Chapter 13 Electric Circuits Fundamentals Floyd Copyright 2007

Chapter 13 Electric Circuits Fundamentals - Floyd © Copyright 2007 Prentice-Hall

Chapter 13 Summary 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 The total impedance is given by The phase angle is given by R L C VS Electric Circuits Fundamentals - Floyd © Copyright 2007 Prentice-Hall

Chapter 13 Summary Variation of XL and XC with frequency At the frequency where XC=XL, the circuit is at series resonance. Below the resonant frequency, the circuit is predominantly capacitive. Above the resonant frequency, the circuit is predominantly inductive. Electric Circuits Fundamentals - Floyd Reactance In a series RLC circuit, the circuit can be capacitive or inductive, depending on the frequency. XC>XL XL>XC XC XL XC=XL Series resonance f © Copyright 2007 Prentice-Hall

Chapter 13 Summary Impedance of series RLC circuits What is the total impedance and phase angle of the series RLC circuit if R= 1. 0 k. W, XL = 2. 0 k. W, and XC = 5. 0 k. W? The total reactance is 3. 16 k. W The total impedance is The phase angle is 71. 6 o The circuit is capacitive, so I leads V by 71. 6 o. R VS Electric Circuits Fundamentals - Floyd L C 1. 0 k. W XL = XC = 2. 0 k. W 5. 0 k. W © Copyright 2007 Prentice-Hall

Chapter 13 Summary Impedance of series RLC circuits What is the magnitude of the impedance for the circuit? R 753 W VS 470 W L C 330 m. H 2000 p. F f = 100 k. Hz Electric Circuits Fundamentals - Floyd © Copyright 2007 Prentice-Hall

Chapter 13 Summary Impedance of series RLC circuits Depending on the frequency, the circuit can appear to be capacitive or inductive. The circuit in the Example-2 was capacitive because XC>XL. X XL XC f Electric Circuits Fundamentals - Floyd © Copyright 2007 Prentice-Hall

Chapter 13 Summary Impedance of series RLC circuits What is the total impedance for the circuit when the frequency is increased to 400 Hz? R 786 W The circuit is now inductive. Electric Circuits Fundamentals - Floyd VS L C 470 W 330 m. H 2000 p. F f = 400 k. Hz © Copyright 2007 Prentice-Hall

Chapter 13 Summary Impedance of series RLC circuits By changing the frequency, the circuit in Example-3 is now inductive because XL>XC X XL XL XC XC f Electric Circuits Fundamentals - Floyd © Copyright 2007 Prentice-Hall

Chapter 13 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. Notice that VC is out of phase with VL. When they are algebraically added, the result is…. VL VC This example is inductive. Electric Circuits Fundamentals - Floyd © Copyright 2007 Prentice-Hall

Chapter 13 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. Algebraic sum is zero. Electric Circuits Fundamentals - Floyd © Copyright 2007 Prentice-Hall

Chapter 13 Summary Series resonance The formula for resonance can be found by setting XC = XL. The result is What is the resonant frequency for the circuit? R 470 W L C 330 m. H 2000 p. F VS 196 k. Hz Electric Circuits Fundamentals - Floyd © Copyright 2007 Prentice-Hall

Chapter 13 Summary Series resonance Ideally, at resonance the sum of VL and VC is zero. VS 5. 0 Vrms 470 W VS 5. 0 Vrms Electric Circuits Fundamentals - Floyd V=0 R What is VR at resonance? By KVL, VR = VS L C 330 m. H 2000 p. F 5. 0 Vrms © Copyright 2007 Prentice-Hall

Chapter 13 Summary 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. X XL Z XC Z=R f Series resonance Electric Circuits Fundamentals - Floyd © Copyright 2007 Prentice-Hall

Chapter 13 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 Electric Circuits Fundamentals - Floyd © Copyright 2007 Prentice-Hall

Chapter 13 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. Circuit response: Vout Resonant circuit L C Vin Vout R f Series resonance Electric Circuits Fundamentals - Floyd © Copyright 2007 Prentice-Hall

Chapter 13 Summary Series resonant filters By taking the output across the resonant circuit, a bandstop (or notch) filter is produced. Circuit response: Vin Vout R Resonant circuit Vout L 1 Stopband 0. 707 C f f 1 fr f 2 BW Electric Circuits Fundamentals - Floyd f 2 © Copyright 2007 Prentice-Hall

Chapter 13 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. Susceptance is the reciprocal of reactance. Admittance is the reciprocal of impedance. Electric Circuits Fundamentals - Floyd © Copyright 2007 Prentice-Hall

Chapter 13 Summary Impedance of parallel RLC circuits The admittance can be used to find the impedance. Start by calculating the total susceptance: The admittance is given by The impedance is the reciprocal of the admittance: The phase angle is VS Electric Circuits Fundamentals - Floyd R L C © Copyright 2007 Prentice-Hall

Chapter 13 Summary Impedance of parallel RLC circuits What is the total impedance of the parallel RLC circuit if R= 1. 0 k. W, XL = 2. 0 k. W, and XC = 5. 0 k. W? First determine the conductance The total admittance is: and total susceptance as follows: 881 W VS Electric Circuits Fundamentals - Floyd R 1. 0 k. W XL = 2. 0 k. W XC = 5. 0 k. W © Copyright 2007 Prentice-Hall

Chapter 13 Summary Sinusoidal response of parallel RLC circuits A typical current phasor diagram for a parallel RLC circuit is IC The total current is given by: +90 o IR The phase angle is given by: -90 o IL What is Itot and q if IR = 10 m. A, IC = 15 m. A and IL = 5 m. A? 14. 1 m. A Electric Circuits Fundamentals - Floyd © Copyright 2007 Prentice-Hall

Chapter 13 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. IC Notice that IC is out of phase with IL. When they are algebraically added, the result is…. Electric Circuits Fundamentals - Floyd IL © Copyright 2007 Prentice-Hall

Chapter 13 Summary Currents in a parallel RLC circuits IC Draw a diagram of the phasors if IR = 12 m. A, IC = 22 m. A and IL = 15 m. A? 20 m. A 10 m. A • Set up a grid with a scale that will allow all of the data– say 2 m. A/div. 0 m. A 10 m. A • Plot the currents on the appropriate axes • Combine the reactive currents 20 m. A • Use the total reactive current and IR to find the total current. In this case, Itot = 16. 6 m. A Electric Circuits Fundamentals - Floyd IR IL © Copyright 2007 Prentice-Hall

Chapter 13 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. Notice that IC is out of phase with IL. When they are algebraically added, the result is…. Electric Circuits Fundamentals - Floyd IC The algebraic sum is zero. IL © Copyright 2007 Prentice-Hall

Chapter 13 Summary Parallel resonance The formula for the resonant frequency in both parallel and series circuits is the same, namely (ideal case) What is the resonant frequency for the circuit? VS R 1. 0 k. W C L 680 m. H 15 n. F 49. 8 k. Hz Electric Circuits Fundamentals - Floyd © Copyright 2007 Prentice-Hall

Chapter 13 Summary Parallel resonance 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 Electric Circuits Fundamentals - Floyd © Copyright 2007 Prentice-Hall

Chapter 13 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. Circuit response: Vout R Vin Passband L C Vout 1. 0 0. 707 Resonant circuit Parallel resonant band-pass filter Electric Circuits Fundamentals - Floyd f f 1 fr f 2 BW © Copyright 2007 Prentice-Hall

Chapter 13 Summary Parallel resonant filters For the band-stop filter, the resonant circuit and resistance are reversed as shown here. Circuit response: Vout C Vin Vout L R 1 Stopband 0. 707 Resonant circuit Parallel resonant band-stop filter f f 1 fr f 2 BW Electric Circuits Fundamentals - Floyd © Copyright 2007 Prentice-Hall

Chapter 13 Summary Key ideas for resonant filters • A band-pass filter allows frequencies between two critical frequencies and rejects all others. • 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. Electric Circuits Fundamentals - Floyd © Copyright 2007 Prentice-Hall

Chapter 13 Series resonance Resonant frequency (fr) Parallel resonance Tank circuit Key Terms 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. A condition in a parallel RLC circuit in which the reactances ideally are equal and the impedance is a maximum. A parallel resonant circuit. Electric Circuits Fundamentals - Floyd © Copyright 2007 Prentice-Hall

Chapter 13 Key Terms Half-power The frequency at which the output power of a frequency resonant circuit is 50% of the maximum value (the output voltage is 70. 7% of maximum); another name for critical or cutoff frequency. Decibel Ten times the logarithmic ratio of two powers. Selectivity A measure of how effectively a resonant circuit passes desired frequencies and rejects all others. Generally, the narrower the bandwidth, the greater the selectivity. Electric Circuits Fundamentals - Floyd © Copyright 2007 Prentice-Hall

Chapter 13 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 Electric Circuits Fundamentals - Floyd © Copyright 2007 Prentice-Hall

Chapter 13 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 Electric Circuits Fundamentals - Floyd © Copyright 2007 Prentice-Hall

Chapter 13 Quiz 3. The frequency represented by the red dashed line is the a. resonant frequency X b. half-power frequency c. critical frequency XL d. all of the above XC f f Electric Circuits Fundamentals - Floyd © Copyright 2007 Prentice-Hall

Chapter 13 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 Electric Circuits Fundamentals - Floyd © Copyright 2007 Prentice-Hall

Chapter 13 Quiz 5. In a series resonant circuit, the resonant frequency can be found from the equation a. b. c. d. Electric Circuits Fundamentals - Floyd © Copyright 2007 Prentice-Hall

Chapter 13 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 Electric Circuits Fundamentals - Floyd © Copyright 2007 Prentice-Hall

Chapter 13 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. Electric Circuits Fundamentals - Floyd © Copyright 2007 Prentice-Hall

Chapter 13 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. Electric Circuits Fundamentals - Floyd © Copyright 2007 Prentice-Hall

Chapter 13 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 Electric Circuits Fundamentals - Floyd © Copyright 2007 Prentice-Hall

Chapter 13 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 Electric Circuits Fundamentals - Floyd © Copyright 2007 Prentice-Hall

Chapter 13 Quiz Answers: 1. c 6. d 2. c 7. b 3. a 8. d 4. a 9. d 5. b 10. a Electric Circuits Fundamentals - Floyd © Copyright 2007 Prentice-Hall