**Circuit Analysis**

**Cover Image Credit – Santeri Viinamäki, MCB Circuit breakers for DIN rail, CC BY-SA 4.0**

**Points of discussion : Circuit Analysis**

**Introduction to circuit analysis****Ideal Circuit elements****Realistic circuit elements****Ideal Energy Sources****Real-World Energy Sources****Important terminologies related to Circuit analysis**

**Introduction to Circuit Analysis**

Circuit analysis is one of the primary and essential modules for Electrical and Electronics Engineering. Before exploring out the concepts and theories of circuit analysis, let us know what a circuit is.

A circuit can be defined as a closed or open loop consists of electrical and electronic components and have interconnection between them. Circuit analysis is the method to determine the necessary current or voltage value at any point of the circuit by studying and analysing the circuit. There are numerous different methods for circuit analysis and used as per suitable conditions.

**What is a DC Circuit? Learn About KCL & KVLs! ***Click Here!*

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**Ideal Circuit Elements**

*An ideal circuit can be defined as a circuit without any losses,* thus the appearance of 100% input power at the output side. An ideal circuit consists of three ideal elements. They are – Resistances, capacitor, Inductor.

**Resistors:**Resistors are passive electrical components used to resist the flow of electrons in a circuit. The voltage across the resistor is expressed by a famous law, known as Ohm’s law. It states that. If V and I respectively denote the voltage value and current, then**“the voltages are directly proportional to the currents”**

**V ∝ I**

**Or, V = IR**

Here R represents the resistance or resistor value. * The unit is given by ohm(Ω). * The following image

represents the resistor –

The following mathematical expression gives the power stored by a resistor.

**P = VI**

**Or, P = (IR) I**

**Or, P = I ^{2}R**

**Or, P = V ^{2} / R**

**Capacitor:**A typical capacitor is a passive electrical equipment which stores electrical energy inside an electric field. It is a two-terminal device. Capacitance is known as the effect of the capacitor. Capacitance has a unit – Farad(F). The capacitor is represented in the circuit by the following image.

The relation between charges and capacitance is given by **Q = CV**, where C is the capacitance value, Q is the Charge, V is the applied voltage.

The current relationship can be derived from the above equation. Let us differentiate both side with respect to time.

**dQ/dt = C dV/dt; C is a constant value**

**Or, I = C dV/dt; as I = dQ/dt.**

Power stored in a capacitor can be described written as

**P = VI**

**Or, P = V C dV/dt**

**Now, the energy is given as U = ∫ p dt**

**Or, U = ∫ V C (dV/dt) dt**

**Or, U = C ∫ V dV**

If we assume that the capacitor was discharged at the beginning of the circuit, then the power comes as **U = ½ CV ^{2}.**

**Inductor:**Inductor is another passive device present in an ideal circuit. It holds energies in a magnetic field. The unit of inductance is given by Henry(H). The relation between voltage and inductance is given below.

**V = L dI/dt**

The reserved energies are returned back to the circuitry in current form. The following image represents the inductor in the circuit.

The power of an inductor is given as **P =VI.**

**Or, P = I * L (dI/dt)**

**Again, the energy U = ∫ p dt**

**Or, U = ∫ I * L (dI/dt) dt**

**Or, U = L ∫ I dI**

**The energy comes as U = ½ LI ^{2}.**

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**Realistic Circuit Elements**

Ideal circuit components are for ideal circuits. They are not applicable in real circuits. However, the main characteristics remain the same for the elements. Elements suffer some loss, have some tolerance values and some abstractions while using it.

The working principles and equations get changed in real domains. Also, some other factors get added during operations. For example, capacitors work differently in high-frequency domains; resistors generate a magnetic field during operations.

**Resistors**: The real-world resistors should be made to obey Ohm’s law as close as they can. The resistance offered by a resistor depends upon the material and shape of the resistor.

A real resistor maybe gets destroyed or burned out due to heat generated by itself. There is a certain tolerance level mentioned for every resistor via the color codes.

**Capacitors:**The realistic capacitors should be made to obey the capacitor’s equation as close as possible. Two conducting surfaces are needed to build a capacitor. They are placed together, and air or any material is filled in between them. The capacitor value is dependent on the surface area of the conductor and the distance between them and upon the permittivity of the inside material. There are various categories of capacitors in the market. Some of them are –,**Electrolytic Capacitors**etc.**Tantalum Capacitors,**

Capacitors are connected with wire at their terminals. That causes resistance and a small amount of impedance. An increase in voltage across the capacitors sometimes damages the insulative materials between the plates.

**Inductors:**The realistic or real-world inductors should be made to obey the inductor equation as close as possible. Inductors are choke of coils. They induce magnetic fields to store electrical-energies.

Inductors are made using the winding wires in a coil-like structure: the more the winding, the stronger the magnetic field. Placing a magnetic material inside the coil would increase the magnetic effect. Now, as these wires are wounded around the material, this causes the generation of resistance. Also, it is needed to be large enough to accumulate the magnetic field. That sometimes causes problems.

**Ideal Energy Sources**

An ideal circuit needs an ideal source of energy. There are two types of ideal energy sources. They are – ideal voltage source and ideal current source.

Ideal Voltage Source: Ideal voltage sources supply a constant amount of voltage for every instant of time. Voltage is constant throughout the source. In reality, there is no ideal source for circuits. It is an assumption to simplify the circuit analysis. The below image represents an ideal voltage source.

Ideal Current Source: Ideal current sources supply currents independent of the variation of voltage in the circuit. An ideal current source is an approximation that does not take place in reality but can be achieved. The below picture represents the ideal current source in a circuit.

**Real energy sources for circuits**

Real electrical or electronic circuits need natural sources of energy. There are some differences between ideal and real-world energy sources though the main principle of supplying the energy to the circuit remains the same. Real-world energy sources have several types. Some are even dependent upon other sources. Like – Voltage controlled current source, Current controlled current source, etc. We will discuss them briefly in this circuit analysis article.

**Voltage Sources:**Real voltage sources come up with an internal resistance, which is consider it to be in series with the voltage source. No matter how negligible the resistance is, it affects the V-I characteristic of the circuit. The voltage source can be of two types –

**Independent Voltage Source****Dependent Voltage Source**

**Independent Voltage Sources: **These voltage sources have no dependency upon any other energy sources of the circuit. It provides a small resistance, which changes the V-I characteristic plot.

**Dependent Voltage Sources: **These voltage sources are dependent upon any other energy sources present in the circuits. They can be classified into two categories

**Voltage Controlled Voltage Source****Current Controlled Voltage Source.**

**Voltage Controlled Voltage Source:**If any other voltage source is controlled by any kind of voltage source, it is known as Voltage controlled voltage source. V0 = AVc gives voltage output; Here, A represents the gain, and Vc is the controlling voltage.**Current Controlled Voltage Source:**If any other voltage source is controlled by different current source in the circuit, it is known as a current-controlled current source. V0 = AIc gives the output; Here, A represents the gain, and Ic controls the current.

**Current Sources:**Real current sources come up with internal resistance. The resistance may be negligible but has its effect throughout the circuit. Current Source can be of two kind.

**Dependent Source****Independent Source**

**Independent Source: **These current sources have no dependency upon any other energy sources of the circuit. It provides a small resistance, which changes the V-I characteristic plot.

**Dependent Current Sources: **These current sources are dependent upon any other energy sources present in the circuits. They can be classified into two categories

**Current Controlled Current Source****Voltage Controlled Current Source.**

**Current Controlled Current Source:**If any other current source controls any current source, then it is known as a current-controlled current source. I0 = AIc gives the output; Here, A represents the gain, and Ic is the controlling current.**Voltage Controlled Current Source:**If any current source is controlled by any other current source in the circuit, it is known as a voltage-controlled current source. I0 = AVc gives the output; Here, A represents the gain, and Vc controls voltage.

**Important terminologies related to circuit analysis**

Circuit analysis is a vast field which includes years of researches by scientist and inventor. It has grown up with lots of theories and terminologies. Let us discuss some of the primaries yet important circuit theory terminologies, which will be required throughout the sections.

**Elements / Components:****Any electrical device present and connected in the circuit is known as Elements or components of the circuit.****Node / Junction:***Nodes are the junctions where two or more elements get connected.***Reference Node:****Reference nodes are arbitrarily selected nodes as a reference point to start the calculation and***analyse the circuit.***Branches:***Branches are the parts of the circuit that connects the nodes. A branch consists of an element like a resistor, capacitors, etc. The number of branches gives us the number of elements in the circuitry.***Loop:***Loop: Loops are enclosed paths whose start point and finishing point are same.***Mesh:***Meshes are the minimal loop within an electrical circuit without any overlapping.***Circuit:****The word ‘circuit’ is originated from the word ‘Circle’. A typical circuit is referred to as the interconnected assemblies of different electrical and electronic equipment.**

**Port:****Port is referred to as the two terminals where the same current flows as the other.****Ground:***Ground is considered as one of the reference nodes and has some characteristics. It is a physical connection that connects to the earth’s surface. It is vital for the safety of the circuit. The below image represents the representation of the ground in a circuit.*

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