DC Circuits: Fundamentals

Fundamental of DC Circuits

• Basics of Electrical & Electronics Engineering (01EE1101) is the subject of the notes.
• The notes describe the fundamentals of DC circuits, according to Marwadi University's Electrical Engineering Department.
• The outline covers current, voltage, EMF, power, energy, resistance, open circuits, short circuits, Kirchhoff's Laws, and nodal and mesh analysis of electrical networks.

Charge (Q)

• Coulomb (C) is the unit
• One coulomb is equal to one ampere second, with 1 coulomb being 6.24 × 10^18 electrons.
• Coulomb is the quantity of electricity flowing past a given point when a current of one ampere passes for one second.
• Charge in coulombs can be calculated as Q = It, where I is the current and t is the time.
• The charge equals Q = ne, where 'n' is the number of electrons and 'e' is the charge of one electron.

Positively and Negatively Charged Objects

• Electrons are removed, making the object electron deficient for positively charged matter.
• Electrons are added, creating an excess of electrons for negatively charged matter.

Current (I)

• Electric pressure, or voltage, leads electrons to move in a direction.
• The movement of free electrons, drift, creates electric current flow and is the rate of movement of charge.
• Ampere (A) is the unit of current.
• The current equals one ampere when the drift of electrons takes place at one coulomb per second.

Electric Potential / Electromotive Force (e.m.f)

• Voltage is the development of electrical energy, such as a battery or dynamo.
• Volt (V) is the unit of electric potential, where one volt equals one joule per coulomb.
• Electrical Potential = Work done/Charge, V=W/Q, V=dw/dq.
• A change in electric potential between two points in an electric circuit is called potential difference.

Electrical Power and Energy

• When a direct current of I amperes traverses an electric circuit with a voltage V, the power in watts P=VI.
• Electrical energy = Power × time = VIt joules.
• Power indicates the rate of change of energy, expressed as Power = energy/time, P = W/t, p = dw/dt.
• Kilowatt hour (kWh) is the unit used for energy.
• 1 kWh equals 1000 watt hour, which is 1000 × 3600 watt seconds or joules, and equals 3,600,000 J.

Resistance (R)

• The flow of electric current experiences friction in any material.
• Resistance opposes current in a conductor.
• Ohm (Ω) is the unit of resistance. One ohm creates a current of 1 ampere when 1 volt is connected.
• Resistance (R) = potential difference (V) / Current (I).
• Resistance is directly proportional to length (R ∝ l) and resistivity (R ∝ ρ), and inversely proportional to area (R ∝ 1/A); R = ρl/A.

Specific Resistance / Resistivity (ρ)

• Resistivity is the Resistance of a material based on material having a single unit dimension.

Ohm's Law

• Ohm's law states current I, flowing in a circuit, is directly proportional to the applied voltage V and inversely proportional to the resistance R, so long as the temperature remains constant.
• Ohm's Law equation: I = V/R, V = IR, or R = V/I.

Limitations of Ohm's Law

• Applies to unilateral networks (diode, transistors, etc.)
• Does not apply to non-linear elements (thyristor, electric arc, etc.)

Factors Affecting Resistance

• The dependence of a material's resistance relies on four factors which include, material, length, cross-sectional area and temperature.
• The higher the resistivity, the greater the resistance of a conductor.
• The longer the conductor, the greater the resistance.
• The greater the area of a conductor, the less the resistance.

Resistors in Series

• Resistors in series are daisy chained together in a single line, so the same current flows through them.
• In a series circuit, the total voltage equals the sum of individual voltages.
• The total resistance is the sum of all the series resistors.

Resistors in Parallel

• Parallel resistors have both terminals respectively connected to each terminal of the other resistor.
• Parallel circuits entail the current having more than one path.
• The total current entering a parallel resistive circuit is the sum of all currents flowing in all the parallel branches.
• The equation for the total resistance in parallel is 1/RT = 1/R1 + 1/R2 + 1/R3 + 1/R4 + ... + 1/Rn

Series and Parallel Circuits: Example 1

• A 9V battery connects to a circuit of four 20Ω and one 10Ω resistors in series, with negligible internal resistance.
• Equivalent resistance equals Req = R1 + R2 + R3 + R4 + R5 = 20 + 20 + 20 + 20 + 10 = 90 Ω.
• Current in the circuit equals I = V / Req = 9 / 90 = 0.1A.
• Potential drop across each resistor: V1 = V2 = V3 = V4 = (0.1) × 20 = 2 V, V5 = (0.1) × 10 = 1 V, and V1 + V2 + V3 + V4 + V5 = 9 V.

Series and Parallel Circuits: Example 2

• Total current IT taken from a 12V supply needs to be calculated.
• R2 and R3 are in series, therefore they can be combined ( (R2 + R3 = 8Ω + 4Ω = 12Ω)
• RA (R2 + R3), R4 as parallel (1/RA = 1/R4 = 1/12 + 1/12 = 0.1667)
• R(combination) = 1/0.1667 = 6Ω
• Resultant resistive circuit consists of R1 in series, then the resistance becomes R(ab) = Rcomb + R1 = 6Ω + 6Ω = 12Ω
• The Circuit Current is equivalent to Circuit Current (I) = V/R = 12/12 = 1 Ampere.

Series and Parallel Circuits: Example 3

• Find the equivalent resistance, REQ for the following resistor combination circuit.
• RA is in series with R7: RA + R7 = 4 + 8 = 12Ω.
• Rₐ= (R₈ × (R₉ + R₁₀)) / (R₈ + R₉ + R₁₀) = 6 × (10 + 2) / 6 + 10 + 2 4Ω.

Comparision between series and parallel circuits

• Series circuits have the same current all the way
• Parallel circuits the current splits at the junction
• Series circuits add resistors in series increases total resistance in circuit.
• Parallel Circuits add resistors in parallel decreases total resistance in circuit
• Total resistance is the sum of all individual resistances in series circuit.
• Total resistance is given by the formula 1/Total resistance = 1/R1 + 1/R2 in a parallel circuit.
• Potential difference across each resistor can be calculated via V = IR in both circuit types
• In a series circuit, the potential difference is shared between the resistors. This is because the energy from the cell (EMF) is shared between resistors.
• In a parallel circuit, the potential difference across each resistor is the same as the PD across the cell (EMF).
• If one of the resistors broke, the series circuit would be broken instantly and no current would flow
• Parallel Circuit’s If one of the resistors broke, the current could still flow through the second resistor, although the current would be smaller because there would now be greater total resistance in the circuit.

Voltage Sources

• Voltage Sources: In general, there will be a current flowing through a voltage source that can be positive, negative, or zero, depending on how the source connects into the circuit.
• Ideal Independent Voltage Source: The ideal independent voltage source maintains a fixed voltage across its terminals regardless of the current through it.
• Ideal Dependent Voltage Source: The ideal dependent (or controlled) voltage source maintains a voltage across its terminals depending on a voltage/current in the circuit.

Current Sources

• Current Sources: In circuits generally, there will be a voltage across a current source that can be positive, negative, or 0 depending on how it connects into the circuit.
• Ideal Independent Current Source: The independent current source maintains a fixed current through its terminals regardless of the voltage.
• Ideal Dependent Current Source: The ideal dependent current source maintains a current through its terminals and depends on voltage/current.

Open Circuit and Short Circuit

• Open Circuit: This exists when there is nothing attached to the terminals. Open circuit means RL = ∞. The voltage across the terminals in this case is the open circuit voltage.
• Short Circuit: the condition occurs when a wire connects between the terminals; in other words, RL = 0. The current flowing through the wire is the short circuit current.

Open Circuit and Short Circuit Differences

• Current through an open circuit is zero; current through a short circuit is infinite.
• An open circuit has infinite resistance, but a short circuit has zero resistance.
• The voltage through a short circuit measures zero, but maximum via an open circuit.
• An ohmmeter displays "0" across a short, while the meter shows "infinity" or "OL" across an open.
• A short circuit involves a low resistance wire across the circuit, while an open circuit has a broken connection.

Open Circuit and Short Circuit: Resistance

• Infinite resistance in an open circuit.
• Zero resistance in a short circuit.
• The symbol Ω = infinite for an open circuit.
• The symbol Ω = zero represents a short circuit.
• An ohmmeter displays '0' (or very small values) ohms in a short circuit.
• Ohmmeters can display 1 or OL in open circuits.

Open Circuit and Short Circuit: Current

• Current needs a path to flow. No electrons flow in the open because there is no path.
• Higher resistance equals lower current.
• Ohm's law equation for current, I= V/R.
• Current equals zero in an open circuit because I = V/infinite = 0.
• Current is infinite in a short circuit, where I = V/0 = Infinite.

Open Circuit and Short Circuit: Voltage

• Voltage across a short circuit registers zero.
• Voltage across open terminals equals the supply voltage.

Kirchhoff's Current Law

• The algebraic sum of currents entering a node is zero.

Kirchhoff's Voltage Law

• The algebraic sum of all voltages around a closed path equals zero.

Kirchhoff's Laws: Definitions

• Network: Interlaced connection of two elements.
• Circuit: Contains at least one pathway or closed path.
• Node: Interlaced connections between two or more elements.
• Junction: Three or more junctions connecting together.
• Branch: Portion between two junction points is called as branch.
• Loop: Closed circuit or closed path.
• Mesh: Most basic element, which cannot be further broken down.

Circuit Analysis by Kirchhoff's Laws: Example

• Calculate the values V₁ and V₂ for the series circuit, with a terminal voltage of 9V, consisting of four 20Ω and one 10Ω resistor, with negligible internal resistance.

Nodal Analysis

• A loop that doesn't include other loops is a mesh.
• Mesh variables include meshes rather than current in determining the circuit.
• Mesh analysis involves the Kirchhoff's Voltage Law (KVL) to determine unknown current in the given circuit.
• It is also known as mesh-current method or loop analysis.
• To analyze mesh's technique, confirm transformation is possible, assign current directions, apply KVL to each mesh in voltage source an simplify, and finally solve simultaneous functions.

Nodal Analysis & Mesh Analysis: Procedure

• Identify the principal nodes and choose one as a reference (ground), then write nodal equations using KCL. Solve to determine voltages.