Transcript Goal: To understand RLC circuits

Goal: To understand RLC
circuits
Objectives:
1) To understand how Impedance compares
to resistance
2) To learn how to calculate Voltage and
Current from Impedance
3) To learn about Resonance
4) To be able to calculate the Phase Angle
of a circuit
5) To learn the differences in calculating
Power in a RLC circuit vs a circuit with
resistors
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
Sample
• XL = wL and XC = 1/(wC)
• You have a 12 Vrms and 60 Hz power
source hooked up in series to a 0.05 H
inductor, 5 Ω resistor, and 0.01 F
capacitor.
• What is the impedance of this circuit?
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 Ω resistor, and 0.01 F
capacitor.
• What is the impedance of this circuit?
• On board
Voltage and Current
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V = IR – before
V = IZ – now
And Vmax = Imax Z, ect
So, for the question before (where
Z = on board) if the voltage is 132 V then
what is the current?
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?
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?
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
Resonance frequency
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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
Since w = 2π f, then f = 1 / [2 π (LC)1/2]
Resonance sample
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So, the resonance frequency occurs at:
w = (LC)-1/2
Or f = 1 / [2 π (LC)1/2]
If you have a 0.5 H inductor and a 0.2 F
capacitor then what is the resonance
frequency?
Resonance sample
• So, the resonance frequency occurs at:
• f = 1 / [2 π (LC)1/2]
• If you have a 0.5 H inductor and a 0.2 F
capacitor then what is the resonance frequency?
• (On board)
• Now suppose we quartered the inductance of
the inductor, what will happen to the resonance
frequency?
Phase Angle
• Remember that Capacitors are 90 degrees
behind in phase and Inductors are 90
degrees ahead (in voltage)!
• 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?
Phase equation
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cos(Φ) = R / Z
Or tan(Φ) = (XL - XC) / R
This is the MAGNITUDE of the phase!
However, if XL < XC then the phase angle is
negative.
• Note, for the tan version if XL < XC then you will
get a negative answer already.
• Another way to do this…
Phasors
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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.
• (but note you are using the tangent anyway this
way…)
Sample
• For the example we did at the start:
• You have a 12 Vrms and 60 Hz power
source hooked up in series to a 0.05 H
inductor, 5 Ω resistor, and 0.01 F
capacitor.
• Find the phase angle (you should have the
value of R and Z from the sample we did
early in class).
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 don’t
use any up (well not significant amounts).
• Pav = Irms Vrms
• But V rms = V cos(Φ)
• So, Pav = Irms V cos(Φ)
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.