Electric Circuits Fundamentals Floyd 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

Summary Variation of XL and XC with frequency In a series RLC circuit, the circuit can be capacitive or inductive, depending on the frequency. XC>XL XL>XC At the frequency where XC=XL, the circuit is at series resonance. Reactance Below the resonant frequency, the circuit is predominantly capacitive. XC XL XC=XL Above the resonant frequency, the circuit is predominantly inductive. f Series resonance

Summary Impedance of series RLC circuits Example-1 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? Example-1 The total reactance is 3.16 kW The total impedance is The phase angle is 71.6o The circuit is capacitive, so I leads V by 71.6o. R L C VS 1.0 kW XL = 2.0 kW XC = 5.0 kW

Summary Impedance of series RLC circuits Example-2 What is the magnitude of the impedance for the circuit? R L C VS 470 W 330 mH 2000 pF 753 W f = 100 kHz

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 XC XL f

Summary Impedance of series RLC circuits Example-3 What is the total impedance for the circuit when the frequency is increased to 400 Hz? Example-3 R L C 470 W 330 mH 2000 pF 786 W VS f = 400 kHz The circuit is now inductive.

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

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.

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.

Summary Series resonance The formula for resonance can be found by setting XC = XL. The result is Example What is the resonant frequency for the circuit? R L C VS 330 mH 470 W 2000 pF 196 kHz

Summary Series resonance Ideally, at resonance the sum of VL and VC is zero. By KVL, VR = VS VS V = 0 Example R L C VS 330 mH 470 W 2000 pF 5.0 Vrms What is VR at resonance? 5.0 Vrms 5.0 Vrms

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

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

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

Summary Series resonant filters By taking the output across the resonant circuit, a band-stop (or notch) filter is produced. Circuit response: Vout R Stopband 1 Vin Vout 0.707 Resonant circuit L C f f1 fr f2 BW f2

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.

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 R L C

Summary Impedance of parallel RLC circuits Example What is the total impedance of the parallel RLC circuit if R= 1.0 kW, XL = 2.0 kW, and XC = 5.0 kW? Example First determine the conductance and total susceptance as follows: The total admittance is: 881 W VS R XL = 2.0 kW XC = 5.0 kW 1.0 kW

Summary Sinusoidal response of parallel RLC circuits Example A typical current phasor diagram for a parallel RLC circuit is IC The total current is given by: +90o IR The phase angle is given by: -90o IL Example What is Itot and q if IR = 10 mA, IC = 15 mA and IL = 5 mA? 14.1 mA

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…. IL

Summary Currents in a parallel RLC circuits Example Solution IC Draw a diagram of the phasors if IR = 12 mA, IC = 22 mA and IL = 15 mA? Solution Set up a grid with a scale that will allow all of the data– say 2 mA/div. IR Plot the currents on the appropriate axes Combine the reactive currents Use the total reactive current and IR to find the total current. IL In this case, Itot = 16.6 mA

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. IC Notice that IC is out of phase with IL. When they are algebraically added, the result is…. IL

Summary Parallel resonance The formula for the resonant frequency in both parallel and series circuits is the same, namely (ideal case) Example What is the resonant frequency for the circuit? R L C VS 680 mH 1.0 kW 15 nF 49.8 kHz

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

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: R Vout Passband Vout 1.0 Vin L C 0.707 Resonant circuit f Parallel resonant band-pass filter f1 fr f2 BW

Summary Parallel resonant filters For the band-stop filter, the resonant circuit and resistance are reversed as shown here. Circuit response: C Vout Stopband Vin Vout L 1 R 0.707 Resonant circuit f Parallel resonant band-stop filter f1 fr f2 BW

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.

Key Terms 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. A condition in a parallel RLC circuit in which the reactances ideally are equal and the impedance is a maximum. A parallel resonant circuit.

Key Terms 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. A measure of how effectively a resonant circuit passes desired frequencies and rejects all others. Generally, the narrower the bandwidth, the greater the selectivity.

Quiz 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

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

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 X XL XC f f

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

Quiz 5. In a series resonant circuit, the resonant frequency can be found from the equation a. b. c. d.

Quiz 6. In an ideal parallel resonant circuit, the total impedance at resonance is zero equal to the resistance equal to the reactance infinite

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.

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.

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

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

Quiz Answers: 1. c 2. c 3. a 4. a 5. b 6. d 7. b 8. d 9. d 10. a