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 = (R 2 + (X L – X C ) 2 ) 1/2 Z is called the Impedance. Note if X L – X C = 0 then Z = R

Sample X L = wL and X C = 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 X L = wL and X C = 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 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 = (R 2 + (X L – X C ) 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 = (R 2 + (X L – X C ) 2 ) 1/2 ? X L = X C and Z = R

Resonance frequency X L = X C So, wL = 1/(wC) Doing some math this means that: w 2 = 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 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 cos(Φ) = R / Z Or tan(Φ) = (X L - X C ) / R This is the MAGNITUDE of the phase! However, if X L < X C then the phase angle is negative. Note, for the tan version if X L < X C then you will get a negative answer already. Another way to do this…

Phasors Is to use phasors… Draw R in the X direction. Then draw X L - X C 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.