1. SINLE PHASE A.C. SERIES and PARALLEL CIRCUITS, Electrical Engg 3rd semester notes EE 3002

 

1. Generations of Alternating Voltages

• Alternating Current (AC) is a type of electrical current that changes direction periodically. Unlike DC (direct current), where the flow of current is constant, AC varies sinusoidally with time.

How AC is generated:

• AC is generated by mechanical energy converting into electrical energy using a generator (alternator).
• The coil of wire in the generator rotates in a magnetic field, inducing a current. As the coil moves, the induced voltage oscillates between positive and negative, creating a sinusoidal waveform.

Diagram Suggestion:
A basic generator setup with a rotating coil inside a magnetic field.

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1.2 Phasor Representation of Sinusoidal Quantities

• A phasor is a way to represent sinusoidal waveforms as rotating vectors.
• The sinusoidal wave $V(t) = V_0 \sin(\omega t + \phi)$ can be represented as a rotating vector (phasor) in the complex plane.
• Phasor: $V = V_0 \angle \phi$, where:
  • $V_0$ is the amplitude (peak value).
  • $\phi$ is the phase angle.

Diagram Suggestion:

• A circle showing a rotating vector representing a sinusoidal waveform with a phase angle and amplitude.

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1.3 R, L, C Circuit Elements, Voltage and Current Response

• R (Resistor): Resistor opposes the flow of current, causing a voltage drop that is in phase with the current.
• L (Inductor): Inductor opposes changes in current, causing the voltage to lead the current by 90°.
• C (Capacitor): Capacitor opposes changes in voltage, causing the current to lead the voltage by 90°.

Voltage and Current in Different Components:

• Resistor (R): $V_R = I R$, current and voltage are in phase.
• Inductor (L): $V_L = I \omega L$, voltage leads current by 90°.
• Capacitor (C): $V_C = I / \omega C$, current leads voltage by 90°.

Diagram Suggestion:

• A phase diagram showing the phase relationships between current and voltage for R, L, and C.

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1.4 R-L, R-C, R-L-C Combination of A.C. Series and Parallel Circuits

In an AC circuit, combinations of resistors, inductors, and capacitors can be connected in series or parallel. Let's break it down:

1.4.1 Impedance (Z)

• Impedance is the total opposition to current in AC circuits, combining resistance (R) and reactance (X).
• For a series circuit:
  $Z = \sqrt{R^2 + X^2}$, where $X$ is the reactance (combination of inductive and capacitive reactance).

1.4.2 Reactance (X)

• Inductive Reactance $X_L = \omega L$ (resists changes in current).
• Capacitive Reactance $X_C = \frac{1}{\omega C}$ (resists changes in voltage).

1.4.3 Impedance Triangle

• A graphical representation of impedance, showing the real part (resistance) and the imaginary part (reactance).
  • The horizontal axis represents resistance (R).
  • The vertical axis represents reactance (X).
  • The hypotenuse represents impedance (Z).

Diagram Suggestion:

• Impedance triangle with labeled R, X, and Z.

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1.4.4 Power Factor (pf)

• The power factor is the ratio of real power to apparent power. It tells us how efficiently the power is being used.
• Power Factor:
  $pf = \cos(\theta) = \frac{R}{Z}$, where $\theta$ is the phase angle.

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1.4.5 Active Power (P)

• Active power (real power) is the power that does actual work, measured in watts (W).
• Formula:
  $P = V I \cos(\theta)$

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1.4.6 Reactive Power (Q)

• Reactive power is the power that alternates between the source and the load, but doesn’t do useful work. Measured in volt-amperes reactive (VAR).
• Formula:
  $Q = V I \sin(\theta)$

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1.4.7 Apparent Power (S)

• Apparent power is the total power supplied to the circuit, combining both active and reactive power. Measured in volt-amperes (VA).
• Formula:
  $S = V I$

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1.4.8 Power Triangle

• The power triangle visually represents the relationship between real power (P), reactive power (Q), and apparent power (S).
  • The horizontal axis is real power (P).
  • The vertical axis is reactive power (Q).
  • The hypotenuse represents apparent power (S).

Diagram Suggestion:

• Power triangle with P, Q, and S labeled.

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1.4.9 Vector Diagram

• A vector diagram is used to represent the phase relationships between voltage and current in an AC circuit.
  • For pure resistance: Voltage and current are in phase.
  • For pure inductance: Voltage leads current by 90°.
  • For pure capacitance: Current leads voltage by 90°.

Diagram Suggestion:

• A vector diagram showing the phase relationship of voltage and current for resistive, inductive, and capacitive circuits.

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1.5 Resonance, Bandwidth, Quality Factor, and Voltage Magnification in Series R-L, R-C, R-L-C Circuit

• Resonance occurs when the inductive reactance equals the capacitive reactance, causing the circuit to resonate at a specific frequency.

  • Resonant frequency $f_0 = \frac{1}{2\pi \sqrt{LC}}$.

• Bandwidth is the range of frequencies over which the circuit can effectively operate.

  • $BW = \frac{f_0}{Q}$, where $Q$ is the quality factor.

• Quality Factor (Q):
  $Q = \frac{f_0}{BW}$. A higher $Q$ means less energy loss and more resonance sharpness.

• Voltage Magnification:
  At resonance, the voltage across the components of the circuit (especially in a series R-L-C circuit) can increase significantly compared to the source voltage.

Diagram Suggestion:

• A graph showing resonance at $f_0$, with bandwidth around it and a sharp peak representing voltage magnification.