Our modern world relies on technology built thanks to electrical engineering and circuitry. Circuit analysis is one of the most important aspects of understanding how these technologies work. The process of circuit analysis involves studying various electrical quantities in the different components. This article is going to explore the concept of circuit analysis, including the methods we can use and the calculations you will need to know, before finally looking at examples of each method.
Circuit Analysis Definition
Circuit analysis is a concept that comes from electrical engineering principles. The main idea behind it is that when you build a circuit, you are going to want to know whether the components selected can handle the voltages and currents they will be subjected to. These calculations are usually made after simplifying the circuit.
Circuit analysis is the mathematical analysis of any electrical circuit.
In other words, it is the calculation of unknown elements within a circuit, such as the voltage or current.
What Tools Are Required to Perform a Circuit Analysis?
Completing a circuit analysis requires us to gather some information and utilize some equations. There are four things that you will need to know in particular about any circuit you are attempting to analyze. The tools are as follows:
You will have to know the schematics of the circuit. Knowing what the circuit looks like on paper will make it a lot easier for you to calculate and record certain values at certain points around the circuit. The best way to do this is to draw a circuit diagram, including every component.
Once you have a schematic of your circuit, you are going to want to simplify all the resistors down into one resistor. That means, any resistor, no matter if it is in series or parallel, should be placed into the value of a single resistor in series with the rest of the circuit.
You will need to know some fundamental equations and laws of physics.
Finally, you'll have to know Kirchhoff's current and voltage laws. These are laws that explain how current and voltage work in different types of circuits.
It should also be noted that in real-life applications, you will need a voltmeter and an ammeter, to measure the voltage and current, respectively. Once the voltage and current are known, every other value can then be calculated.
Using Circuit Analysis Tools
Each of the tools listed above is important for performing good circuit analysis. The tools themselves also have important aspects. A brief description of each is included in this section.
Circuit Schematics
A circuit schematic (often called a circuit diagram) is, quite simply, a diagram of a circuit, along with all of its connections and components. Most of the time, they are drawn before making the physical circuit so that engineers can figure out the necessary components. You will see various of examples of circuit diagrams throughout this article, note the different components connected by vertical and horizontal conductor lines.
Resistor Simplification
Oftentimes, a circuit will consist of multiple resistors connected in various ways. To successfully perform circuit analysis, we must simplify them as much as possible, analogous to simplifying fractions in a math equation. That can be done in the following ways.
Start with the resistor that is the furthest away from the main circuit.
Replace any group of resistors in a loop with a single resistor. It's important to recognize the type of circuit (series or parallel), before performing the calculations, as the equations differ.
Repeat the first two steps until there is only one resistor in the circuit.
- If the resistors are in series (i.e. next to each other), you add the value of each resistor together: \(R_{\mathrm{series}}=R_1 + R_2.\)
- If the resistors are in parallel, the rule for finding the total resistance is as follows: \(\frac{1}{R_{\mathrm{parallel}}}=\frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}.\)
Let's apply these steps in an example to simplify multiple resistors into a single resistor.
We have the following circuit consisting of five resistors.
As previously discussed, all simplifications must start from the resistors furthest away from the voltage source.
Here it's \(B\) and \(E\). These two resistors are connected in series, therefore we use equation
$$R_{\mathrm{series}}= R_{\mathrm{1}} + R_{\mathrm{2}}$$
$$ R_{\mathrm{BE}}= R_{\mathrm{B}} + R_{\mathrm{E}} $$
$$R_{\mathrm{BE}}= 5 \, \mathrm{\Omega} + 2 \, \mathrm{\Omega}$$
$$R_{\mathrm{BE}}= 7 \, \mathrm{\Omega}$$
Now resistors \(B\) and \(E\) are replaced by a new resistor \(BE\) parallel to resistor \(D\) with resistance equal to 7 \(\mathrm{\Omega}\). To find the new resistance between these two parallel resistors, we use
$$R_{\mathrm{parallel}} = \frac{R_1 \, R_2}{R_1 + R_2}$$
$$R_{\mathrm{BED}} = \frac{R_{\mathrm{BE}} \, R_{\mathrm{D}}}{R_{\mathrm{BE}} + R_{\mathrm{D}}}$$
$$R_{\mathrm{BED}} = \frac{ 7 \, \Omega \cdot 10 \, \Omega}{7 \, \Omega + 10 \, \Omega}$$
$$R_{\mathrm{BED}} = \frac{70 \, \Omega}{17 \, \Omega}$$
$$R_{\mathrm{BED}} = 4.1 \, \Omega$$
The same steps are then repeated for the remaining two resistors \(C\) and \(A\), obtaining
$$R_{\mathrm{CBED}} = \frac{4 \, \Omega \cdot 4.1 \, \Omega}{4 \, \Omega + 4.1 \, \Omega} = 2 \,\Omega, $$
for another parallel circuit, and
$$R_{\mathrm{ACBED}}= 3 \, \mathrm{\Omega} + 2 \, \mathrm{\Omega} = 5 \, \Omega$$
for the final series circuit. The final circuit will now consist of a singular resistor equivalent to \(5 \, \Omega\) resistance.
Fundamental Equations
Fortunately, we only need to worry about ideal circuits, rather than circuits with varying resistances or voltages. This means there is only a few equations to keep in mind. Firstly, we must know Ohm's law
$$ V= I \, R, $$
where \(V\) is the voltage in volts (\(\mathrm{V}\)), \(I\) is the current in amperes (\(\mathrm{A}\)), and \(R\) is the resistance in ohms (\(\mathrm{\Omega}\)). It allows us to calculate the resistance at a point in a circuit.
We also must calculate the power in a resistor, which can be calculated with
$$P= I \, V $$
where \(P\) is the power measured in watts (\(\mathrm{W}\)), \(I\) and \(V\) are the same variables as defined above.
Kirchhoff's Current and Voltage Laws
Besides Ohm's law, there are two main laws we have to know for circuit analysis, Kirchhoff's current and voltage law.
Kirchhoff's current law states that the amount of current entering a circuit is the same as the amount of current leaving a circuit.
Mathematically, it can be expressed as
$$ I_{\mathrm{entering}} = I_{\mathrm{leaving}}.$$
We use this when it comes to calculating the current entering or leaving a branch of a parallel circuit.
Kirchhoff's voltage law states that the sum of potential differences in a loop is equal to zero.
Mathematically, it can be expressed as
$$ V_{\mathrm{rise}} + V_{\mathrm{drop}} = 0. $$
In other words, the total voltage that enters a circuit is divided between all the components until there is zero voltage at the end of the circuit.
Circuit Analysis Types
There are three main methods that we can use to perform circuit analysis. Each of these three methods will return the same results.
The first method is to directly apply the fundamental laws that we discussed earlier. This is a combination of both Ohm's law and Kirchhoff's voltage and current laws. This is the most likely technique you will use to analyze a circuit at a high school level.
The second technique we can use to perform circuit analysis is called the node voltage method, and is based on Kirchhoff's current law. This technique requires us to use two equations.
The third and final technique we can use is called the mesh current method, and it also uses a system of two equations.
Both the second and the third methods are extremely efficient and elegant ways to streamline circuit analysis. With small circuits, this level of efficiency is not really substantial, however as soon as the circuit in question becomes large, they can be extremely useful for quickly calculating any necessary values within said circuit.
The simple circuits you will be dealing with at a high school level aren't too common in everyday life. Meanwhile, complex circuits are all around us. A familiar example would be a PC motherboard. If we wanted to perform circuit analysis on such a large circuit, simplifying the hundreds of components and applying an approach such as Kirchhoff's laws could be done, however, wouldn't be efficient or long-lasting. As a result, simulators and software tools have been developed to perform circuit analysis that can handle highly complex circuits like these automatically.
Circuit Analysis Techniques
Now that we know what the different types of circuit analysis require, let's take a look at how they each work, starting with applying the fundamental laws.
Applying the Fundamental Laws
Performing a circuit analysis by applying the fundamental laws is deceptively simplistic. If you follow these steps, then you will be able to perform circuit analysis on any given circuit.
Using the sign convention for passive components, label the voltages and the currents.
Select an independent variable to create the simplest equations. In the case of circuits, you need to pick between selecting either the voltage or the current. The way you choose is by figuring out how many unknowns you have in each case. If you have more unknown voltages, it would be beneficial for you to choose to use current as the independent variable, and vice versa.
You have to write these equations using either Kirchhoff's current law or Kirchhoff's voltage law, and in some cases, you may end up using both! When performing this step, make sure that every element in your circuit is included in at least one of the equations.
Once you have your system of equations, you should be able to solve algebraically to find every unknown value you want to know.
The Node Voltage Method
The node voltage method is another method we can use for performing circuit analysis, and it's based on Kirchhoff's current law.
Node voltage is the potential difference between two nodes on a circuit. A node is a point on the circuit where two or more branches meet, or connect.
The main challenge that rises from using the node voltage method is that we need to perform twice as many equations as there are components in the circuit. To complete the node voltage method, we have to do the following steps:
Assign one node as the reference node. At this point, the voltage is zero, and we measure the voltage across each other node from the ground node. Once we have assigned the reference node, we assign names to each of the other nodes.
Solve the easiest nodes first. The easiest node to solve is the node with a power source connected directly to it.
Next, calculate Kirchhoff's current law for each node. On top of this, you should also calculate the resistance using ohm's law, and immediately write the current in terms of resistance for each node.
Once you've calculated and written Kirchhoff's current laws regarding the current using Ohm's law, you will find you have a system of equations. Solving this system of equations will provide you with the voltage across a node.
Finally, after calculating both of the node voltages, you can solve for any unknown currents that you want to find using Ohm's law.
The Mesh Current Method
The final method we can use for performing circuit analysis is the mesh current method. This method of circuit analysis is similar to the node voltage method, in the sense that it requires us to solve twice the number of elements in the circuit of equations. The mesh current method is based on Kirchhoff's voltage law.
Before we go any further, we need to introduce some new terminology: loops and meshes.
Regarding the mesh current method, a loop is any closed path around a circuit, that does not cross over itself. You pick a starting point and draw a loop until you get back to that same starting point.
A mesh in the mesh current method is any loop that does not contain a smaller loop within itself.
From the definitions above, we can see that all meshes are loops, but not all loops are meshes! But what steps do we need to take to use the mesh current method?
When we are using the mesh current method, we only really want to deal with meshes. Therefore, our first step is to identify the meshes in the circuit.
Once you have identified every mesh in the circuit, assign each with a current variable, using a consistent direction around the mesh for each.
Complete Kirchhoff's current law around each mesh. This will give you a system of equations.
With our system of equations, we can solve to get the current within each mesh.
And finally, once you have the individual mesh currents, you can solve for individual element currents and voltages using Ohm's law.
Circuit Analysis Examples
Let's look at an example problem for each of these methods, following all the steps stated earlier.
Fundamental Laws Method Example
To demonstrate the straight forwardness of the fundamental laws method, let's use the resistor example from earlier.
After simplifying the seemingly complicated circuit involving five resistors, we found that the \(10 \, \mathrm{V}\) voltage source is producing enough current to drive \(5 \, \Omega\). Taking that into account, the circuit diagram can be simplified into the one visible below.
Now we can apply Ohm's law
$$ V = I \, R$$
to find that the current around the circuit is
$$ I = \frac{V}{R} = \frac{10 \, \mathrm{V}}{5 \, \mathrm{\Omega}}$$
$$ I = 2 \, \mathrm{A}$$
Node Voltage Method Example
Now let's look at a more complicated example. In this case, we will use the node voltage method; however, it's up to you to decide which method to use, as both will lead to the same result.
The circuit diagram in the figure below consists of three resistors and two voltage sources. Find the current flowing through \(R_1\) and \(R_3\).
When deciding which method to use, count the number of nodes and meshes present. The one with the least elements will require fewer equations and will be more efficient!
Here, the circuit has one principal node (the other one on the bottom is considered a reference node) and two meshes, so the node voltage method is the more logical choice. Based on the definition of the method, we will use the Kirchhoff's current law.
The one unknown value of voltage we are aiming to find is located at the principal node, so let's label it \(V_{\mathrm{N}}\). At this node, the voltage is the highest, in contrast to the two voltage sources provided,
$$ V_{\mathrm{N}} > 8 \, \mathrm{V} > 4 \, \mathrm{V} $$
as we are assuming that the current is leaving from this point.
Considering current is only leaving the node, the \(I_{\mathrm{entering}}\) is zero. Mathematically, it can be expressed as follows
$$ I_1 + I_2 + I_3 = 0.$$
Now we can re-express each current using Ohm's law, where the voltage will be the potential difference between the principal node and each voltage source as follows:
$$ \frac{V_{\mathrm{N}} - V_1}{R_1} + \frac{V_{\mathrm{N}}-V_2}{R_2} + \frac{V_{\mathrm{N}}-V_3}{R_3} = 0.$$
After plugging in our values from the circuit diagram we obtain
$$ \frac{V_{\mathrm{N}} - 4 \, \mathrm{V}}{3 \, \Omega} + \frac{V_{\mathrm{N}}-0 \, \mathrm{V}}{6 \, \Omega} + \frac{V_{\mathrm{N}}-8 \, \mathrm{V}}{3 \, \Omega} = 0.$$
The common denominator in this case is \(6 \, \Omega\), so the first and third terms are multiplied by \(2\) so that equation simplifies into:
$$ 2 \, V_{\mathrm{N}} - 8 \, \mathrm{V} + 2 \, V_{\mathrm{N}} - 16 \, \mathrm{V} + V_{\mathrm{N}} = 0 $$
$$ 5 \cdot V_{\mathrm{N}} = 24 \, \mathrm{V}$$
$$ V_{\mathrm{N}} = 4.8 \, \mathrm{V} $$
Now we can use this newfound value to find the current flowing through each resistor by once again implementing the Ohm's law to obtain the following values:
$$ I_1 = \frac{4.8 \, \mathrm{V} - 4 \, \mathrm{V} }{3 \, \Omega} = 0.3 \, \mathrm{A}$$
and
$$ I_2 = \frac{4.8 \, \mathrm{V} - 8 \, \mathrm{V} }{3 \, \Omega} = -1.1 \, \mathrm{A}.$$
The negative sign in front of the \(I_2\) current indicates that the direction we initially assumed is opposite to the actual direction of the current and should be flipped.
Mesh Current Method Example
Let's solve the same example as in the node voltage method, only using the mesh current method, to confirm that results in both cases are identical.
Even though we established that in this particular circuit, the node voltage method is more effective, doing the analysis using the mesh current method should give us the same results. We can see the same circuit diagram as before in the figure below, but this time it showcases the two meshes. Let's find the value for current in each mesh. Based on the definition of the method, we will use the Kirchhoff's voltage law.
Let's analyze each mesh, separately.
In the first mesh,
- The current is flowing in the clockwise direction; therefore, the source voltage \(V_S\) will have a negative sign;
- The voltage in the first resistor can be expressed using Ohm's law and will have a positive sign because of the positive direction of the current;
- The second resistor is shared between the two meshes and has two currents (\(I_1\) and \( I_2\)) flowing through it, so both will contribute to the voltage, and once again can be expressed using Ohm's law;
- It's important to note, when working in the first mesh, the \(I_1\) is taken as positive and \(I_2\) is considered negative. The signs will flip when working with in the second mesh.
Mathematically, all these observations can be expressed as follows
$$ V_S + V_1 + V_2 = 0$$
$$ V_S + R_1 \, I_1 + R_2 \cdot (I_1 - I_2) = 0$$
$$ -4 \, \mathrm{V} + 3 \, \Omega \cdot I_1 + 6 \, \Omega \cdot (I_1 - I_2)= 0$$
After simplifying this expression, we obtain the first equation in our system of equations
$$ 9 \, I_1 - 6 \, I_2 = 4 \, \mathrm{V}$$
where \(I_1\) and \(I_2\) are the two unknown values.
In the second mesh,
- The loop starts from the shared resistor \(R_2\), so we express the voltage using the Ohm's law and consider both currents that flow through it, remembering the sign convention mentioned earlier;
- Voltage across \(R_3\) is expressed using Ohm's law;
- Voltage in the second source is given and will have a positive sign as the loop enters the voltage source from the positive end.
All of these observations lead to the following expression
$$ 6 \, \Omega \cdot (I_2 - I_1) + 3 \, \Omega \cdot I_2 + 8 \, \mathrm{V} = 0 \, \mathrm{V}$$
which simplifies into our second equation
$$ 9 \, I_2 - 6\, I_1 = -8. $$
By combining the two equations containing the two unknowns, we find the currents flowing through each mesh to be
$$ I_1 = -\frac{4\, \mathrm{V} }{15\, \Omega } = - 0.3 \, \mathrm{A}$$
and
$$ I_2 = -\frac{16\, \mathrm{V} }{15\, \Omega } = - 1.1 \, \mathrm{A}$$
which are the same as the results obtained in the example above. The negative sign indicates that the actual direction of the current in each mesh is opposite to what we initially chose.
Analysis of Electrical Circuits - Key takeaways
- Circuit analysis is the mathematical analysis of any electrical circuit.
- A circuit diagram can be analyzed by breaking down the schematics of a circuit, simplifying the resistors into one, and applying fundamental laws of physics.
- There are three main circuit analysis techniques: the fundamental laws, using the node voltage method or the mesh current method.
- Ohm's law is a fundamental law used to analyze electrical circuits.
- Mathematically, Ohm's law can be expressed as \(V = I \, R\).
- The node voltage method is based on Kirchhoff's current law.
- The mesh current method is based on Kirchhoff's voltage law.
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