FAD1022 L14-L16 — AC Analysis

Introduction to alternating current (AC) circuits, phasor representation, and reactance concepts.

Lecture Files

  • L14 — Alternating Current
  • L15 — Phasor Diagram
  • L16 — Concepts of Reactance

L14 — Alternating Current

Learning Objectives

  • Explain the concepts of alternating current (AC) voltage and current.
  • Describe the sinusoidal nature and mathematical representation for current and voltage in AC.
  • Calculate the average value and root mean square (RMS) value of an AC signal given its waveform or equation.

AC vs DC

  • Direct Current (DC): flows in only one direction. Current flows from +ve to −ve terminal; electron flow is from −ve to +ve terminal. The brightness of a lamp bulb is constant.
  • Alternating Current (AC): reverses its direction periodically and changes its magnitude with time. Current and electron flow alternate direction. The brightness of a lamp bulb flickers.

[!note] Why AC instead of DC? AC won the "War of Currents" (Tesla vs Edison, late 1800s) because it is more efficient for long-distance power transmission. In a DC-only world:

  • Power lines would lose too much energy due to resistance.
  • Every home would need massive power plants nearby.
  • Transformers wouldn't work — no voltage conversion for efficient transmission.

AC Signal — Sinusoidal Current & Voltage

The general equations to represent AC current and voltage:

$$I(t) = I_0 \sin(\omega t)$$

$$V(t) = V_0 \sin(\omega t)$$

Where:

  • $I(t)$ : instantaneous current
  • $V(t)$ : instantaneous voltage
  • $I_0$ : current peak (maximum current)
  • $V_0$ : voltage peak (maximum voltage)
  • $\omega$ : angular frequency (rad/s)
  • $t$ : time (s)

[!note] The sinusoidal AC can also be expressed in the form of a cosine graph.

Angular frequency relation:

$$\omega = \frac{2\pi}{T} = 2\pi f$$

  • $T$ : period (time taken to complete one cycle) in seconds
  • $f$ : frequency (number of complete cycles in 1 second) in Hz

Example 1 — Writing the equation for AC current:

  • From graph: $I_0 = 4\ \text{A}$, $T = 12\ \text{s}$
  • $\omega = \frac{2\pi}{12} = \frac{\pi}{6}\ \text{rad/s}$
  • Equation: $I(t) = 4\sin\left(\frac{\pi}{6}t\right)$

Example 2 — Writing the equation for AC voltage:

  • From graph: $V_0 = 15\ \text{V}$, $T = 0.1\ \text{s}$
  • $\omega = \frac{2\pi}{0.1} = 62.83\ \text{rad/s}$
  • Equation: $V(t) = 15\sin(62.83t)$

Average Value & RMS Value

Average Value: The average value of a sinusoidal AC signal over a full cycle is zero, because the positive and negative halves cancel out. However, power is still delivered.

Root Mean Square (RMS) Value: RMS is a way to find the "effective" value of something that changes over time. For AC, it gives the equivalent DC value that would deliver the same power.

The RMS process (S → M → R):

  1. Square all the values (make everything positive)
  2. Mean (average) those squared values
  3. Take the Root (square root) of that average

For a sinusoidal current $I(t) = I_{\max}\sin(\omega t)$:

$$I_{\text{rms}} = \frac{I_{\max}}{\sqrt{2}} = 0.707, I_{\max}$$

$$V_{\text{rms}} = \frac{V_{\max}}{\sqrt{2}}$$

[!tip] RMS is the AC equivalent of DC power. Your home power supply is 230V RMS; the actual peak voltage is 325V! Electric bill calculations use RMS values, not peak values.

Power in AC circuits:

$$P = V_{\text{rms}} I_{\text{rms}}$$

Example: A 100W light bulb operates at 230V RMS.

  • $I_{\text{rms}} = \frac{P}{V_{\text{rms}}} = \frac{100}{230} \approx 0.435\ \text{A}$
  • $I_{\max} = I_{\text{rms}} \times \sqrt{2} = 0.435 \times \sqrt{2} \approx 0.615\ \text{A}$

Practice Questions

Question 1 AC Current is expressed by $I(t) = 52\sin(38\pi t)$, where $t$ is in seconds. Calculate:

  • (a) amplitude of the current
  • (b) RMS value of the current
  • (c) period of the cycle
  • (d) frequency
  • (e) current at $t = 10\ \text{ms}$ and $t = 35\ \text{ms}$

Question 2 A series circuit consisting of a resistor $R = 150\ \Omega$ is connected to an AC voltage source of $V(t) = 250\sin(\omega t)$. Calculate the RMS and maximum current through the resistor.

Past Year 2023/2024 — (b) The current in an AC circuit is given as $I(t) = 2.5\sin(100\pi t)$. Determine:

  • (i) the RMS current
  • (ii) the current when $t = 20\ \text{ms}$

L15 — Phasor Diagram

Learning Objectives

  • Define phase differences, phase angle, lead and lag in AC circuits.
  • Understand phasor diagrams and their usefulness in AC circuit analysis.

Recap

  • AC current and voltage: $I(t) = I_0 \sin \omega t$, $V(t) = V_0 \sin \omega t$
  • Sinusoidal AC can also be expressed as a cosine graph.

Phasor Diagram

  • A phasor diagram is a vector that rotates anticlockwise to represent an AC signal.
  • It shows sinusoidally varying quantities (alternating current and voltage).
  • It determines the phase angle — the phase difference between current and voltage in an AC circuit.
  • General sinusoidal form: $A(t) = A_m \sin(\omega t + \phi)$
  • The phasor rotates anticlockwise at angular velocity $\omega$ from the positive x-axis.
  • The vertical projection of the rotating phasor traces the sinusoidal waveform in the time domain.

Phase Angle ($\phi$)

  • Symbol: $\phi$ (phi)
  • Accounts for the shift when the sine wave does not start at $t = 0$.
  • Left shift (lead): $I(t) = I_0 \sin(\omega t + \phi)$ → positive $\phi$
  • Right shift (lag): $I(t) = I_0 \sin(\omega t - \phi)$ → negative $\phi$
  • Summary:
    • In-phase: $A(t) = A_m \sin(\omega t)$, $\phi = 0^\circ$
    • Positive phase (lead): $A(t) = A_m \sin(\omega t + \phi)$
    • Negative phase (lag): $A(t) = A_m \sin(\omega t - \phi)$

Leading & Lagging

  • Leading: A signal reaches its peak or zero-crossing earlier than the reference.
  • Lagging: A signal reaches its peak or zero-crossing later than the reference.
  • The phase angle is the magnitude of the lead or lag.

Example 1 — Voltage leads current by $\frac{\pi}{2}$:

  • $V(t) = V_0 \sin\left(\omega t + \frac{\pi}{2}\right)$
  • $I(t) = I_0 \sin(\omega t)$
  • At $t=0$, the V phasor points at $90^\circ$ and the I phasor at $0^\circ$.

Example 2 — Current leads voltage by $\frac{\pi}{2}$:

  • $I(t) = I_0 \sin(\omega t)$
  • $V(t) = V_0 \sin\left(\omega t - \frac{\pi}{2}\right)$
  • At $t=0$, the I phasor points at $0^\circ$ and the V phasor at $-90^\circ$ ($270^\circ$).

Past Year Question (2022/2023)

  • Sketch sinusoidal waves from a given phasor diagram at $t=0$ and state which signal leads.
  • (i) V leads I by $\frac{\pi}{2}$.
  • (ii) I leads V by $\frac{\pi}{4}$.

L16 — Concepts of Reactance

Lecture 16 covers the behavior of resistors, capacitors, and inductors in AC circuits, phase relationships between voltage and current, and the calculation of reactance.

Learning Objectives

  1. Explain the behavior of resistors, capacitors, and inductors in AC circuits.
  2. Understand phase relationships between voltage and current in:
    • Pure resistive circuit (PRC)
    • Pure capacitive circuit (PCC)
    • Pure inductive circuit (PLC)
  3. Calculate reactance for inductors and capacitors.

Impedance, $Z$

Impedance is the quantity that measures the opposition of a circuit to AC flow.

$$Z = \frac{V_{\text{rms}}}{I_{\text{rms}}} = \frac{V_0}{I_0}$$

  • It is a scalar quantity and its unit is ohm ($\Omega$).
  • In a DC circuit, impedance behaves like resistance.

01 — Pure Resistive Circuit (PRC)

A pure resistor means that there is no capacitance and self-inductance effect in the AC circuit.

Phase relationship:

  • $I = I_0 \sin(\omega t)$
  • $V_R = V_0 \sin(\omega t) = V$
  • The phase difference between $V$ and $I$ is $\Delta\phi = \omega t - \omega t = 0$
  • In a pure resistor, the current $I$ is always in phase with the voltage $V$.

Impedance in a pure resistor: $$Z = \frac{V_{\text{rms}}}{I_{\text{rms}}} = \frac{V_0}{I_0} = R$$

02 — Pure Capacitive Circuit (PCC)

A pure capacitor means that there is no resistance and self-inductance effect in the AC circuit.

Phase relationship:

  • $V_C = V = V_0 \sin(\omega t)$
  • $I = I_0 \sin\left(\omega t + \frac{\pi}{2}\right)$ or $I = I_0 \cos(\omega t)$
  • The phase difference: $\Delta\phi = \omega t - \left(\omega t + \frac{\pi}{2}\right) = -\frac{\pi}{2}$ rad
  • In a pure capacitive circuit:
    • The voltage $V$ lags behind the current $I$ by $\pi/2$ radians.
    • The current $I$ leads the voltage $V$ by $\pi/2$ radians.

Capacitive Reactance, $X_C$: Capacitive reactance is the opposition of a capacitor to the alternating current flow.

$$X_C = \frac{1}{2\pi f C} = \frac{V_{\text{rms}}}{I_{\text{rms}}} = \frac{V_0}{I_0}$$

  • $X_C$ is known as capacitive reactance.
  • $f$: frequency of AC source; $C$: capacitance of the capacitor.
  • Capacitive reactance is a scalar quantity with unit ohm ($\Omega$).
  • $X_C \propto \frac{1}{f}$ — capacitive reactance is inversely proportional to frequency.

03 — Pure Inductive Circuit (PLC)

A pure inductor means that there is no resistance and capacitance effect in the AC circuit.

Phase relationship:

  • $V = V_0 \cos(\omega t)$ or $V = V_0 \sin\left(\omega t + \frac{\pi}{2}\right)$
  • $I = I_0 \sin(\omega t)$
  • The phase difference: $\Delta\phi = \left(\omega t + \frac{\pi}{2}\right) - \omega t = \frac{\pi}{2}$ rad
  • In a pure inductive circuit:
    • The voltage $V$ leads the current $I$ by $\pi/2$ radians.
    • The current $I$ lags behind the voltage $V$ by $\pi/2$ radians.

Inductive Reactance, $X_L$: Inductive reactance is the opposition of an inductor to the alternating current flow.

$$X_L = 2\pi f L = \frac{V_{\text{rms}}}{I_{\text{rms}}} = \frac{V_0}{I_0}$$

  • $X_L$ is known as inductive reactance.
  • $f$: frequency of AC source; $L$: self-inductance of the inductor.
  • Inductive reactance is a scalar quantity with unit ohm ($\Omega$).
  • $X_L \propto f$ — inductive reactance is directly proportional to frequency.

CIVIL Mnemonic

A memory aid for remembering which quantity leads in capacitor and inductor circuits:

  • C (Capacitor): I leads V
  • L (Inductor): V leads I (or I lags V)

Past Year Questions

Past Year 23/24 — A5 A capacitor with capacitive reactance $X_C = 69.2\ \Omega$ is connected to an AC voltage source $V(t) = 200 \sin(120\pi t)$. Calculate the value of capacitance. (3 marks)

Past Year 23/24 — (c) The value of inductive reactance in an AC circuit at frequency $30\ \text{Hz}$ is $X_L = 1.5\ \Omega$.

  • (i) Calculate the value of new $X_L$ if frequency is increased to $90\ \text{Hz}$.
  • (ii) Explain the change in the value of $X_L$ for a frequency of $90\ \text{Hz}$ and state the relationship between frequency and $X_L$. (6 marks)

Past Year 22/23 — A4 An inductor has a reactance of $150\ \Omega$ in a $60\ \text{Hz}$ AC circuit. Calculate the inductance of the inductor. (3 marks)

Key Concepts

  • AC Circuits — AC vs DC, sinusoidal waveforms
  • Alternating Current — instantaneous, average, and RMS values
  • AC Voltage and Current — phase relationships
  • Phasors — rotating vectors representing AC quantities
  • Phasor Diagrams — graphical analysis of AC circuits
  • Impedance — complex resistance in AC circuits
  • Reactance — capacitive ($X_C$) and inductive ($X_L$) reactance
  • Frequency Dependence — how reactance varies with frequency
  • Pure Circuits — PRC, PCC, and PLC

Diagrams

RMS Calculation Process

graph LR
    A[("RMS Calculation")] --> B["Step 1: Square"]
    B --> C["Step 2: Mean"]
    C --> D["Step 3: Root"]
    D --> E["Irms = Imax / sqrt(2)"]
    
    style A fill:#e7f5ff,stroke:#1971c2
    style B fill:#ffe8cc,stroke:#d9480f
    style C fill:#fff4e6,stroke:#e67700
    style D fill:#d3f9d8,stroke:#2f9e44
    style E fill:#c5f6fa,stroke:#0c8599

Phase Relationships in Pure AC Circuits

graph TB
    subgraph resistive["Pure Resistive (PRC)"]
        R1["V and I In Phase"] --> R2["Phase Angle = 0"]
    end
    subgraph capacitive["Pure Capacitive (PCC)"]
        C1["I Leads V by pi/2"] --> C2["Xc = 1/(2 pi f C)"]
    end
    subgraph inductive["Pure Inductive (PLC)"]
        L1["V Leads I by pi/2"] --> L2["Xl = 2 pi f L"]
    end
    
    resistive ~~~ capacitive
    capacitive ~~~ inductive

Summary

This module introduces AC circuit analysis using phasor methods. Students learn to represent sinusoidal voltages and currents as rotating vectors, calculate RMS values, and understand the frequency-dependent behavior of capacitors and inductors through reactance concepts. Phasor diagrams provide a visual tool for analyzing AC circuit relationships. Lecture 15 covers phasor diagrams, phase angle, and lead/lag concepts. Lecture 16 develops the concepts of resistance, reactance, and impedance in pure R, C, and L circuits, including the CIVIL mnemonic for phase relationships.

Lecturer

Nurul Izzati (NIA) — PASUM Physics Lecturer

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