current measured, or a constant current can be applied, and the resulting voltage measured.

The equivalent resistance is then given by

$R_{eq}=\frac{v}{i}$

Since a resistive element has a straight v-i characteristic, only one measurement is needed. Note that if the element is not purely resistive, this method will give an erroneous result.

Degenerate Resistors

Short Circuit

Consider the v-i characteristic for a resistor:

$v=iR$

If the resistance is zero, we have

$v=i\times 0$

This limiting case is called a short-circuit. We can consider a short circuit to be a voltage source with a zero value. Just as current in a voltage source is arbitrary (depends completely on the rest of the circuit), so is the current in a short circuit (whatever i is, when multiplied by the zero in the v-i characteristic, it will be zero). We therefore have the following equivalence:

Open Circuit

Consider the i-v characteristic for a resistor:

$i=\frac{1}{R}v$

If the resistance tends to infinity, we have

$i=\lim {\mathrm{\backslash limits}}_{R\to \mathrm{\infty }}\left(\frac{1}{R}v\right)=0$

This limiting case is called an open-circuit. We can consider an open circuit to be a current source with a zero value. Just as voltage across a current source is arbitrary (depends completely on the rest of the circuit), so is the voltage across an open circuit. We therefore have the following equivalence:

Elements in Series

Circuit elements are said to occur in series when they are directly connected end-to-end with no branching nodes in between them. Circuit elements are in parallel if they share a common starting and ending node.

There are two formulas that apply to series and parallel combinations of passive circuit elements. Which formula applies depends on the element and the combination.

1. $x=\sum _n^{}{x}_n$
2. $x={\left(\sum _n^{}\frac{1}{x_n}\right)}^{-1}$

Where x is the quantity being considered. The first equation applies to series resistors, while the second applies to parallel resistors. The same formula are applicable for inductors when their mutual inductance is neglected. For capacitors, the formula have to be interchanged.

Resistors

Resistors appearing in series can be converted into a single resistor, $r_{series}$ according to the following equation:

Resistors in a series configuration

$r_{tot}=\sum _n^{}{r}_n$

Where n is the number of resistors in series. Multiple resistors appearing in series then, can be converted conceptually into a single resistor whose resistance is simply the sum of the parts.

Voltage Sources

Voltage sources given in series can be added together to form a single source, $v_{series}$ given by the equation:

$v_{series}=\sum _n^{}{v}_n$

It is to be noted that the magnitude of voltage should be a signed integer depending on the polarity of the voltage source. The convention for polarity can vary from one reference to another. But usually the voltage source whose positive terminal is connected to other element in the assumed direction of the circuit is considered as positive. If the aforementioned condition is with the negative terminal, the polarity is considered to be negative

Current Sources

Current sources may not appear in series, as doing so would violate Kirchoff's Current Law (explained below).

Elements in Parallel

This section will talk about how to condense circuit elements that exist in parallel. "Parallel" is defined as elements that share common endpoints.

Resistors

If multiple resistors are parallel to each other, we can calculate out the conceptual resultant resistor as follows:

A diagram of several resistors appearing in parallel with each other.

$r_{tot}={\left(\sum _n^{}\frac{1}{r_n}\right)}^{-1}$

In the special case of 2 resistors in parallel the following notation is used:

$r_1\Vert r_2=\frac{r_1{r}_2}{r_1+r_2}$.

Voltage Sources

Placing voltage sources in parallel has no effect on their voltage, and, theoretically, has no effect at all, as a proper voltage source is capable of producing infinite current. However, as the Sources subheading of Section 1 notes, no voltage source can offer unlimited current, and the most common voltage sources, batteries, generally have fairly low current limits.

So, for practical purposes, voltage sources are placed in parallel to offer more current. Assuming two identical voltage sources, such as a pair of "AA" batteries, two cells in parallel offer twice the current of one of the cells. This can be used to either power devices with larger current draws or to extend battery life. (Doubled current potential means double battery life with a given load.)

Current Sources

When Current Sources are in parallel, they may be replaced with a single source with the output:

$i_{parallel}=\sum _n^{}{i}_n$

Kirchoff's Laws

Kirchoff has two important laws that govern electrical circuits: the current law (KCL) and the voltage law (KVL). These laws, along with Ohm's law are the three fundamental formulas that are needed to analyze circuits. Without these three laws, many of the more advanced techniques and situations that we are going to discuss in this book would not be possible.

Kirchoff's Current Law (KCL)

Kirchoff's current law (KCL) states that the sum of all the currents entering into a single node must equal zero. To come to this conclusion, we need to work out a current convention:

Positive Current

current flowing into a node

Negative Current

current flowing out of a node

The opposite convention may also be used, but the user needs to make sure that they use the same current flow convention throughout an entire problem, or the answers will be wrong. The point cannot be stressed enough that when doing circuit work, conventions must be specified and they must be followed exactly. Failure to do so will cause all the calculations to be wrong from that point forward.

now, let us examine some circuits:

1A->           <-i
o--------o-------o
         N

in the above diagram, 1 Amp of current is moving into node N. We can then perform the summation, and solve for current i:

$1A+i=0$ and therefore: $i=-1$

A negative current flows away from the node, as per our convention, and therefore we can update our schematic:

1A->          1A->
o--------o-------o
         N

Granted, this is a simple example, but it gets the point across.

Kirchoff's Voltage Law (KVL)

Kirchoff's voltage law (KVL) states that the sum of the voltages around a closed loop must all equal zero. Again, we should come up with a convention, although this one will be a little bit more complicated.

Forward Current

Current is flowing from the negative terminal of an element to the positive terminal.

Backwards Current

Current is flowing from the positive terminal of an element to the negative terminal.

Once we have our notions of forward and backward current flow decided upon, we can then write out our convention for voltage increases and decreases:

1. Forward current on a source creates positive voltage, or a voltage increase.
2. Forward current on a load element creates a voltage drop, or a negative voltage.
3. Backwards current on a passive load element also creates a voltage drop

This convention is more tricky, so we will examine a few small examples first:

1A->  5ohm
o----/\/\/\----o
+      v       -

Here, the current is flowing from the positive terminal of the resistor to the negative terminal. By our convention, this is a "Backwards Current" flow, and therefore over the resistor we have a voltage drop:

$v=-(1A)(5\mathrm{\Omega })=-5V$

This voltage drop corresponds to the fact that on the left side of our schematic there is more electrical potential then on the right side of the schematic.

now, let's look at a whole circuit:

     5ohm
 +--/\/\/\--+
+|  + vr -  |
( )12V      |
-|     <-i  |
 +----------+

To figure out the voltage drop across the resistor using only Ohm's law, we would need to know the current in the loop, i. This circuit has 2 unknowns now: the voltage across the resistor (vr), and the current, i. Using KVL, we can sum the voltage contributions of both the source and the resistor, as such:

$12V+vr=0\to vr=-12V$

Keeping in mind that this is a resistor, and therefore it is a voltage drop across the resistor, we can reverse the sign to show the voltage drop of the resistor:

$vr=12V$

Using Ohm's law now, we can calculate i because we know the resistance of the resistor, and the voltage across the terminals of the resistor:

$12V=i(5\mathrm{\Omega })\to i=\frac{12V}{5\mathrm{\Omega }}=2.4A$

Current Divider

Current dividers and voltage dividers are two types of circuits with similar intentions: to decrease current or voltage by a certain factor, using only resistors. This page will talk about current dividers and voltage dividers.

Definition

A current divider is formed by connecting resistors in parallel. The current through any single resistor can be found by:

$I_i=I_{in}\frac{\frac{1}{Ri}}{\sum _{k=1}^n\frac{1}{R_k}}$

or equivalently, using conductances:

$I_i=I_{in}\frac{G_i}{\sum _{k=1}^n{G}_k}$

Voltage Divider

Definition

A voltage divider is created by connecting resistors in series. The voltage of resistor i in an n-resistor voltage divider is:

$V_i=V_{in}\frac{Ri}{\sum _{k=1}^n{R}_k}$

Voltage division can be used to adapt 220-240V AC to 110-120V AC (to allow 120V US devices to run on 220V). However, voltage division can be inefficient since the resistors have to dissipate large amounts of heat. More efficient adapters use transformers.

Construction

A simple voltage divider diagram