ECE 249 Unit 1 Fundamentals of DC circuits

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1. Resistance (R)

  • Definition: Opposition to the flow of electric current.
  • Unit: Ohm (Ω).
  • Principle: Governed by Ohm's Law, where V=IR. This means that the voltage (V) across a resistor is directly proportional to the current (I) flowing through it, with resistance (R) as the proportionality constant.
  • Working: Governed by Ohm's Law, where V=IR. This means that the voltage (V) across a resistor is directly proportional to the current (I) flowing through it, with resistance (R) as the proportionality constant.
  • Uses:  Control current flow in circuits, divide voltage, and limit current in various applications.
  • Advantages
        • Simple and reliable components.
        • Easily available and low-cost.
Disadvantages
      • Energy loss as heat (dissipation), leading to inefficiency in high-power applications.
      • Limited power handling capacity can lead to overheating.
Formula: The resistance of a conductor can also be calculated using the formula:
                                                                        R=V/I

  • ​This can also be related to the physical properties of the material:

    ​R=ρ(L/A)

    Where:

    R = Resistance (in ohms)

    ρ = Resistivity of the material (in ohm-meters)

    L = Length of the conductor (in meters)

    A = Cross-sectional area of the conductor (in square meters)

Resistor


2. Inductance (L)

  • Definition: Inductance is the property of an electrical conductor by which a change in current flowing through it induces an electromotive force (EMF) in the same conductor or in a nearby conductor..
  • Unit: Henry (H).
  • Principle: Faraday's Law states that a change in magnetic flux can induce an EMF.
  • Working: When current changes, it creates a changing magnetic field, inducing voltage that opposes the change in current (Lenz's Law).
  • Uses:
      • Inductors: In filters, energy storage, and tuning circuits.
      • Transformers: Transfer electrical energy between circuits.
  • Advantages
      • Useful in AC applications to control current flow.
      • Can store energy temporarily.
  • Disadvantages
      • Can create phase shifts between current and voltage.
      • Size and weight can be an issue in some applications.

Formula: V=LdidtV = L \frac{di}{dt}
  • Where:
    • VV = Induced voltage (Volts, V)
    • LL = Inductance (Henries, H)
    • didt\frac{di}{dt} = Rate of change of current (Amperes per second, A/s)

  • Energy Stored in an Inductor:
  • W=12LI2
  • Where:
    • W= Energy (Joules, J)
    • I = Current through the inductor (Ampere,A)

Inductors

3. Capacitance

  • Definition: Capacitance is the ability of a system to store an electric charge.
  • Unit: Farad (F)
  • Principle: Defined by the relationship Q=CVQ = CV, where QQ is the charge stored, CC is the capacitance, and VV is the voltage across the capacitor.
  • Working: A capacitor consists of two conductive plates separated by an insulating material (dielectric). When voltage is applied, charge accumulates on the plates, creating an electric field.
  • Uses:
    • Energy storage in power supply circuits.
    • Timing applications in oscillators and filters.
  • Advantages:
    • Fast charging and discharging capabilities.
    • Low energy loss in ideal capacitors.
  • Disadvantages:
    • Limited charge storage capacity.
    • Leakage current over time can reduce effectiveness.

  • Formula:
C=QV
  • Energy Stored:
W=12CV2
  • Derivation: Capacitance is defined as charge per unit voltage, and energy is derived from integrating the voltage over the charge.

4. Voltage

  • Definition: Voltage is the electric potential difference between two points in a circuit, representing the work done to move a charge.

  • Unit: Volt (V)

  • Principle: Voltage drives current through a circuit, enabling the flow of electrical energy.

  • Working: Voltage is generated by sources such as batteries and generators, creating a potential difference that causes current to flow.

  • Uses:
    • Power supply for electronic devices and appliances.
    • Measurement of electrical energy in circuits.

  • Advantages:
    • Fundamental for circuit design and analysis.
    • Easily measurable with voltmeters.

  • Disadvantages:
    • High voltages can be dangerous and require safety precautions.

  • Formula: V=IRV = IR
    • Work Done:
    W=QVW = QV
    • Derivation: Voltage is defined through Ohm's Law and is also related to work done on a charge.

5. Current

  • Definition: Current is the flow of electric charge through a conductor.

  • Unit: Ampere (A)

  • Principle: Current is the amount of charge that flows past a point in a circuit per unit time.

  • Working: Current flows due to a potential difference and is influenced by the resistance in the circuit.

  • Uses:

                    Essential for powering devices, from light bulbs to motors.

                    In signal processing, current carries information.


Advantages:

                  Essential for circuit functionality and performance.

                  Can be controlled and measured easily.


Disadvantages:

Excessive current can cause overheating and damage components.


Formula: 

                                             I=QtI = \frac{Q}{t}
  • Where:

    • I       = Current (Amperes, A)
    • = Charge (Coulombs, C)
    • tt= Time (seconds, s)


Derivation: Current is defined as charge divided by time, representing the flow of electrons.


6. Power

  • Definition: Power is the rate at which electrical energy is transferred or converted.
  • Unit: Watt (W)
  • Principle: Power can be calculated as the product of voltage and current.
  • Working: Power indicates how quickly energy is being used or generated in a circuit.
  • Uses:
    • Ratings for electrical devices (e.g., light bulbs, appliances).
    • Monitoring and managing energy consumption in homes and industries.
  • Advantages:
    • Key for designing efficient systems and understanding energy use.
  • Disadvantages:
    • High power levels can be dangerous and require careful handling.
  • Formulas:
    • Basic Formula:
    P=VI
    • Alternative Forms:


    P=I2R    P=V2RP = \frac{V^2}{R}
    • Derivation: Power is derived from the relationship between voltage, current, and resistance in Ohm’s Law.

  • Definition: Energy is the ability to perform work. In electrical circuits, it is the product of power (the rate of doing work) and time.

  • Formula:

    E=P×tE = P \times t

    Where:

    • EE is energy (in joules, J)
    • PP is power (in watts, W)
    • tt is time (in seconds, s)


7.Ohm’s Law

Ohm’s Law establishes a fundamental relationship between voltage, current, and resistance in an electrical circuit.

  • Definition: It states that the current (II) passing through a conductor between two points is directly proportional to the voltage (VV) across the two points and inversely proportional to the resistance (RR).

  • Working:

    • In a circuit, Ohm’s Law explains how increasing voltage increases current if resistance is constant, and how increasing resistance decreases current for the same voltage.

     Uses:

  • Designing electrical circuits, such as calculating resistor values or determining power requirements in devices.

  • Formula:

    V=I×RV = I \times R

    Where:

    • VVis voltage (in volts, V)
    • II is current (in amperes, A)
    • RR is resistance (in ohms, Ω)

  • Advantages:

    • Simple relationship makes it easy to calculate the missing variable.
    • Widely applicable to resistive circuits.

  • Disadvantages:

    • Not valid for non-linear devices (e.g., diodes, transistors).
    • Does not apply when temperature affects resistance (in materials like semiconductors).

  • Steps for Calculation:

    1. Identify known values of VV, 
      I      , or
      R      .
    2. Rearrange the formula to solve for the unknown.
    3. Plug in the values and solve.


8. Kirchhoff’s Laws

Kirchhoff’s Current Law (KCL)

  • Definition: The total current entering a junction equals the total current leaving the junction, based on the conservation of charge.

  • Formula:

    ∑Iin=∑Iout\sum I_{in} = \sum I_{out}

  • Working:

    • KCL ensures that current is conserved at a node or junction point in a circuit. All incoming currents must equal the sum of all outgoing currents.

  • Uses:

    • Analyzing current flow in complex circuits with multiple branches.

Kirchhoff’s Voltage Law (KVL)

  • Definition: The sum of all voltages in a closed loop is zero, which reflects energy conservation.

  • Formula:

    ∑V=0\sum V = 0

  • Working:

    • KVL maintains that the total voltage gains and losses in any closed loop must balance out, ensuring no net voltage in a full circuit loop.

  • Uses:

    • Voltage analysis in loops, often in AC or DC circuits, and in feedback systems.

  • Advantages:

    • Allows for circuit analysis of even complex networks by systematically applying laws.

  • Disadvantages:

    • Analysis can become complicated in circuits with multiple loops or branches.

9. Voltage Division Rule

  • Definition: In a series circuit, the voltage division rule helps calculate the voltage drop across individual resistors based on their resistance.

  • Formula:

    VRi=Vtotal×RiRtotalV_{R_i} = V_{total} \times \frac{R_i}{R_{total}}

    Where:

    • VRiV_{R_i} is the voltage across resistor
      R_i
    • VtotalV_{total}is the total applied voltage
    • RiR_iis the resistance of the ii-th resistor
    • RtotalR_{total} is the total resistance in series

  • Working:

    • Voltage is divided across resistors in proportion to their resistance. Resistors with higher resistance drop more voltage, according to the total voltage and resistances.

  • Uses:

    • Voltage dividers are used in signal conditioning circuits, sensor interfaces, and measurement circuits.

  • Advantages:

    • Simplifies voltage calculations in series circuits.

  • Disadvantages:

    • Only works in series circuits; cannot be applied to parallel circuits.

5. Current Division Rule

  • Definition: The current division rule determines the current through each branch of a parallel circuit.

  • Formula:

    IRi=Itotal×RtotalRiI_{R_i} = I_{total} \times \frac{R_{total}}{R_i}

    Where:

    • IRiI_{R_i}is the current through resistor Ri.​
    • ItotalI_{total} is the total current
    • RiR_iis the resistance of the branch
    • RtotalR_{total} is the equivalent resistance of the parallel circuit

  • Working:

    • In a parallel circuit, current is split across the branches inversely proportional to the resistance of each branch. A branch with lower resistance will have a higher current.

  • Uses:

    • Used to analyze current distribution in power distribution systems and multi-load parallel circuits.

  • Advantages:

    • Facilitates quick current analysis in parallel circuits.

  • Disadvantages:

    • Only applies to parallel circuits, not series circuits.

6. Dependent and Independent Sources

  • Independent Source: A constant voltage or current source that is unaffected by other elements in the circuit.

  • Dependent Source: A voltage or current source whose value depends on another variable in the circuit (like voltage or current at another location).

  • Working:

    • Independent sources provide a consistent supply (e.g., batteries), while dependent sources change based on other circuit conditions (e.g., in amplifiers).

  • Uses:

    • Independent sources are used in power supplies for circuits, while dependent sources are common in amplifiers, transistors, and operational circuits.

  • Advantages:

    • Independent sources: Reliable power supply.
    • Dependent sources: Mimic real-world devices like transistors and operational amplifiers.

  • Disadvantages:

    • Dependent sources add complexity to circuit analysis.

7. Mesh Analysis

  • Definition: Mesh analysis uses Kirchhoff’s Voltage Law (KVL) to calculate the current in each loop of a planar circuit.

  • Working:

    • By writing a KVL equation for each independent loop (mesh), you can solve for unknown currents. Mesh analysis involves writing equations for each loop and solving them simultaneously.

  • Uses:

    • Applied in circuits where multiple loops need to be analyzed, such as amplifier circuits, filter networks, and multi-loop power systems.

  • Advantages:

    • Reduces complex circuits into manageable equations.

  • Disadvantages:

    • Can only be applied to planar circuits where no wires cross.

  • Steps to Conduct Mesh Analysis
    • To apply mesh analysis to a particular circuit network the following steps need to be followed :-
    • Check if the network is planar or not. If rearrangement possible rearrange to make the network planar.
    • Identify the total number of meshes in the circuit network.
    • Assign mesh currents to each mesh.
    • Develop the KVL equation in each mesh.
    • Solve the KVL equations to find the mesh currents.

  • Rules of Mesh Analysis

  • Direction of the flow of current inside the mesh can be in any direction either clockwise or anticlockwise. In most cases , it is taken to be clockwise as it is simpler.
  • Direction should remain same for all the meshes.
  • The number of equation required to sole an electrical network with the help of Mesh formula is:-
    • e=M=b-(N-1)
    • Where,
      • e is the number of equations
      • M is the number of meshes
      • b is the number of branches
      • N is the number of nodes
    • We can say, number of equations = number of meshes
  • When a resistor is common to two meshes then the current of the mesh for which we are writing the equation that current is given high value and the current of the other mesh is subtracted from it.

Example: -Find the current I(current through 10 ohm resistor) using Mesh Analysis.



First assign mesh (or) loop currents to the two meshes (ABCA and BDCB) as i1 and i2. Assume the current directions as clockwise mesh ABCA

- 30i1 - 10(i1 - į2) + 120 = 0

-30i1 – 10 i1 + 10 i2 + 120 = 0

-40 i1 + 10 i2+ 120 = 0

120 = 40 i1 – 10 i2 …………(1)

Mesh BDCB

-50i2- 60 - 10(i2 – i1) = 0

-50i2 -10 i2 + 10 i1 - 60 = 0

-60 i2 + 10 i1 − 60 = 0

-60 = -10 i1 + 60 i2 …………..(2)

Multiplying equ (2) by 4 and adding the equ (2) in to equ (1)

120 = 40i1 - 10i2

-240= -40i1 +240i2

________________

-120 = 230i2

i2 = -120/230 = -0.521 A

The negative sign indicates that the direction of i2 is anticlockwise.

The i, value is substituted in equ (1)

40 i1 - 10 (-0.521) = 120

40 i1 + 5.21 = 120

40 i1 = 114.79

i1 = 114.79/40 = 2869 A

The actual direction of flow of mesh currents is shown in above fig

The mesh currents i1 = 2.869 A

i2 = 0.521 A

Current in branch CAB is

i1 = 2.869 A

Current in branch CDB is

i2 = 0.521 A

Current in branch BC is = i1 + i2

= 2.869 +0.521

= 3.39 A

8.Thevenin’s Theorem

  • Definition: Thevenin’s theorem simplifies any linear circuit to an equivalent circuit with a single voltage source and series resistance.

  • Working:

    • To find the Thevenin equivalent, remove the load resistor, calculate the open-circuit voltage, and find the equivalent series resistance by shorting all voltage sources.

  • Uses:

    • Widely used in circuit simplification, particularly in fault analysis, to model complex networks in power distribution systems.

  • Advantages:

    • Simplifies complex circuits for easier analysis.

  • Disadvantages:

    • Involves multiple steps and can be cumbersome for large circuits.

9. Norton’s Theorem

  • Definition: Norton’s theorem states that any linear circuit can be replaced with an equivalent circuit composed of a current source in parallel with a resistor.

  • Working:

    • To find the Norton equivalent, calculate the short-circuit current and the equivalent parallel resistance. This simplifies the circuit for further analysis.

  • Uses:

    • Useful in analyzing current-based systems, such as communication circuits or signal processing circuits.

  • Advantages:

    • Works similarly to Thevenin’s theorem but simplifies current analysis.

  • Disadvantages:

    • Requires similar steps as Thevenin’s theorem, which can be tedious in large circuits.

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