Phasor Diagram Examples for Inductors in AC Circuits: Worked Solutions

📚 Quick Study Guide

• ⚡ In an AC circuit with only an inductor, the voltage ($V_L$) leads the current ($I_L$) by 90 degrees.
• 📐 A phasor diagram represents sinusoidal quantities (like voltage and current) as rotating vectors.
• 📈 The length of the phasor represents the amplitude (peak value) of the sinusoidal quantity.
• 🔄 The angular velocity of the phasor corresponds to the angular frequency ($\omega$) of the AC source.
• 💡 Inductive Reactance ($X_L$) is given by the formula: $X_L = \omega L = 2\pi f L$, where $L$ is the inductance and $f$ is the frequency.
• 📝 The relationship between voltage and current in an inductor is $V_L = I_L X_L$.
• 🧭 In a phasor diagram, if current is taken as the reference (horizontal axis), then the voltage phasor will be 90 degrees counter-clockwise from the current phasor.

🧪 Practice Quiz

1. Which of the following statements is true regarding the phase relationship between voltage and current in a purely inductive AC circuit?
   1. Voltage and current are in phase.
   2. Voltage lags current by 90 degrees.
   3. Voltage leads current by 90 degrees.
   4. Voltage leads current by 180 degrees.
2. In a phasor diagram for a purely inductive circuit, if the current phasor is along the positive x-axis, where is the voltage phasor oriented?
   1. Along the positive x-axis.
   2. Along the negative x-axis.
   3. Along the positive y-axis.
   4. Along the negative y-axis.
3. What does the length of a phasor represent in a phasor diagram?
   1. The frequency of the AC signal.
   2. The phase angle of the AC signal.
   3. The amplitude (peak value) of the AC signal.
   4. The average value of the AC signal.
4. An inductor has an inductance of 10 mH and is connected to a 50 Hz AC source. What is the inductive reactance ($X_L$)?
   1. $\pi \Omega$
   2. $2\pi \Omega$
   3. $3.14 \Omega$
   4. $10 \Omega$
5. If the voltage across an inductor is $V_L = 10V$ and the inductive reactance is $X_L = 5\Omega$, what is the current through the inductor ($I_L$)?
   1. 0.5 A
   2. 2 A
   3. 5 A
   4. 50 A
6. What happens to the inductive reactance ($X_L$) if the frequency of the AC source is doubled?
   1. $X_L$ is halved.
   2. $X_L$ remains the same.
   3. $X_L$ is doubled.
   4. $X_L$ is quadrupled.
7. In a series RL circuit, how would the phasor diagram look compared to a purely inductive circuit?
   1. The voltage would still lead the current by 90 degrees.
   2. The voltage would lag the current by 90 degrees.
   3. The voltage would lead the current by an angle less than 90 degrees.
   4. The voltage would lead the current by an angle greater than 90 degrees.