Question

Draw a diagram showing inductance in an AC circuit with the voltage and current waveform.

Hint:

To easily remember lagging phase angles: think of the word ELI. In an Inductor ($L$), Emf (Voltage) leads the Inductive current. Because current lags voltage by exactly $90^\circ$ in a pure inductor, the average power dissipated over a complete cycle is zero ($P_{avg} = V_{rms} I_{rms} \cos(90^\circ) = 0\text{ W}$).

Answer 1

Step 1: Defining a Purely Inductive AC Circuit:
A purely inductive AC circuit consists of an ideal inductor with self-inductance $L$ (measured in Henrys) connected across an alternating voltage source. Because the inductor opposes changes in current, the circuit current lags behind the applied voltage by exactly $90^\circ$ ($\pi/2$ radians).

Step 2: Component-by-Component Description of the Diagram:
Since drawing the diagram directly is not required, below is a complete, detailed visual breakdown of the three key diagrams used to represent this circuit:

• The Circuit Schematic Diagram (L-AC Loop):
  • AC Source Symbol: A circle containing a sine-wave symbol ($\sim$), representing the alternating voltage source: $$v(t) = V_m \sin(\omega t)$$ Where $V_m$ is the peak voltage and $\omega = 2\pi f$ is the angular frequency.
  • Inductor Symbol: Connected in series across the AC source, shown as a continuous, spiral-coiled wire labeled with self-inductance $L$.
  • Current Arrow: Shows current $i(t)$ flowing through the loop: $$i(t) = I_m \sin(\omega t - 90^\circ)$$ Where $I_m = V_m / X_L$ is the peak current.
• The Phasor Diagram (Phase Vector Plot):
  • Reference Vector (Voltage): Draw a horizontal arrow pointing to the right along the positive X-axis, labeled $\vec{V$}.
  • Lagging Vector (Current): Draw a vertical arrow pointing straight down along the negative Y-axis, labeled $\vec{I$}.
  • Phase Angle Representation: Draw a right-angle arc between the two arrows, showing an angle of $\theta = 90^\circ$ (or $\pi/2$ radians) with an arrow pointing clockwise from $\vec{V}$ to $\vec{I}$. This visually represents that current lags the voltage by $90^\circ$.
• The Waveform Diagram (Time-Domain Plot):
  • Axes:
    • Horizontal X-Axis: Labeled as Time ($t$) or electrical angle ($\omega t$), marked with milestones at $0$, $90^\circ$ ($\pi/2$), $180^\circ$ ($\pi$), $270^\circ$ ($3\pi/2$), and $360^\circ$ ($2\pi$).
    • Vertical Y-Axis: Represents instantaneous values, with positive values on top ($+V_m, +I_m$) and negative values below ($-V_m, -I_m$).
  • Voltage Waveform ($v$):
    • A standard sine wave starting at $(0,0)$.
    • It rises to its positive peak ($+V_m$) at $90^\circ$, falls back to zero at $180^\circ$, reaches its negative peak ($-V_m$) at $270^\circ$, and returns to zero at $360^\circ$ to complete one cycle.
  • Current Waveform ($i$):
    • Because current lags voltage by $90^\circ$, it starts at its negative peak ($-I_m$) when $\omega t = 0$.
    • It rises to cross the zero line going upwards at $90^\circ$ (exactly when voltage is at its positive peak).
    • It reaches its positive peak ($+I_m$) at $180^\circ$, falls to cross zero going downwards at $270^\circ$, and returns to its negative peak ($-I_m$) at $360^\circ$.