| Resistors |  |
Prerequisites:
Resistors are the most common components in a circuit. They are the basic elements that are included in the model of more complex components, while they can be used as a stand alone component in a circuit. Make sure to understand everything about resistors as they will be used a lot.
Resistor is a component that shows resistance against the current passing through them. To understand it better, assume a source of water with a constant water pressure. if you attach a hose to it, depending on the diameter of the hose channel, the hose shows a different resistance on the water current. In a thinner hose the water passes more difficultly and the output volume of water is less. Similarly for a constant voltage, a resistor with higher resistance passes less current through itself. A resistor is always shown with the letter "R" and with the symbol below:
The
unit of resistance is Ohm shown with the symbol W.
The reverse of resistance is called Conductance and its unit is
shown with 
. The
conductor, which is basically the same component as resistor, is shown
with "G". The
relation between conductance and resistance along with voltage and current in a resistor is shown with
basic but very important formulas below:
V=RxI is known as Ohm's law, where it says that the voltage across a resistor is the product of the resistance and the current going through the resistor. V is the value of voltage across the resistor, I is the value of current through the resistor and R is the value of resistance shown by the resistor, while G is the conductance. This formula simply shows that for a constant voltage, a higher resistance passes a smaller current through it. Yet for a constant current, the higher resistance results in a bigger voltage across it and for a constant resistance, the more voltage across the resistor results in more current and vise-versa.
Electrical components in general are either passive or active. A passive component doesn't have any gain or control over the signal passing through it. In contrast an active component shows gain or control over the signal.
A resistor is a passive component. Also it is a consumer meaning that it only consumes the electrical power and doesn't generate any power. In a passive consumer resistor, the current always flows from the higher voltage to the lower voltage, and the value of such resistor is always positive.
Energy and Power in a Resistor
You remember from the Energy and Power section the formulas for power and energy also mentioned below:
and 
Now using these formulas with the resistor equation above, we can get important formulas for power and energy in resistors:
Using these equations the energy and power dissipation in a resistor can be calculated, which in reality is dissipated in the environment as thermal energy. Therefore the more power consumed in a resistor, the more heat is produced. This is exactly how they make electrical heaters, electrical ovens or soldering guns. You may think that more resistance results in more power dissipation. But that's not always true. Because usually the voltage across a resistor is provided from a constant source, and more resistance means less current. To understand it better refer to the formula of power in terms of voltage. You simply see that for a constant voltage, more resistance is translated in less power and vise-versa. Only in the case of constant current through a resistance a higher resistance results in a higher power, better seen from the power formula in terms of current. But the first case, the constant voltage, is more likely as most of the sources we use in our systems are constant voltage sources such as battery or power plug voltage.
Note that the power consumed in a resistor is a Real Power, versus the other kind of power which is imaginary. As also explained in the energy and power section, this is the kind of power that is really consumed in a circuit and dissipated in the environment as heat. Only resistive components consume real power and every thing else consumes imaginary power.
Equivalent Resistance of a Resistor Network
Resistors can be put together to shape a resistor network, in series, parallel, triangle or star configurations. Such a network can seem very confusing and hard to calculate for its parameters. But resistor networks can be simplified up to a single resistor. When there is a network of resistors between two nodes, no matter how complex the network is, the entire network cab be replaced with a single resistor. This is because the behavior of a resistor network is exactly the same as a resistor. To simplify a resistor network one should know how to replace simpler network components with their equivalent resistors, or convert them to another network that helps simplifying the circuit. Below equivalent resistors for different famous configurations are discussed.
Equivalent Resistance of Series Resistors
One of the basic connections between resistors is series configuration. Figure below shows resistors in series.
The equivalent resistance of series resistors shown with Req is simply equal to the sum of all the resistances in this circuit.
Let's see why is that. You remember from KCL law that the current in all the components in series is the same. This fact is shown in the figure below:
And also from KVL law, you know that: V = V1 + V2 + ... + Vn-1 + Vn. From the resistor equation at the beginning of the page, we can write: V1 = R1.I, V2 = R2.I to the last resistance. Therefore we can replace the voltage vales and get the equation below:
And hence we extract the equations below:
The equivalent resistance of the series resistors will behave exactly the same as the series network. Therefore: V = Req.I where V is the voltage across the entire branch and I is the current through the branch. It is easily seen that:
One useful result from this equation is that when there is an infinite resistance in series with other resistors, such as an open circuit between two nodes, the equivalent resistance is infinity, again similar to an open circuit. Another result from this is that if there is a resistance much larger than the sum of all other resistances in series, the final result is almost equal to the value of that large resistance, neglecting the sum of other small resistances.
Equivalent Resistance of Parallel Resistors
Figure below shows resistors in parallel. To show parallel components in general two parallel components are placed on the sides of this symbol: ||. For example when R1 and R2 resistors are in parallel, it is shown as R1||R2. Again from KVL and KCL laws we know that voltage across all the parallel components is the same (here V) and the current going into the parallel network is equal to the sum of all the currents flowing in the parallel branches, which is: I = I1 + I2 + ... + In-1 + In.
From the resistor equation we get: I1 = V/R1 , I2 = V/R2 to the last resistor. Replacing these in the sum of currents, we get:
One good result from this formula is that when there is a zero resistance in parallel to anything, the equivalent resistance of this circuit is zero. Or no matter how many resistances we have in parallel, when there is a zero Ohm in parallel to the resistors, we can replace the entire circuit with a short. Also another result is that when there is a reasonably small resistor in parallel to other resistors, the equivalent resistance of the circuit is almost equal to the resistance of that small resistor, neglecting the small effect of other much larger resistors located in parallel.
Lets take a look at a very common example, two resistors in parallel as shown in the figure below:
The equivalent resistance is calculated as below:
Star and Triangle Conversion
The conversion between star and triangle configurations is a very useful solution to simplify complex resistor networks. In the figure below these configurations are shown.
With proper resistor values for each configuration, they behave exactly the same. Therefore one can replace one configuration with another, where it results in easier calculations. Below it is shown how to extract conversion formulas for start to triangle configuration and back. The final equations are enough for the conversion. But extracting these formulas is also a great practice to solve for resistor networks.
Like the solutions for series and parallel configurations, you can assume different voltages across each resistor and current through them. Then in order to get two similar circuits you can put the voltages and currents on terminals of two configurations equal and calculate for the resistor equations. Although this way works, it will take a tone of very complicated equations to solve. Always try to find the best way to solve a circuit as improper equations can make it so hard to solve a circuit.
In the case of any circuit, when you have the choice to assume voltages or currents for the circuit, it is better to make as many as you can constant, like zero value and solve for the remaining voltages and currents. This way you get fewer equations that are less complex. In the case of these configuration, we assume V2 and V3 to be equal to zero in both circuits, achieving the circuits below:
Now you can see how easier the circuits look having proper assumptions. Let's solve the star circuit first to get its equivalent resistance between the remaining two nodes. Figure below shows the steps taken to get the equivalent resistance of the star configuration.
First, R2 and R3 are in parallel, therefore we replace them with their equivalent resistance shown as R2||R3. Then you see that two resistors in series remain. The equivalent resistance between the two node is equal to Req = R1 + R2||R3. Now for the triangle configuration, you can see that both sides of R'1 are connected to the same voltage, zero. Therefore the voltage across the resistor is zero and from the formula V=R.I, it means that there is no current going through this resistor. This leaves the resistor ineffective and we can eliminate it.
Note this: when two nodes are connected to the same voltage, it is exactly like shorting these two together. Because no matter what is between these two nodes, because of the lack of energy across them, no current goes true them and both nodes will have the same properties, similar as two shorted nodes.
This simply means that the ends of the resistor R'1 are shorted. Therefore as mentioned in the parallel equation above, we simply replace this with a short. Figure below shows the remaining circuit for the triangle configuration and its equivalent resistance, which is simply shown as R'2||R'3.
Now as mentioned before, both circuits must show the same properties. Therefore under the same conditions mentioned above (V2 = V3 = 0) the equivalent resistance of both circuits must be the same. This results in the first equation relating the resistances of both circuits together. Similarly under two more condition, V1 = V3 = 0 and V1 = V2 = 0, we get two more equations. They are all summarized below:
Now solving these equations gives us formulas to convert a star configuration to triangle and back. To convert from a start to triangle configuration, we use the formulas below:
To convert from triangle to star configuration, we use the formulas below:
| Written by: Mehdi Sadaghdar | |