Goal: To understand RLC circuits

1
Goal To understand RLC circuits

• Objectives
• Impedance
• Voltage and Current from Impedance
• Resonance
• Phase Angle
• Power

2
Impedance

• Impedance is the effective resistance of a RLC
  circuit.
• However, each part of the circuit is in a
  different part of the cycle or in a different
  phase.
• So, the exact voltages (and therefore
  resistances) across any component in the circuit
  varies.
• However, there is a overall solution.
• Z (R2 (XL XC)2)1/2
• Z is called the Impedance.
• Note if XL XC 0 then Z R

3
Sample

• XL wL and XC 1/(wC)
• You have a 10 Vrms and 50 Hz power source hooked
  up in series to a 0.04 H inductor, 5 O resistor,
  and 0.01 F capacitor.
• What is the impedance of this circuit?

4
Sample

• XL wL and XC 1/(wC)
• You have a 10 Vrms and 50 Hz power source hooked
  up in series to a 0.04 H inductor, 5 O resistor,
  and 0.01 F capacitor.
• What is the impedance of this circuit?
• How many people got 5 O?

5
Sample

• XL wL and XC 1/(wC)
• You have a 10 Vrms and 50 Hz power source hooked
  up in series to a 0.04 H inductor, 5 O resistor,
  and 0.01 F capacitor.
• What is the impedance of this circuit?
• How many people got 5 O you forgot that w 2pf
• So, XL wL 2pf L 2p 50Hz 0.4 H 12.57 O
• XC 1/(wC) 1/(2pf C) 0.32 O
• And Z (R2 (XL XC)2)1/2
• (25 (12.57 0.32)2)1/2
• 13.2 O

6
Voltage and Current

• V IR before
• V IZ now
• So, for the question before (where
• Z 13.2 O) if the voltage is 132 V then what is
  the current?

7
Voltage and Current

• V IR before
• V IZ now
• So, for the question before (where
• Z 13.2 O) if the voltage is 132 V then what is
  the current?
• V IZ, so I V/Z 132 / 13.2 10 A

8
Resonance

• Resonance is when you set the frequency such that
  you get the maximum current.
• What must be true about the resistance if the
  current is maximized?

9
Resonance

• Resonance is when you set the frequency such that
  you get the maximum current.
• What must be true about the impedance if the
  current is maximized?
• Impedance must be minimized!
• When do you get the minimum impedance for Z (R2
  (XL XC)2)1/2?

10
Resonance

• Resonance is when you set the frequency such that
  you get the maximum current.
• What must be true about the impedance if the
  current is maximized?
• Impedance must be minimized!
• When do you get the minimum impedance for Z (R2
  (XL XC)2)1/2?
• XL XC and Z R

11
Resonance frequency

• XL XC
• So, wL 1/(wC)
• Doing some math this means that
• w2 1/ (LC)
• So, the resonance frequency occurs at
• w (LC)-1/2

12
Resonance sample

• So, the resonance frequency occurs at
• w (LC)-1/2
• If you have a 0.5 H inductor and a 0.2 F
  capacitor then what is the resonance frequency?

13
Resonance sample

• So, the resonance frequency occurs at
• w (LC)-1/2
• If you have a 0.5 H inductor and a 0.2 F
  capacitor then what is the resonance frequency?
• w (0.5 0.2)-1/2
• 3.2 Hz
• Now suppose we quartered the inductance of the
  inductor, what will happen to the resonance
  frequency?

14
Resonance sample

• So, the resonance frequency occurs at
• w (LC)-1/2
• If you have a 0.5 H inductor and a 0.2 F
  capacitor then what is the resonance frequency?
• w (0.5 0.2)-1/2
• 3.2 Hz
• Now suppose we quartered the inductance of the
  inductor, what will happen to the resonance
  frequency?
• It will double to 6.4 Hz

15
Phase Angle

• Remember that Capacitors are 90 degrees ahead in
  phase and Inductors are 90 degrees behind!
• What will the phase of the circuit be?
• Well, that will be decided by which of the two is
  the most dominant.
• If the two are equal, then the phase is 0.
• But what about any other case?

16
Phase equation

• cos(F) R / Z
• This is the MAGNITUDE of the phase!
• However, if XL lt XC then the phase angle is
  negative.
• Another way to do this

17
Phasors

• Is to use phasors
• Draw R in the X direction.
• Then draw XL - XC in the Y direction.
• Draw in the hypotenuse between those.
• The angle between the hypotenuse and R is the
  phase angle.
• And if the angle is downwards it is negative.

18
Sample

• For the example we did at the start
• You have a 10 Vrms and 50 Hz power source hooked
  up in series to a 0.04 H inductor, 5 O resistor,
  and 0.01 F capacitor.
• Find the phase angle (you should have the value
  of R and Z).

19
Sample

• For the example we did at the start
• You have a 10 Vrms and 50 Hz power source hooked
  up in series to a 0.04 H inductor, 5 O resistor,
  and 0.01 F capacitor.
• Find the phase angle
• Z 13.2, so R/Z 5/13.2 0.379
• So, cos(F) 0.379
• And F 67.7 degrees
• Since XL is greater than XC this will be a
  positive angle.

20
Power

• What about the power used in a RLC circuit?
• The only part of the circuit using power is the
  resistor.
• The other two transfer the power but dont use
  any up (well not significant amounts).
• Pav Irms Vrms
• But V rms V cos(F)
• So, Pav Irms V cos(F)

21
Conclusion

• We have learned how to find the impedance of a
  RLC circuit.
• We learned how to use that impedance to find the
  voltage and current for the RLC circuit.
• We learned how to find the resonance frequency
  for a RLC circuit.
• We learned how to find the phase angle and power
  used by a RLC circuit.