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- 1. Chapter 13 © Copyright 2007 Prentice-Hall Electric Circuits Fundamentals - Floyd Chapter 13
- 2. Chapter 13 © Copyright 2007 Prentice-Hall Electric Circuits Fundamentals - Floyd Summary VS R L C Series RLC circuits When a circuit contains an inductor and capacitor in series, the reactance of each tend to cancel. The total reactance is given by tot L C X X X The total impedance is given by 2 2 tot tot Z R X The phase angle is given by 1 tan tot X R
- 3. Chapter 13 © Copyright 2007 Prentice-Hall Electric Circuits Fundamentals - Floyd Summary Variation of XL and XC with frequency Reactance f XC XL In a series RLC circuit, the circuit can be capacitive or inductive, depending on the frequency. At the frequency where XC=XL, the circuit is at series resonance. Series resonance XC=XL Below the resonant frequency, the circuit is predominantly capacitive. Above the resonant frequency, the circuit is predominantly inductive. XC>XL XL>XC
- 4. Chapter 13 © Copyright 2007 Prentice-Hall Electric Circuits Fundamentals - Floyd Summary VS R L C What is the total impedance and phase angle of the series RLC circuit if R= 1.0 kW, XL = 2.0 kW, and XC = 5.0 kW? 1.0 kW XC = 5.0 kW The total reactance is 2.0 k 5.0 k 3.0 k tot L C X X X W W W The total impedance is 2 2 2 2 1.0 k +3.0 k tot tot Z R X W W 3.16 kW XL = 2.0 kW The circuit is capacitive, so I leads V by 71.6o . The phase angle is 1 1 3.0 k tan tan 1.0 k tot X R W W 71.6o Impedance of series RLC circuits
- 5. Chapter 13 © Copyright 2007 Prentice-Hall Electric Circuits Fundamentals - Floyd Summary What is the magnitude of the impedance for the circuit? R L C VS 330 mH f = 100 kHz 470 W 2000 pF 2 2 100 kHz 330 H 207 L X fL m W 1 1 796 2 2 100 kHz 2000 pF C X fC W 207 796 589 tot L C X X X W W W Impedance of series RLC circuits 2 2 = 470 589 Z W W 753 W
- 6. Chapter 13 © Copyright 2007 Prentice-Hall Electric Circuits Fundamentals - Floyd Summary XL f XC X Depending on the frequency, the circuit can appear to be capacitive or inductive. The circuit in the Example-2 was capacitive because XC>XL. XL XC Impedance of series RLC circuits
- 7. Chapter 13 © Copyright 2007 Prentice-Hall Electric Circuits Fundamentals - Floyd Summary R L C VS What is the total impedance for the circuit when the frequency is increased to 400 Hz? 330 mH f = 400 kHz 470 W 2000 pF 2 2 400 kHz 330 H 829 L X fL m W 1 1 199 2 2 400 kHz 2000 pF C X fC W The circuit is now inductive. Impedance of series RLC circuits 829 199 630 tot L C X X X W W W 2 2 = 470 630 Z W W 786 W
- 8. Chapter 13 © Copyright 2007 Prentice-Hall Electric Circuits Fundamentals - Floyd Summary XL f XC X By changing the frequency, the circuit in Example-3 is now inductive because XL>XC XL XC Impedance of series RLC circuits
- 9. Chapter 13 © Copyright 2007 Prentice-Hall Electric Circuits Fundamentals - Floyd Summary Voltages in a series RLC circuits The voltages across the RLC components must add to the source voltage in accordance with KVL. Because of the opposite phase shift due to L and C, VL and VC effectively subtract. 0 Notice that VC is out of phase with VL. When they are algebraically added, the result is…. VC VL This example is inductive.
- 10. Chapter 13 © Copyright 2007 Prentice-Hall Electric Circuits Fundamentals - Floyd Summary Series resonance At series resonance, XC and XL cancel. VC and VL also cancel because the voltages are equal and opposite. The circuit is purely resistive at resonance. 0 Algebraic sum is zero.
- 11. Chapter 13 © Copyright 2007 Prentice-Hall Electric Circuits Fundamentals - Floyd Summary Series resonance The formula for resonance can be found by setting XC = XL. The result is 1 2 r f LC What is the resonant frequency for the circuit? R L C VS 330 mH 470 W 2000 pF 1 2 1 2 330 μH 2000 pF r f LC 196 kHz
- 12. Chapter 13 © Copyright 2007 Prentice-Hall Electric Circuits Fundamentals - Floyd Summary Series resonance What is VR at resonance? Ideally, at resonance the sum of VL and VC is zero. R L C VS 330 mH 470 W 2000 pF 5.0 Vrms 5.0 Vrms V = 0 VS By KVL, VR = VS 5.0 Vrms
- 13. Chapter 13 © Copyright 2007 Prentice-Hall Electric Circuits Fundamentals - Floyd Summary XL f XC X Impedance of series RLC circuits The general shape of the impedance versus frequency for a series RLC circuit is superimposed on the curves for XL and XC. Notice that at the resonant frequency, the circuit is resistive, and Z = R. Z Series resonance Z = R
- 14. Chapter 13 © Copyright 2007 Prentice-Hall Electric Circuits Fundamentals - Floyd Summary Series resonance Summary of important concepts for series resonance: • Capacitive and inductive reactances are equal. • Total impedance is a minimum and is resistive. • The current is maximum. • The phase angle between VS and IS is zero. • fr is given by 1 2 r f LC
- 15. Chapter 13 © Copyright 2007 Prentice-Hall Electric Circuits Fundamentals - Floyd Summary Series resonant filters An application of series resonant circuits is in filters. A band-pass filter allows signals within a range of frequencies to pass. R L C Vout Vin Resonant circuit f Vout Series resonance Circuit response:
- 16. Chapter 13 © Copyright 2007 Prentice-Hall Electric Circuits Fundamentals - Floyd Summary Series resonant filters By taking the output across the resonant circuit, a band- stop (or notch) filter is produced. R Vout Vin f Vout Circuit response: L C Resonant circuit f1 fr f2 BW Stopband 0.707 1 f2
- 17. Chapter 13 © Copyright 2007 Prentice-Hall Electric Circuits Fundamentals - Floyd Summary Conductance, susceptance, and admittance Recall that conductance, susceptance, and admittance were defined in Chapter 10 as the reciprocals of resistance, reactance and impedance. Conductance is the reciprocal of resistance. 1 G R Admittance is the reciprocal of impedance. Susceptance is the reciprocal of reactance. 1 B X 1 Y Z
- 18. Chapter 13 © Copyright 2007 Prentice-Hall Electric Circuits Fundamentals - Floyd Summary Impedance of parallel RLC circuits The admittance can be used to find the impedance. Start by calculating the total susceptance: tot L C B B B The admittance is given by 2 2 tot Y G B The phase angle is 1 tan tot B G The impedance is the reciprocal of the admittance: 1 tot Z Y R L C VS
- 19. Chapter 13 © Copyright 2007 Prentice-Hall Electric Circuits Fundamentals - Floyd Summary What is the total impedance of the parallel RLC circuit if R= 1.0 kW, XL = 2.0 kW, and XC = 5.0 kW? 1.0 kW XC = 5.0 kW First determine the conductance and total susceptance as follows: The total admittance is: 2 2 2 2 1.0 mS + 0.3 mS 1.13 mS tot tot Y G B 881 W XL = 2.0 kW Impedance of parallel RLC circuits R VS 1 1 1.0 mS 1.0 k G R W 1 1 0.5 mS 2.0 k L L B X W 1 1 1.13 mS Z Y 1 1 0.2 mS 5.0 k C C B X W 0.3 mS tot L C B B B
- 20. Chapter 13 © Copyright 2007 Prentice-Hall Electric Circuits Fundamentals - Floyd Summary Sinusoidal response of parallel RLC circuits A typical current phasor diagram for a parallel RLC circuit is +90o 90o IC IR IL The total current is given by: 2 2 tot R C L I I I I What is Itot and if IR = 10 mA, IC = 15 mA and IL = 5 mA? 45 mA 1 tan CL R I I The phase angle is given by: 2 2 10 mA + 15 mA 5.0 mA tot I 14.1 mA
- 21. Chapter 13 © Copyright 2007 Prentice-Hall Electric Circuits Fundamentals - Floyd Summary Currents in a parallel RLC circuits The currents in the RLC components must add to the source current in accordance with KCL. Because of the opposite phase shift due to L and C, IL and IC effectively subtract. Notice that IC is out of phase with IL. When they are algebraically added, the result is…. IC IL 0
- 22. Chapter 13 © Copyright 2007 Prentice-Hall Electric Circuits Fundamentals - Floyd Summary Currents in a parallel RLC circuits Draw a diagram of the phasors if IR = 12 mA, IC = 22 mA and IL = 15 mA? • Set up a grid with a scale that will allow all of the data– say 2 mA/div. 0 mA 10 mA 20 mA 10 mA 20 mA IR IC IL • Plot the currents on the appropriate axes • Combine the reactive currents • Use the total reactive current and IR to find the total current. In this case, Itot = 16.6 mA
- 23. Chapter 13 © Copyright 2007 Prentice-Hall Electric Circuits Fundamentals - Floyd 0 Summary Parallel resonance Ideally, at parallel resonance, IC and IL cancel because the currents are equal and opposite. The circuit is purely resistive at resonance. The algebraic sum is zero. Notice that IC is out of phase with IL. When they are algebraically added, the result is…. IC IL
- 24. Chapter 13 © Copyright 2007 Prentice-Hall Electric Circuits Fundamentals - Floyd Summary The formula for the resonant frequency in both parallel and series circuits is the same, namely 1 2 r f LC What is the resonant frequency for the circuit? 1 2 1 2 680 μH 15 nF r f LC 49.8 kHz Parallel resonance R L C VS 680 mH 1.0 kW 15 nF (ideal case)
- 25. Chapter 13 © Copyright 2007 Prentice-Hall Electric Circuits Fundamentals - Floyd Summary Summary of important concepts for parallel resonance: • Capacitive and inductive susceptance are equal. • Total impedance is a maximum (ideally infinite). • The current is minimum. • The phase angle between VS and IS is zero. • fr is given by 1 2 r f LC Parallel resonance
- 26. Chapter 13 © Copyright 2007 Prentice-Hall Electric Circuits Fundamentals - Floyd Summary Parallel resonant filters Parallel resonant circuits can also be used for band-pass or band-stop filters. A basic band-pass filter is shown. R Vin Vout Parallel resonant band-pass filter C L Resonant circuit 0.707 f1 fr f2 BW f Vout Passband Circuit response: 1.0
- 27. Chapter 13 © Copyright 2007 Prentice-Hall Electric Circuits Fundamentals - Floyd Summary Parallel resonant filters For the band-stop filter, the resonant circuit and resistance are reversed as shown here. f Vout f1 fr BW Stopband 0.707 1 R Vin Vout Parallel resonant band-stop filter C L Resonant circuit Circuit response: f2
- 28. Chapter 13 © Copyright 2007 Prentice-Hall Electric Circuits Fundamentals - Floyd Summary Key ideas for resonant filters • A band-stop filter rejects frequencies between two critical frequencies and passes all others. • Band-pass and band-stop filters can be made from both series and parallel resonant circuits. •The bandwidth of a resonant filter is determined by the Q and the resonant frequency. •The output voltage at a critical frequency is 70.7% of the maximum. •A band-pass filter allows frequencies between two critical frequencies and rejects all others.
- 29. Chapter 13 © Copyright 2007 Prentice-Hall Electric Circuits Fundamentals - Floyd Series resonance Resonant frequency (fr) Parallel resonance Tank circuit A condition in a series RLC circuit in which the reactances ideally cancel and the impedance is a minimum. The frequency at which resonance occurs; also known as the center frequency. Key Terms A condition in a parallel RLC circuit in which the reactances ideally are equal and the impedance is a maximum. A parallel resonant circuit.
- 30. Chapter 13 © Copyright 2007 Prentice-Hall Electric Circuits Fundamentals - Floyd Half-power frequency Decibel Selectivity The frequency at which the output power of a resonant circuit is 50% of the maximum value (the output voltage is 70.7% of maximum); another name for critical or cutoff frequency. Ten times the logarithmic ratio of two powers. Key Terms A measure of how effectively a resonant circuit passes desired frequencies and rejects all others. Generally, the narrower the bandwidth, the greater the selectivity.
- 31. Chapter 13 © Copyright 2007 Prentice-Hall Electric Circuits Fundamentals - Floyd Quiz 1. In practical series and parallel resonant circuits, the total impedance of the circuit at resonance will be a. capacitive b. inductive c. resistive d. none of the above
- 32. Chapter 13 © Copyright 2007 Prentice-Hall Electric Circuits Fundamentals - Floyd Quiz 2. In a series resonant circuit, the current at the half-power frequency is a. maximum b. minimum c. 70.7% of the maximum value d. 70.7% of the minimum value
- 33. Chapter 13 © Copyright 2007 Prentice-Hall Electric Circuits Fundamentals - Floyd Quiz 3. The frequency represented by the red dashed line is the a. resonant frequency b. half-power frequency c. critical frequency d. all of the above XL f XC X f
- 34. Chapter 13 © Copyright 2007 Prentice-Hall Electric Circuits Fundamentals - Floyd Quiz 4. In a series RLC circuit, if the frequency is below the resonant frequency, the circuit will appear to be a. capacitive b. inductive c. resistive d. answer depends on the particular components
- 35. Chapter 13 © Copyright 2007 Prentice-Hall Electric Circuits Fundamentals - Floyd Quiz 5. In a series resonant circuit, the resonant frequency can be found from the equation a. b. c. d. max 0.707 r f I 1 2 r f LC 1 2 r f LC r BW f Q
- 36. Chapter 13 © Copyright 2007 Prentice-Hall Electric Circuits Fundamentals - Floyd Quiz 6. In an ideal parallel resonant circuit, the total impedance at resonance is a. zero b. equal to the resistance c. equal to the reactance d. infinite
- 37. Chapter 13 © Copyright 2007 Prentice-Hall Electric Circuits Fundamentals - Floyd Quiz 7. In a parallel RLC circuit, the magnitude of the total current is always the a. same as the current in the resistor. b. phasor sum of all of the branch currents. c. same as the source current. d. difference between resistive and reactive currents.
- 38. Chapter 13 © Copyright 2007 Prentice-Hall Electric Circuits Fundamentals - Floyd Quiz 8. If you increase the frequency in a parallel RLC circuit, the total current a. will not change b. will increase c. will decrease d. can increase or decrease depending on if it is above or below resonance.
- 39. Chapter 13 © Copyright 2007 Prentice-Hall Electric Circuits Fundamentals - Floyd Quiz 9. The phase angle between the source voltage and current in a parallel RLC circuit will be positive if a. IL is larger than IC b. IL is larger than IR c. both a and b d. none of the above
- 40. Chapter 13 © Copyright 2007 Prentice-Hall Electric Circuits Fundamentals - Floyd Quiz 10. A highly selectivity circuit will have a a. small BW and high Q. b. large BW and low Q. c. large BW and high Q. d. none of the above
- 41. Chapter 13 © Copyright 2007 Prentice-Hall Electric Circuits Fundamentals - Floyd Quiz Answers: 1. c 2. c 3. a 4. a 5. b 6. d 7. b 8. d 9. d 10. a