RMS and Average Quantities

• For $v(t)$ and $i(t)$ we use RMS; for $p(t)$, we use average

• For purely resistive loads, $p_{\text{avg}} = V_{\text{rms}} I_{\text{rms}}$
• For purely reactive loads (inductor/capacitor), $p_{\text{avg}} = 0$
• $\phi$ is the phase between V and I

Types of Electric Power

• ALWAYS USE RMS VALUES FOR CURRENT AND VOLTAGE OR YOU WILL GET THE QUESTION WRONG!
• Real/Active Power $(P)$: Actual power consumed, measured in watts (W)

• Reactive Power $(Q)$: Power stored/released by inductors and capacitors, measured in Volt-Amps-Reactive (VAR)

• Apparent Power $(|S|)$: Combination of reactive power and true power, measured in Volt-Amps (VA)
  • Magnitude of complex power, $S$

Complex Power

• Shows the total power running through the AC circuit, measured in Volt-Amps

Power Factor

• Ratio of real power to apparent power
• Shows how effectively electrical power is being used

Mutual Inductance

• The property of two coils or inductors that enables energy transfer through a shared magnetic field

• When the current in one coil changes, it generates a time-varying magnetic field that links with the second coil, inducing an emf

• Dependent on

  • The number of turns in each coil
  • The core material (permeability)
  • The geometrical arrangement of the coils
  • The degree of magnetic coupling

• Magnetic flux linkage definition:

Where:

• $M$ is the mutual inductance in Henries (H)

• $\Phi_{21}$ is the magnetic flux in coil 2 due to current in coil 1

• $\Phi_{12}$ is the magnetic flux in coil 1 due to current in coil 2

• $N_1, N_2$ are the number of turns in coils 1 and 2

• $I_1, I_2$ are the respective coil currents

• Induced voltage in coil 1 from changing current in coil 2:

• Induced voltage in coil 2 from changing current in coil 1:

• When the two coils are part of the same circuit, the mutual inductance also affects the total voltage across each inductor

• Voltage across the first inductor:

• Voltage across the second inductor:

• The sign depends on the relative orientation of the coils, determined by the dot convention
  • A positive sign is used if current enters the dotted terminal of both coils and negative otherwise

Ideal Transformer

• Theoretical device that models perfect magnetic coupling between two coils to step up or step down voltage/current while maintaining constant apparent power. It assumes:

  • No energy losses (no winding resistance, no core losses)
  • Perfect magnetic coupling (coupling coefficient $k = 1$)
  • All magnetic flux produced by the primary winding links with the secondary winding
  • Infinite core permeability (zero magnetizing current)

• An ideal transformer is used to:

  • Match impedances
  • Change voltage levels between circuits
  • Isolate different parts of a system electrically

• Turns ratio definition:

Where $a$ is the turns ratio (primary to secondary).

• Voltage transformation:

• Current transformation:

• Conservation of power:

Example 05.1.

(Example 13.1 [Alexander & Sadiku, Fundamentals of Electric Circuits, 7th ed., p. 585])

Example 05.2.

(From Quiz 3 - Ch11)

References

• AC Steady-State Power, Mutual Inductance, and (course handout)