AC Circuit Study Notes

Universiti Teknologi MARA (UiTM) - Topic 21: AC Circuit

21.1: Introduction to AC Circuit

• An AC circuit consists of a combination of circuit elements and an AC generator or source.

• The output voltage of an AC generator is sinusoidal and varies with time according to the equation: $\Delta v = \Delta V_{max} \sin (2\pi f t)$ where:

  • $\Delta v$ = instantaneous voltage

  • $\Delta V_{max}$ = maximum voltage of the generator

  • $f$ = frequency at which the voltage changes, measured in Hz

21.2: Resistors in an AC Circuit

21.2.1: Behavior of a Resistor

• In a simple circuit consisting of an AC source and a resistor:

  • The current ($i$) and voltage ($\Delta V$) across the resistor reach their maximum values simultaneously.

  • Current and voltage are said to be in phase meaning they oscillate in sync.

21.2.2: Power Dissipation

• The rate of electrical energy dissipation in the circuit is given by the formula: $P = i^2 R$ where:

  • $P$ = power in watts

  • $i$ = instantaneous current

  • $R$ = resistance in ohms

• The heating effect from an AC current with a maximum value of $I_{max}$ differs from that of a DC current of the same value.

21.3: rms Current and Voltage

• The rms (root mean square) current is defined as the direct current that would dissipate the same amount of energy in a resistor as an AC current.

• Similarly, AC voltages can be expressed in terms of their rms values.

21.3.1: Finding rms Values

1. The area under the curve of $i^2$ is proportional to the average power dissipation.

2. The average value of $i^2$ can be expressed as:
   $\text{{Average value of }} i^2 = \frac{1}{2}I_{max}^2$

21.4: Power in an AC Circuit

• The average power dissipated across a resistor in an AC circuit carrying a current $I$ is given by:
  $P<em>{av} = I</em>{max}^2 R$

21.5: Notation Used in This Chapter

• Voltage and Current Notations:

  • Instantaneous Voltage: $\Delta V$

  • Maximum Voltage: $\Delta V_{max}$

  • rms Voltage: $\Delta V_{rms}$

  • Instantaneous Current: $i$

  • Maximum Current: $I_{max}$

  • rms Current: $I_{rms}$

21.6: Ohm’s Law in an AC Circuit

• Ohm’s Law can be applied in AC circuits using rms values:
  b $\Delta V<em>{R,rms} = I</em>{rms} R$

• This applies equally to the maximum values of $\Delta V$ and $i$.

Example 21.1: Finding rms Current

• Problem: An AC voltage source has an output:
  $\Delta V = (2.00 \times 10^{2} V) \sin(2 \pi f t)$
  Connected to a resistor $R = 1.00 \times 10^{2} \Omega$. Find the rms voltage and rms current.

• Solution:

  • Maximum Voltage: $\Delta V_{max} = 2.00 \times 10^{2} V$

  • rms Voltage: $\Delta V<em>{rms} = \frac{\Delta V</em>{max}}{\sqrt{2}}$

  • rms Current:
    $I<em>{rms} = \frac{\Delta V</em>{rms}}{R}$

21.7: Power in an RLC Circuit

• Important Principle:

  • No power loss occurs in pure capacitors and pure inductors.

  • In a pure capacitor, energy is alternately stored and returned to the circuit.

  • In pure inductors, energy is stored when work is done against the back emf, then returned when current decreases.

Example 21.5: Average Power in an RLC Circuit

• Problem: A series RLC AC circuit has:

  • resistance: $R = 2.50 \times 10^{2} \Omega$

  • inductance: $L = 0.600 H$

  • capacitance: $C = 3.50 \mu F$

  • frequency: $f = 60.0 Hz$

  • maximum voltage: $\Delta V_{max} = 1.50 \times 10^2 V$

• Solution:
  To calculate average power:

• Find the rms current $I<em>{rms}$ and then use: $P</em>{av} = I<em>{rms} \Delta V</em>{rms} \cos \phi$

  • where $\cos \phi$ is the power factor.

21.8: Capacitors in an AC Circuit

21.8.1: Charging and Discharging

• When connected to an AC source, a capacitor allows current to charge its plates but with a time-dependent behavior.

  • Initially, there’s high current; as charge builds, voltage increases and current decreases.

21.8.2: Voltage and Current Behavior

• In a capacitor, voltage lags behind current by 90 degrees.

21.8.3: Capacitive Reactance

• The opposition to current flow by a capacitor in AC is termed capacitive reactance, given by:
  $X_C = \frac{1}{2\pi f C}$

• Ohm’s Law for capacitors states:
  $\Delta V<em>{C,rms} = I</em>{rms} X_{C}$

Example 21.2: Capacitive Circuit Calculations

• Problem: A circuit with a capacitor $C = 8.00 \mu F$ connected to an AC generator outputting rms voltage of $1.50 \times 10^2 V$ at $f = 60.0 Hz$.

• Solution:
  Calculate capacitive reactance and rms current:

1. $X_C = \frac{1}{2\pi (60.0 Hz)(8.00 \times 10^{-6} F)} = 332 \Omega$

2. Use Ohm’s law to find rms current:
   $I<em>{rms} = \frac{V</em>{C,rms}}{X_C} = 0.452 A$

21.9: Inductors in an AC Circuit

21.9.1: Behavior of Inductors

• The current is impeded due to the back emf generated by the inductor.

• Voltage across the inductor leads the current by 90 degrees.

21.9.2: Inductive Reactance

• Inductive reactance is defined as:
  $X_L = 2\pi f L$

• Ohm’s law for inductors indicates that:
  $\Delta V<em>{L,rms} = I</em>{rms} X_{L}$

Example 21.3: Inductive Circuit Calculations

• Problem: Consider a purely inductive AC circuit with $L = 25.0 mH$ and rms voltage of $1.50 \times 10^2 V$.

• Solution:

1. Find inductive reactance:
   $X_L = 2\pi (60.0 s^{-1})(25.0 \times 10^{-3} H) = 9.42 \Omega$

2. Find rms current:
   $I<em>{rms} = \frac{1.50 \times 10^2 V}{X</em>L} = 15.9 A$

21.10: The RLC Series Circuit

21.10.1: RLC Series Configuration

• The circuit combines resistance, inductance, and capacitance.

• The current varies sinusoidally over time.

21.10.2: Voltage-Current Relationships

• Instantaneous voltage across the resistor is in phase with current (90 degrees lead from inductor, 90 degrees lag from capacitor).

Phasor Diagrams

21.11: Diagrams of Voltage Relationships

• To manage the different voltage phases, phasor diagrams can be used to visually represent voltage in the circuit.

21.12: Calculating Impedance

• Total impedance ($Z$) represented as:
  $Z = \sqrt{R^2 + (X<em>L - X</em>C)^2}$

• Applied to AC circuits through parts of the Ohm's Law, $\Delta V<em>{max} = I</em>{max} Z$.

21.13: Example for RLC Circuits

• An example illustrates solving for RLC circuits, considering resistance $R$, inductance $L$, and capacitance $C$ in relation to frequency $f$ and voltage. Materials such as reactance and phase calculation principles apply.

21.14: Resonance in AC Circuits

• Resonance occurs at a frequency where current achieves its maximum when inductive reactance equals capacitive reactance ($X<em>L = X</em>C$).

• Phases influence resonance conditions in AC circuits.

• Examples include radio tuning and metal detectors, demonstrating practical applications of resonance and changes in current due to alterations in circuit characteristics.

Example 21.6: Circuit Resonance Calculation

• Consider a series RLC circuit, finding capacitance for maximum rms current based on the circuit resistance and inductance parameters.

• Approach includes applying the resonance frequency calculation, demonstrating effective application of theoretical principles to real-world circuit behavior.