Physics 2420 In-Class Problems: AC Circuits & Phasor Diagrams

1. Show that the RMS value of the pure AC part of a wave is given by
   .

   V_DC is the DC offset of the wave V(t).

2. The diagram below shows a triangular wave (often called a sawtooth wave). Over one cycle, the equation for the wave is
   
   

   Find the DC offset. Find the RMS value of the wave. Find the AC part of the wave. Find the RMS value of the pure AC part of the wave.


3. The diagram below shows an exponentially-decaying wave. Over one cycle, the equation for the wave is for 0 £ t < T.
   
   1. Find the DC offset.
   2. Examine the DC offset in the limit that KT is large.
   3. Examine the DC offset in the limit that KT is small. The expansion will be of use.
   4. Find RMS value of the wave. Examine the limits of the RMS value. The expansion will be of use.
   5. Find the RMS value of the purely AC part of the wave. Examine the limits.
   

4. A single line carries two signals, as shown in the diagram below, V₁ = 10.0sin(100t) and V₂ = 7.0sin(10000t) where t is in seconds. A capacitor of 1.00 μF and a 1000-Ω resistor are connected in series. What is the voltage signal observed over the resistor?
   
   

5. A single line carries two signals, as shown in the diagram below, V₁ = 10.0sin(100t) and V₂ = 7.0sin(10000t) where t is in seconds. An inductor of 1.00 H and a 1000-Ω resistor are connected in series. What is the voltage signal observed over the resistor?
   
   

6. The AC generator in the diagram below supplies 120 V (RMS) at 60.0 Hz. With the switch open as in the diagram, the current leads the generator emf by 20.0°. With the switch in position 1 the current lags the generator emf by 10.0°. With the switch in position 2 the RMS current is 2.00 A. Find the values of R, L, and C.
   
   

7. An LCR circuit consists of an inductor of 1.25 × 10^-2 H, a capacitor of 2.75 μF, and a resistor of 5.0 Ω connected in series with an AC emf. The emf delivers a voltage ε = ε_maxsin(Ωt), with ε_max = 0.60 V and Ω = 6.0 × 10³ rad/s.
   
   1. What is the impedance of this circuit?
   2. What is the maximum current in this circuit?
   3. What is the phase angle of the current?
   4. What is the natural frequency of this circuit?
   5. What is the ratio of the current in part (b) to the current at resonance?

8. An LCR circuit consists of an inductor of 1.25 × 10^-2 H, a capacitor of 2.75 μF, and a resistor of 5.0 Ω all connected in parallel with an AC emf. The emf delivers a voltage ε = ε_maxsin(Ωt), with ε_max = 0.60 V and Ω = 6.0 × 10³ rad/s.
   
   1. What is the impedance of this circuit?
   2. What is the maximum current produced by the battery?
   3. What is the phase angle of the current?
   4. What is the natural frequency of this circuit?
   5. What is the ratio of the current in part (b) to the current at resonance?

9. For the diagram below, V(t) = 5.00sin(t) Volts, Ω = 1000 s^-1, R₁ = 100 Ω, R₂ = 50.0 Ω, and C = 20.0 μF. Determine the voltage drop over, and current through, each element.
   
   

10. For the diagram below, V(t) = 5.00sin(t) Volts, Ω = 1000 s^-1, R = 50.0 Ω, L = 20.0 mH, and C = 20.0 μF. Determine the voltage drop over, and current through, each element.
    
    

11. For the diagram below, the peak voltage from the power supply is 6.00 Volts. The frequency is f = 300 Hz, R₁ = 30.0 Ω, L = 40.0 mH, and C = 25.0 μF. The current from the power supply leads the voltage by 2.50. Determine R₂.
    
    

12. The diagram below shows a circuit with an AC and a DC source connected in series with a capacitor and a resistor.
    
    1. Use the Principle of Linear Superposition to obtain an expression for the current through the resistor as a function of time.
    2. What is the DC offset of the current?
    3. What happens to the DC offset in the limit of large t/RC?
    4. Explain in terms of the RC filter how a suitably chosen capacitor in an AC circuit acts to isolate a load (such as the resistor) from a DC source.
    

13. The diagram below shows a circuit with an AC and a DC source connected in series with an inductor and a resistor.
    
    1. Use the Principle of Linear Superposition to obtain an expression for the current through the resistor as a function of time.
    2. What is the DC offset of the current?
    3. What happens to the DC offset in the limit of large tL/R?
    4. What happens to the pure AC part of the current when ΩL/R >> 1?
    5. Explain in terms of the RL filter how a suitably chosen inductor acts to isolate a load (such as the resistor) from an AC source.
    

Questions? mike.coombes@kwantlen.ca