Unit 1: Fundamentals of DC and AC circuits

1. Basic Electrical Quantities and Components

1.1 Voltage (V)

• Definition: Voltage, also known as potential difference or electromotive force (EMF), is the work done per unit charge. It is the electrical pressure that causes charge (current) to flow in a circuit.
• Analogy: Similar to water pressure in a pipe. Higher pressure causes more water to flow.
• Unit: Volt (V). 1 Volt = 1 Joule / Coulomb.
• Measurement: Measured across two points in a circuit using a voltmeter (connected in parallel).

1.2 Current (I)

• Definition: Current is the rate of flow of electric charge (electrons) through a conductor.
• Analogy: Similar to the flow rate of water in a pipe.
• Formula: I = dQ/dt, where I is current, Q is charge, and t is time.
• Unit: Ampere (A). 1 Ampere = 1 Coulomb / second.
• Measurement: Measured through a component using an ammeter (connected in series).

1.3 Resistance (R)

• Definition: Resistance is the property of a material that opposes the flow of electric current. It dissipates electrical energy in the form of heat.
• Formula: R = ρ * (L/A), where ρ (rho) is the resistivity of the material, L is the length, and A is the cross-sectional area.
• Unit: Ohm (Ω).
• Symbol:

1.4 Inductance (L)

• Definition: Inductance is the property of an electrical conductor by which a change in current through it induces an electromotive force (voltage) in both the conductor itself (self-inductance) and in any nearby conductors (mutual inductance). It opposes a change in current.
• Energy Storage: Stores energy in a magnetic field.
• Voltage-Current Relationship: The voltage across an inductor is proportional to the rate of change of current through it.
  
  TEXT

    v(t) = L * (di(t)/dt)
    

• Unit: Henry (H).
• Symbol:

1.5 Capacitance (C)

• Definition: Capacitance is the ability of a component to store electrical energy in an electric field. It is the ratio of the change in an electric charge in a system to the corresponding change in its electric potential. It opposes a change in voltage.
• Energy Storage: Stores energy in an electric field.
• Current-Voltage Relationship: The current through a capacitor is proportional to the rate of change of voltage across it.
  
  TEXT

    i(t) = C * (dv(t)/dt)
    

• Unit: Farad (F).
• Symbol:

1.6 Power (P) and Energy (W or E)

Power (P)

• Definition: Power is the rate at which energy is consumed or produced in a circuit.
• Formulas:
  • General: P = V * I
  • For resistors: P = I² * R or P = V² / R
• Unit: Watt (W). 1 Watt = 1 Joule / second.
• Sign Convention:
  • Power Absorbed: If current flows from the positive (+) terminal to the negative (-) terminal of an element, power is being absorbed or dissipated by that element (e.g., a resistor). P = +VI.
  • Power Supplied: If current flows out of the positive (+) terminal of an element, power is being supplied by that element (e.g., a battery). P = -VI.

Energy (W)

• Definition: Energy is the capacity to do work. It is power integrated over time.
• Formula: W = P * t
• Unit: Joule (J). A common commercial unit is the kilowatt-hour (kWh). 1 kWh = 3.6 x 10⁶ Joules.

2. Fundamental Laws and Circuit Simplification

2.1 Ohm's Law

• Statement: The voltage across a resistor is directly proportional to the current flowing through it, provided the temperature and other physical conditions remain unchanged.
• Formula: V = I * R
• Graphical Representation: The V-I characteristic for a resistor is a straight line passing through the origin, with the slope equal to the resistance R.

2.2 Kirchhoff’s Laws

Kirchhoff’s Current Law (KCL)

• Principle: Based on the law of conservation of charge.
• Statement: The algebraic sum of currents entering a node (or a closed boundary) is zero.
• Formula: Σ I_entering = Σ I_leaving

• Example:

  At the node shown, KCL gives:
  I1 + I4 = I2 + I3 or I1 - I2 - I3 + I4 = 0

Kirchhoff’s Voltage Law (KVL)

• Principle: Based on the law of conservation of energy.
• Statement: The algebraic sum of all voltages around any closed loop or mesh in a circuit is equal to zero.
• Formula: Σ V_rises = Σ V_drops
• Sign Convention: When traversing a loop:
  • Moving from - to + across a component is a voltage rise.
  • Moving from + to - across a component is a voltage drop.

• Example:

  Applying KVL to the loop (starting from the bottom-left corner and moving clockwise):
  -Vs + V_R1 + V_R2 = 0 or Vs = V_R1 + V_R2

2.3 Voltage Division Rule (VDR)

• Application: Used to find the voltage across a resistor in a series circuit without first calculating the current.
• Derivation: In a series circuit with resistors R1, R2, ..., Rn and total voltage Vs, the total resistance is Req = R1 + R2 + ... + Rn. The current is I = Vs / Req. The voltage across a specific resistor Rx is Vx = I * Rx. Substituting I, we get the formula.
• Formula (for two resistors): For a voltage Vs across two series resistors R1 and R2, the voltage V2 across R2 is:
  
  TEXT

    V2 = Vs * (R2 / (R1 + R2))
    

2.4 Current Division Rule (CDR)

• Application: Used to find the current flowing through a resistor in a parallel circuit without first calculating the voltage.
• Derivation: In a parallel circuit with total current Is splitting between branches, the voltage V across all branches is the same. V = Is * Req. The current Ix through a specific resistor Rx is Ix = V / Rx. Substituting V, we get the formula.
• Formula (for two resistors): For a total current Is splitting between two parallel resistors R1 and R2, the current I2 through R2 is:
  
  TEXT

    I2 = Is * (R1 / (R1 + R2))
    

  
  Note: The numerator contains the other resistor's value.

3. Circuit Analysis Methods

3.1 Dependent and Independent Sources

• Independent Sources: Provide a specified voltage or current that is completely independent of other circuit variables.

  • Independent Voltage Source: Maintains a specified voltage across its terminals. An ideal source has zero internal resistance.
  • Independent Current Source: Maintains a specified current through its terminals. An ideal source has infinite internal resistance.

• Dependent (or Controlled) Sources: The value of the source depends on a voltage or current elsewhere in the circuit. They are essential for modeling active devices like transistors and amplifiers.

  • Voltage-Controlled Voltage Source (VCVS): V = μ * Vx
  • Current-Controlled Voltage Source (CCVS): V = r * Ix
  • Voltage-Controlled Current Source (VCCS): I = g * Vx
  • Current-Controlled Current Source (CCCS): I = β * Ix

3.2 Mesh Analysis

• Basis: Applies KVL to find unknown currents.

• Procedure:

  1. Identify Meshes: A mesh is a loop that does not contain any other loops within it.
  2. Assign Mesh Currents: Assign a distinct current variable (e.g., i1, i2) to each mesh, typically in a clockwise direction.
  3. Apply KVL: Write a KVL equation for each mesh.
     • The voltage drop across a resistor is I * R.
     • For a resistor shared between two meshes (e.g., R between mesh 1 and mesh 2), the current through it is the difference of the two mesh currents (i1 - i2 or i2 - i1, depending on the direction of KVL traversal).
  4. Solve: Solve the resulting system of simultaneous linear equations to find the mesh currents.

• Example:

  • Mesh 1 (left loop): Start at the bottom left, go clockwise.
    -V1 + R1*i1 + R2*(i1 - i2) = 0
(R1 + R2)*i1 - R2*i2 = V1 (Equation 1)
  • Mesh 2 (right loop): Start at the bottom of R2, go clockwise.
    R2*(i2 - i1) + R3*i2 + V2 = 0
-R2*i1 + (R2 + R3)*i2 = -V2 (Equation 2)
  • Solve these two equations for i1 and i2.

3.3 Nodal Analysis

• Basis: Applies KCL to find unknown node voltages.

• Procedure:

  1. Identify Nodes: A node is a point where two or more circuit elements connect.
  2. Select Reference Node: Choose one node as the reference node (ground), which is defined to have a potential of 0V.
  3. Assign Node Voltages: Assign a voltage variable (e.g., V1, V2) to each of the other non-reference nodes. These voltages are relative to the reference node.
  4. Apply KCL: Write a KCL equation for each non-reference node. Assume all unknown currents are leaving the node. The current flowing from node A to node B through a resistor R is (VA - VB) / R.
  5. Solve: Solve the resulting system of simultaneous linear equations to find the node voltages.

• Example:

  • Reference Node: Bottom wire is 0V.
  • Node 1 (V1): Apply KCL.
    (V1 - Vs)/R1 + V1/R2 + (V1 - V2)/R3 = 0
  • Node 2 (V2): Apply KCL.
    (V2 - V1)/R3 + V2/R4 - Is = 0
  • Rearrange and solve these two equations for V1 and V2.

4. Network Theorems

4.1 Thevenin’s Theorem

• Statement: Any linear two-terminal circuit can be replaced by an equivalent circuit consisting of a single voltage source (V_th) in series with a single resistor (R_th). This simplifies analysis of a complex circuit connected to a load.

• Procedure:

  1. Identify Load: Identify the portion of the circuit to be analyzed (the "load," R_L), and remove it from the circuit.
  2. Calculate V_th (Thevenin Voltage): Find the open-circuit voltage across the two terminals where the load was connected. This can be found using mesh, nodal analysis, or other simplification techniques.
  3. Calculate R_th (Thevenin Resistance): Find the equivalent resistance looking back into the open-circuited terminals.
     • Turn off all independent sources:
     • Independent Voltage Sources are replaced by a short circuit (0V).
     • Independent Current Sources are replaced by an open circuit (0A).
     • Do NOT turn off dependent sources. If they are present, you must use a test source (voltage or current) at the terminals to find R_th = V_test / I_test.
  4. Draw Equivalent Circuit: The Thevenin equivalent circuit is V_th in series with R_th. The load R_L can now be reconnected to this simplified circuit to easily calculate load voltage and current (I_L = V_th / (R_th + R_L)).

4.2 Norton’s Theorem

• Statement: Any linear two-terminal circuit can be replaced by an equivalent circuit consisting of a single current source (I_n) in parallel with a single resistor (R_n).

• Procedure:

  1. Identify Load: Identify and remove the load resistor R_L.
  2. Calculate I_n (Norton Current): Find the short-circuit current that flows through the terminals where the load was connected (i.e., place a short circuit across the terminals and find the current through it).
  3. Calculate R_n (Norton Resistance): The Norton resistance R_n is calculated in the exact same way as the Thevenin resistance R_th. R_n = R_th.
  4. Draw Equivalent Circuit: The Norton equivalent circuit is I_n in parallel with R_n. The load R_L can be reconnected to find the load current using the current divider rule.

• Source Transformation (Thevenin-Norton Equivalence):

  • A Thevenin circuit can be converted to a Norton circuit, and vice-versa.
  • V_th = I_n * R_n
  • I_n = V_th / R_th
  • R_th = R_n

5. Fundamentals of AC Circuits

5.1 Alternating Current (AC) and Voltage

• Definition: Alternating Current (AC) is an electric current which periodically reverses direction, in contrast to Direct Current (DC) which flows only in one direction.
• Waveform: While AC can have many waveforms (square, triangular), the most common is the sinusoidal waveform.
• Mathematical Representation: A sinusoidal voltage or current can be expressed as:
  
  TEXT

    v(t) = Vm * sin(ωt + φ)
    i(t) = Im * sin(ωt + φ)
    

  
  where:
  • v(t) or i(t) is the instantaneous value at time t.
  • Vm or Im is the maximum value, or Amplitude.
  • ω is the angular frequency in radians/sec.
  • φ (phi) is the phase angle in radians or degrees.

5.2 Definitions of Amplitude and Phase

• Amplitude (Vm, Im): The peak or maximum value of the waveform, measured from the zero axis. Also called peak voltage or peak current. The peak-to-peak voltage is 2 * Vm.
• Period (T): The time taken to complete one full cycle of the waveform. Unit: seconds (s).
• Frequency (f): The number of cycles completed per second. It is the reciprocal of the period. f = 1/T. Unit: Hertz (Hz).
• Angular Frequency (ω): The rate of change of the phase angle. ω = 2πf = 2π/T. Unit: radians/sec.
• Phase (φ): The phase angle represents the horizontal shift of the waveform from a reference point (usually t=0). It describes the position of the waveform at the starting point.
  • In Phase: Two signals are in phase if their phase difference is 0°. They reach their maximum and minimum points at the same time.
  • Out of Phase (Leading/Lagging): If v1(t) = Vm*sin(ωt) and v2(t) = Vm*sin(ωt - θ), then v2 lags v1 by θ, or v1 leads v2 by θ.

5.3 Average and RMS Value of an AC Signal

Average Value

• Definition: The average value of any periodic waveform is its integral over one period, divided by the period.
  
  TEXT

    V_avg = (1/T) * ∫[from 0 to T] v(t) dt
    

• For a Symmetrical Sinusoid: The average value over one full cycle is zero, because the positive and negative half-cycles cancel each other out.
• For a Half-Cycle (Half-Wave Rectified): The average value is calculated over the first half-cycle.
  
  TEXT

    V_avg = (2/π) * Vm ≈ 0.637 * Vm
    

RMS (Root Mean Square) Value

• Concept: The RMS value of an AC signal is its "effective" value. It is the equivalent DC value that would deliver the same average power to a resistor as the AC signal.
• Definition: It is found by taking the square Root of the Mean of the Square of the signal's instantaneous values.
  
  TEXT

    V_rms = sqrt( (1/T) * ∫[from 0 to T] v(t)² dt )
    

• Formula for a Sinusoidal Waveform: For a pure sine wave, the RMS value is the peak value divided by the square root of 2.
  
  TEXT

    V_rms = Vm / √2 ≈ 0.707 * Vm
    

  
  Similarly, I_rms = Im / √2.
• Importance: Standard AC voltmeters and ammeters are calibrated to read RMS values. Power calculations in AC circuits are almost always done using RMS values. For example, the power dissipated in a resistor is P = (V_rms)² / R = (I_rms)² * R.