Electrical Circuits: Types, Laws, and Analysis Methods
Item 55: Electrical Circuits
Introduction
A circuit or an electrical network is an interconnection of electrical elements tied together in a closed path so that an electrical current can flow. These electrical elements are resistors, inductors, capacitors, voltage sources, and current sources. All these elements are characterized by two terminals and a known current-voltage relationship between the terminals.
Resistance strictly adheres to Ohm’s law: V = R ยท I
In the inductor or coil, there is a voltage across its terminals proportional to the change of current through it (V = L x di / dt). In the capacitor, the current that enters one of its terminals is proportional to the variation of voltage between them (i = C x dv / dt). An ideal voltage source provides a nominal voltage independent of the current that flows through it. In an ideal current source, the current flowing through its terminals is independent of the voltage between them.
Resistance, inductance, and capacitance are passive components and thus dissipate energy. Besides, their current-voltage relationship is linear. Sources are active components. They can provide power to the circuit.
An electrical circuit is an interconnection of any number of these components. These are an idealization of the components technologically feasible. However, to describe a real element, we can always build a model with a combination (serial, parallel, or mixed), that we will later explain, of complex elements to the degree of approximation we desire.
For the analysis of electrical circuits, in addition to the current-voltage relationships in each of its possible components, we have Kirchhoff’s laws, either as a deduction from Maxwell’s equations of the electromagnetic field or as a result of the principles of conservation of load and energy:
- Law of the Voltages (Kirchhoff’s Voltage Law – KVL): The sum of voltage drops along a loop in a circuit is zero. (Follows the principle of conservation of energy).
- Law of the Currents (Kirchhoff’s Current Law – KCL): The sum of incoming currents at a node is zero. (Follows the principle of conservation of charge).
Ohm’s Law
It states that the intensity of electric current running through an electrical circuit is directly proportional to the voltage applied between its ends and inversely proportional to the resistance of the circuit. It can be expressed mathematically in the following equation: I = V / R or V = I x R.
(Insert drawing of an electronic circuit)
Kirchhoff’s Laws
Based on Ohm’s law. To implement these laws in the calculation of electronic circuits, we must consider a number of terms used in electronic circuits:
- Electrical Network: A set of generators and receivers linked together by wires.
- Node: Point of connection of three or more conductors.
- Branch: Portion of a circuit between 2 nodes.
- Loop: Closed circuit formed by several branches connected to each other.
(Insert drawing of a mixed circuit and indicate all points)
First law: The sum of currents that reach a point of connection of multiple conductors is equal to the sum of currents that deviate from it.
(Insert drawing of branches with the central node) I1 + I2 = I3 + I4 + I5.
Second law: In any closed circuit, the total electromotive force (sum of the emf) is equal to the total voltage drop (sum of the voltage drops) in the resistors.
Series Electrical Circuits
An electrical circuit is called a series connection, or simply in series, when there is a connection of elements such that the same current intensity flows through all of them. This is done with the purpose of generating, transporting, or modifying electronic signals.
The characteristics of the series circuit are:
- The current that travels the circuit is the same throughout its stretch.
- Applying Ohm’s law, we have R1 x I = V1, V2 = R2 x I, V3 = R3 x I.
- If we substitute the values of V for their expressions, we get: Vt = R1 x I + R2 x I + …
- The total power is the sum of all partial powers consumed by each resistor: P1 = R1 x I, P2 = R2 x I …. Pt = P1 + P2 + P3.
When making a lighting circuit, this method is rarely used because if one component breaks, the others will not work.
(Insert drawing of a series circuit with 3 resistors)
Parallel Electrical Circuits
A series of parallel components are where, due to the topology of the circuit that connects them, all are subject to the same voltage or potential difference across its terminals or ends. For a number of resistors in parallel, we have that, applying a current source to the circuit and according to Kirchhoff’s law of currents, the current supplied by the source equals the current drawn by the resistors.
(Insert drawing of a parallel circuit)
Features of a parallel circuit:
- The total current consumed by the circuit is the sum of all partial currents that consume all branches of the circuit. It = I1 + I2 + I3.
- The voltage is the same at all points of the resistors. Vt = V1 = V2 = V3.
- The currents that go through each branch are I1 = Vt / R1 … and the total resistance is 1 / Rt = 1 / R1 + 1 / R2 …
- The total power equals the sum of all partial powers consumed by the resistors that comprise the circuit. Pt = P1 + P2 …. + Pn
Mixed Electrical Circuits
A combination of electrical elements connected in series and parallel. The vast majority of electronic circuits consist of mixed circuits. The most reliable way to resolve the circuit is to simplify it, which we will explain later. To simplify, we reduce the circuit to a simple circuit to facilitate the calculations.
(Insert drawing of a mixed circuit)
In an electric circuit composed of several or many components, we have two possibilities for the topology of the circuit:
- That in the circuit, we can create subsets of items with serial or parallel associations.
- That there are certain associations of irreducible components to series or parallel.
Reducible to Sets of Mixed Serial and Parallel Associations
Consider the next circuit (see figure):
(Insert figure of a reducible circuit)
In the previous circuit, R4 and R3 are in parallel (R4 // R3). R2 is in series with the set (R4 // R3). R5 is in parallel with the above R2 + (R4 // R3), and finally, R1 is in series with the foregoing R5 // (R2 + (R4 // R3)). The equivalent resistance seen from its terminals will be: Req = R1 + (R5 // (R2 + (R3 // R4))). This result can be seen by redrawing the circuit (see figure).
(Insert figure of the redrawn circuit)
Mixed Associations of Subsets Not Reducible to Series and/or Parallel
Consider the following loop:
(Insert figure of a non-reducible circuit)
The elements cannot be grouped as parallel or serial compositions. To solve it, we will have to use either a general method or, in this case, the star-delta transformation. Within private partnerships, there are two of great importance due to their widespread use in electrical wiring. These are called star and delta (or triangle) connections.
(Insert drawing of a star with 3 resistors and a triangle)
General Methods of Analysis of Circuits
Below are two general methods for analyzing any electrical circuit. These methods are based on Kirchhoff’s laws and consistently implement these laws throughout the topology of the circuit. In principle, these methods are defined for resistive circuits but are generalized to circuits with sinusoidal sources using phasors and impedances.
Mesh Current Analysis
We define a node at any point that connects more than two components. We define a branch as the path connecting two nodes. We define a loop as any path over a circuit starting from a node and returning to it without passing twice over any other node in the circuit. We define a mesh as that loop that contains no other loop within it. The meshes appear as windows in the schematic representation of the circuit. Applying Kirchhoff’s law of voltages to each of the meshes, we have that the sum of voltage drops in each mesh is equal to the sum of the voltage sources in each of these meshes. With this, we get a system with as many equations as meshes in the circuit, with unknown mesh currents. This method is most appropriate when the voltage sources prevail in the circuit.
(Insert drawing of a circuit with 3 meshes I1, I2, and I3)
Nodal Voltage Analysis
A circuit with n nodes requires n-1 equations for the voltages at the n-1 nodes, one node serves as a reference. Since a voltage is defined between two nodes, we need to identify the voltages at the n-1 nodes in relation to a reference node. In general, we usually choose the lower node of the circuit as a reference. If the circuit includes a grounded node, that node is taken as a reference. For example, consider the following circuit:
(Insert figure of a circuit for nodal analysis)
To determine the voltage on a node, we use Kirchhoff’s law for currents at each node in the circuit, except the reference node. This set of equations allows us to find the voltages at each node. We can choose any reference arbitrarily. However, it is convenient to choose one that has the highest number of branches connected. If you must choose between two nodes with the same number of branches connected, usually choose the lower node. We usually assume that the voltage at the reference node is zero. Then apply the equations of Kirchhoff’s law of currents at node a, by equating the currents that come in with those that leave it. It is also important to note that the current i1 leaving the node is I1 = Va – Vb / R1.
In general, this method solves very easily circuits where current sources dominate.
Conclusion …