# LC Resonant Circuits Dr. Roger King June Introduction

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 LC Resonant Circuits Dr. Roger King June 01 Introduction Second-order systems are important in a wide range of applications including transformerless impedance-matching networks, frequency-selective networks, and control system models. Second-order systems occur whenever two independent energy-storing elements are present. The study of first- and second-order networks is important because of their useful properties, and because they are frequently used as approximations to the main features of even more complex networks. Resonance is usually defined to occur when the imaginary part of the impedance or admittance of a network zeros at a specific resonant frequency, and is usually accompanied by a very large increase in voltage and/or current in some part of the network. Strongly resonant behavior will occur in a network composed of a low-loss inductor and a low-loss capacitor. It can also occur in a two-capacitor or two-inductor system, but only if a controlled source is also present. The purpose of this paper is the review of resonant behavior in a one-capacitor one-inductor system. The primary interest here is the resonance phenomenon, although some note is made of the overdamped condition where the resonance is no longer obvious, but the system behavior remains distinctly second order. Series vs. Parallel LC Resonance Fig. 1 displays two series-resonant LC circuits. Fig. 1(a) shows the classic textbook form of the circuit, including a series-connected inductor, capacitor, and a resistor representing the sum of all internal energy losses in the circuit elements along with any added damping. An independent voltage source is also included. The circuit response may be considered either the inductor current i or the capacitor voltage v. Fig. 1(b) is the same circuit, except that the voltage source and damping resistor are replaced with their Norton equivalent. The key feature of both circuits in Fig. 1 is that with their sources zeroed, the capacitor, inductor and damping resistor are all in series. Fig. displays two parallel-resonant LC circuits. Fig. (a) is the classic form of the circuit. Here the inductor, capacitor and damping resistance are all in parallel, along with an independent current source. Fig. (b) is a Thevenin equivalent to Fig. (a). The key feature of both circuits in Fig. is that with their sources zeroed, the capacitor, inductor and damping resistor are all in parallel. Again, the circuit response may be considered either the inductor current i or the capacitor voltage v. 1 of 14

2 Fig. 1 (a) Series-resonant LC network with voltage source. (b) Series-resonant LC network with current source. Fig. (a) Parallel-resonant LC network with current source. (b) Parallel-resonant LC network with voltage source. The series-resonant configuration arises naturally in a circuit having a voltage source in which the resistive losses in the inductor windings give the predominant damping effect. It is typically found that in LC resonant circuits the losses in the capacitor are much smaller than those in the inductor. The parallel-resonant configuration arises when the circuit has a current source and the predominant damping effect is the inductor core loss, which is modeled by an equivalent resistance in parallel with the inductor. An LC resonant circuit having some damping resistance in series with the inductor as well as additional damping resistance in parallel with the inductor will also be considered in a later section of this paper. This arrangement is neither series nor parallel resonant, but a reasonable approximate series- or parallel-equivalent may be made. of 14

3 Table 1 Parameters of an LC Resonant Network Symbol O Q Parameter Name (units) Resonant Frequency (radians/sec) Characteristic Resistance (ohms) Quality Factor (unit less) Damping Ratio (unit less) Damping Factor (unit less) o Z o Equation 1 LC L C Q 1 1 Q o Q o Q Zo R s Q Rp Z o Comments series res. parallel res. d Damped Resonant Frequency (radians/sec) d o 1 d o for a lightly damped network Characterization of a Resonant Network Any of the networks in Figs. 1 and can be described by listing the values of L, C and R S (or R P ), although this does not give immediate insight into how the network will behave. A more intuitive description is given by stating three equivalent parameters of the network: the resonant frequency, the characteristic resistance, and the quality factor ( O,, Q). Definitions of these three parameters are given in Table 1. As will be seen, O,, and Q give an immediate sense of what the natural response will be like, and how the network will interact with its independent source. The definitions of O and are the same for both the series- and parallel-resonant networks. The resonant frequency O is the radian frequency at which the circuit will ring if it has no energy dissipation and is excited by some initial stored energy. The characteristic resistance is the expected ratio of voltage amplitude to current amplitude in the ringing response. Resistors represent energy-dissipative elements which tend to damp the ringing of the network. There are three different measures of this same thing: quality factor (Q), damping factor ( ) and damping ratio ( ). As Table 1 indicates, these are mutually interrelated. However, the relationship of damping to the circuit element values is different for series- and parallel-resonant networks. The equations for the Q factor are given in Table 1; the expressions for damping factor and damping ratio are easily derived from these. It will also be seen that a damped network will ring at a frequency lower than O, given by the damped resonant frequency d. 3 of 14

4 Table. Three Different Damping Conditions Damping Type Undamped (lossless system) Underdamped Critically Damped Overdamped Quality Factor - Q Q Q > 0.5 Q = 0.5 Q < 0.5 Damping Ratio - Damping Factor - O O O The network equations for Figs. 1 and will be solved in the time-domain and the frequency-domain. The form of the solutions will change depending upon the amount of damping. The three cases to be distinguished are under-damping, critical-damping, and over-damping. Applications of the resonant LC network as a narrow-band filter or impedance transformer often prefer light or zero damping, in which case equations are usually written in terms of Q. In control system applications, the expected damping levels often range from just below to just above critical damping, and the expressions are written in terms of. Overdamping will be considered only briefly for the sake of completeness. The damping conditions for each of these three cases are described in Table. Time-Domain Equations for the Resonant Network For the series-resonant network of Fig. 1(a), write the loop equation in inductor current i. The substitution i = C dv/ may be used to get an equivalent equation in v. These equations are: d i Q d v Q di dv o i Z o dv s o v o v s (1) Note that the series-resonant definition for Q has been used. Fig. 1(b) is best analyzed by converting the Norton form of the independent source into its Thevenin equivalent. For the parallel-resonant network of Fig. (a), write the node equation in capacitor voltage v. The substitution v = -L di/ may be used to get an equivalent equation in i. These equations are: d i d v Q Q di dv o i o i s o v o Z o di s () Note the parallel-resonant definition for Q has been used. Fig. (b) is best analyzed by converting the Thevenin form of the independent source into its Norton equivalent. 4 of 14

5 It may be noted that (1) and () are essentially the same, except for their specific independent source terms. The method for solving (1) or () for any specific driving term and initial conditions is the following. With the driving term (right hand side of equation (1) or ()) set to zero, find the solution of the resulting homogeneous equation. This is the natural response. Then the particular solution to the specific driving term is found. This is known as the forced response. These two solutions are added together to form the general solution. The initial conditions for the network are applied to the general solution to determine the specific values of any unknown constants. The natural response is instructive: It shows what the resonant network will do by itself, given as a starting point the stored energy implied by the initial capacitor voltage and inductor current. These initial conditions are defined as follows: V o I o v(0) i(0) (3) The homogeneous equations in (1) and () are all the same, as shown below using the generic variable x. d x d x Q dx o x 0 or o dx o x 0 (4) The natural solution is given by: x t e ot [A 1 sin d t A cos d t] for 0 <1 x t A 3 e ot A 4 ( o t) e ot for =1 x t A 5 e 1 ot A 6 e 1 ot for >1 where d o 1 (5) The solutions in (5) separate into the three cases of underdamped, critically damped, and overdamped. The underdamped case ( < 1) is the interesting one. 5 of 14

6 For the series-resonant circuit of Fig. 1(a), and assuming initial conditions V O and I O, (5) is rewritten to give the natural behaviors of the capacitor voltage and inductor current. In each case, constant A is determined by applying the initial condition (I O or V O ). A 1 is determined by applying the initial rate-of-change of current or voltage. This can be calculated by finding the initial inductor voltage and initial capacitor current from the given initial conditions and the circuit in Fig. 1(a). The results are, for the underdamped case ( < 1): i t e ot v t e ot 1 1 V O I O 1 and I O V O 1 sin d t I O cos d t sin d t V O cos d t for 0 1 and damped series resonance (6) These results can be reasonably approximated in the case of light damping ( < 0.1 or Q>5): i t e ot V O I O sin o t I O cos o t and v t e ot [I O V O ] sin o t V O cos o t for 0.1 and damped series resonance (7) For the case of zero damping ( =0 or Q i t V O sin o t I O cos o t and v t I O sin o t V O cos o t for 0, undamped resonance (8) For the parallel-resonant circuit of Fig. (a) the result is similar to (6), but not identical. i t e ot v t e ot 1 1 V O I O 1 and I O V O 1 sin d t I O cos d t sin d t V O cos d t for 0 1 and damped parallel resonance (9) These results can be reasonably approximated in the case of light damping ( < 0.1 or Q>5): 6 of 14

7 i t e ot V O I O sin o t I O cos o t and v t e ot [I O V O ] sin o t V O cos o t for 0.1 and damped parallel resonance (10) For the case of zero damping ( =0 or Q the behaviors of the parallel-resonant and series-resonant circuits become the same. Both are given by (8). Careful comparison of (6)-(10) shows that both underdamped series- and parallel-resonance produce decaying sinusoidal oscillations in current and voltage if started with some stored energy. However, there are subtle differences in the two ringing behaviors. The less the damping (higher Q), the less these differences. If the two cases are both considered undamped altogether, the difference disappears entirely. This is explored further by PSpice simulation of three resonant circuits, each having an initial stored energy of 1 J, a natural resonant frequency of 1 rad/s, and an impedance of 1. See Fig. 3. One is series resonant with Q=5, another parallel resonant with Q=5, and the third has very little damping (Q=1000). The capacitor voltage and inductor current for each is plotted in Fig. 3, lower waveforms. The voltage/current waveforms for the series- and parallel-resonant circuits with Q=5 are seen to be close, but slightly different. Fig. 3. Three resonant circuits with initial energy of 1 J and resonant frequency of 1 rad/s. One is series resonant with Q=5, one is parallel resonant with Q=5, and the third has Q= of 14

8 The upper waveforms in Fig. 3 plot the energy stored in each circuit as it rings down. These suggest that the reason for the small differences in the voltage and current waveforms for the parallel- and series-resonant circuits with same Q factor is that the energy losses occur at different times in the ringing cycles, but that as one would expect for two circuits with the same Q, the total energy loss over a complete cycle is the same for both. The high-q circuit is provided as a reference, and has very little energy loss over the time span shown. Some references will note that Q may be defined as times peak energy stored in the circuit divided by energy dissipated over one cycle. This is true if the energy dissipated over one cycle is calculated based on the voltage/current levels remaining the same throughout the cycle. This statement is not true for Fig. 3 with Q = 5 because the voltage/current amplitudes are rapidly decaying. Series-Parallel Equivalents for High-Q Resonant Circuits Sometimes a resonant network includes both series and parallel damping, as illustrated in Fig. 4, left side. In this case, the network is strictly neither series- nor parallel-resonant. However, if the Q factor accounting for all of the damping is higher than about 5, two approximate equivalents are possible. These are shown in Fig. 4 center (series equivalent) and left side (parallel equivalent). The approximate equivalents are given as follows: R P R S R P Q R S Z O P R P (series equivalent) or R P R S Q S R P R S (parallel equivalent) where R S Q P Q S Q eff R P R S 1 1 Q 1 S Q P for Q eff 5 (11) Fig. 4. Resonant network with both series- and parallel-damping, along with an approximate series equivalent, and an approximate parallel equivalent. 8 of 14

9 Fig. 5. Parallel-resonant network, with two approximately equivalent options for placement of the damping (or load) resistance. Converting all of the damping resistances to either a series or a parallel equivalent is a useful approximation when analyzing a lossy inductor, which may show both ohmic losses in its windings (series damping resistance) and core loss related to its magnetic core material (parallel damping resistance). Fig. 5 shows additional options for placement of the damping resistance in a parallel-resonant network. These approximate equivalents are derived by comparing the impedances (or admittances) between the two nodes indicated. Given a high effective Q in each case, the following approximate equivalents may be found. R C1 1 R R for Q 5 C C 1 C L 1 L 1 L (1) The equivalents in Fig. 5 are particularly useful when the resonant network is used as a signal processing circuit. In this application, an input may be applied as a voltage or current source, and the damping resistance may actually represent a resistive load device. Fig. 5 suggests that splitting the capacitor or inductor into two series-connected parts and extracting the output in parallel with one of those two parts would be useful if the load resistance is too small to get a high Q when connected across the whole network. Other useful tapping schemes may be found for a series-resonant network. In using Fig. 5 and (1), the inductor is assumed to be split into two parts having zero mutual coupling. Resonant Networks as Filters Signal-processing filters may be described by their steady-state sinusoidal responses. For this purpose, refer to Fig. 1(a) and calculate its phasor impedance, or to Fig. (a) for its phasor admittance. These results are the following: 9 of 14

10 Z j R S j L 1 j C 1 Q j ( O O ) for series resonant network (impedance) Y j 1 R P j C 1 j L Y O 1 Q j ( O O ) for parallel resonant network (admittance) (13) Note that Y O is () -1. The input signal to a series-resonant circuit may be applied as a voltage source, with the output signal extracted as the voltage drop on R S, C, L, or the series combination of C and L. The damping resistance R S is the composite value which includes losses in the inductor and capacitor, as well as any inserted load resistance. The input signal to a parallel-resonant circuit may be applied as a current source, with the outputs extracted as similar choices of element currents. In addition, a load resistance may be inserted into either of these resonant circuits through the technique of tapping-down the capacitor or inductor. This leads to a wide variety of possibilities. A sample set will be illustrated, but many others are used in practice. Fig. 6 shows four filters derived from the series resonant network. In each of these, the single resistor represents the composite of the external load resistance and internal inductor losses, converted into their series equivalent values. Each of the filters in Fig. 6 has the same loop impedance, given by (13) for a series resonant network. Therefore, each of the four transfer functions can be derived by dividing the impedance of the element(s) across which the output voltage is extracted by the loop impedance. A Spice simulation was used to plot the transfer function magnitude for each of the four filters. These are shown in Fig. 7. For these Spice Fig. 6 Four filters derived from the series-resonant network. 10 of 14

11 Fig. 7 Frequency response for the four filters of Fig. 6, all with f O = 1 Hz, Q = 0.5, 5, and 50. Upper left: bandpass filter. Lower left: bandstop filter. Upper right: lowpass filter. Lower right: highpass filter. simulations, o = (1 Hz), Z o = 1, and Q values of 0.5, 5, and 50 were used. The transfer functions plotted in Fig. 7 are V o /V s in db notation. The frequency response plots for the lowpass and highpass filters suggest that Q values ranging from critical damping to slight underdamping (Q = 0.5 to 1) should be chosen for these types. It can be noted that the peaking in these frequency responses is 0 log(q) above the passband value for values of Q exceeding 5 or so. This is generally undesirable for most applications. There are a number of practical cases where the lowpass filter appears on a dc power bus, either deliberately or created through the interaction of wiring inductance with bus bypass capacitors. In these cases, there is often insufficient damping (excessive Q) of the filter, and it becomes a design problem to add damping by some means to lower the Q to a reasonable value. The bandpass and bandstop filters are intended to act upon a narrow range around the center frequency o. In this type of filter, a high Q is generally desired. It may be seen that the wih of the band of frequencies passed by the bandpass filter, or rejected by the bandstop filter, is inversely proportional to the filter s Q. If this bandwih is defined as the range of frequency for which the bandpass filter transfer function is above -3 db, or the bandstop filter is 11 of 14

12 below -3 db, it may be calculated as follows. First, 3 db was chosen somewhat arbitrarily because it is the equivalent of, or ( ) 1, a round number, but is an industry standard choice. The lower cutoff frequency 1, and upper cutoff frequency, are then calculated for the loop impedance (13): Z j Z o 1 Q j o 1 Q j ( o ) 1 o 1 1 4Q o 1 1 4Q BW 1 Q Q Q 1 or o Q o Q (14) The approximations given for and are valid for high Q. Bandpass and bandstop filters are used to select, or eliminate, a relatively narrow range of signal components centered on. In these filters, Q is selected in accordance with the desired bandwih. Resonant Networks as Impedance Transformers Resonant networks are often used as impedance transformers. In this application, an external load is attached to the resonant network, which presents a transformed equivalent input resistance to the signal source. This is useful when the signal has a narrow bandwih, and allows the resonant network to combine impedance transformation with filtering. Two possible L-networks are shown in Fig. 8. There are at least six others in which the two arms of the L may be composed of L and C, or two series-connected inductors or capacitors. In Fig. 8, the signal sources are shown as idealized voltage or current sources: If the source s internal impedance is significant, it also becomes part of the network. The two networks in Fig. 8 will contribute both resonant filtering and load resistance transformation. The analysis of Fig. 8, network A, may be done by writing an expression for its input impedance Z eq. See equation (15), first line. Network A is parallel-resonant, with the load resistance providing the damping. The approximation in (15) is introduced by assuming that Fig. 8 Two impedance-transforming L-networks. Network A: Real part of input impedance <. Network B: Real part of input impedance >. 1 of 14

13 Q is high. The second line of (15) shows that the approximate input impedance of the network is the same as that of a series-resonant network damped by the transformed load resistance. Note that operation in the vicinity of o is assumed. Z eq j L 1 j C j L j j L j 1 ( C) ( C) C Z eq j L 1 j C Z o for Q o C 5 near o the input resistance is R eq Z o Q for Network A C (15) The effective input resistance is seen to be a fraction of the load resistance. Equation (15) also shows that the transformation turns-ratio is equivalently 1:Q. For frequencies off-resonance, the input impedance also includes significant reactive components, indicated by the terms in brackets in (15), second line. A similar analysis of network B in Fig. 8 produces similar results. In this case, the load resistance provides series-resonant damping. The effective input admittance is found to be: Y eq j C 1 j L j C j L R L ( L) Y eq j C 1 j L Z o for Q L 5 near o the input resistance is R eq Z o Q for Network B j C j L ( L) (16) As before, the approximation in the first line of (16) is introduced by assuming that the Q is high. The effective input admittance, second line, is that of a parallel-connected LC network, together with a reflected input conductance. The resistance equivalent of the input conductance is given in line five of (16). Again, the network reflects the load resistance in a manner of a transformer with a turns ratio of Q:1. A straightforward design procedure can be seen from (15) and (16). The resonant frequency of the network is its operating frequency, with a narrow working bandwih. The characteristic resistance ( ) needs to be set equal to the geometric mean of the intended load and reflected equivalent resistances ( Z o R eq ). However, the Q factor is constrained by the ratio of the load and reflected resistances. Q cannot be independently chosen to achieve bandwih objectives, or to ensure a realizable system. Although the L-network is widely used, it is too inflexible to achieve many required results. It should be noticed that the L-network matching criteria can be used to find the series- and parallel-resonant equivalents to networks with several damping resistances, given in (11) and Fig of 14

14 Fig. 9 Pi-type resonant matching network, along with two approximately equivalent load resistor placements. Finally, the pi-network will be considered. There are numerous 3-element pi- and tee-networks that can be used to get a more flexible resonant matching network, but one example of the pi-network will be sufficient. Fig. 9 (left side) shows a resonant pi-network for load resistance matching, along with two approximately equivalent load resistor placements. The network shown is assumed to be lightly damped (Q>5), and resonance is expected at the frequency corresponding to L and the series equivalent of C 1 and C. It can be seen that the load is tapped down the series combination of C 1 and C, so the equivalents shown in Fig. 5 may be used to generate the two alternate placements of equivalent damping resistance (middle of Fig. 9) and (right of Fig. 9). Using (1), the equivalent damping resistance values are: C R 1 C L C 1 and C R C L C 1 C RL C 1 (17) The equivalent input admittance of Fig. 9, left side, is then calculated using (17). Y eq 1 C 1 C j C1 j C LC 1 (18) C R L C 1 It can be seen from (18) that the reflected value of the load resistance is. The second term in (18) represents the equivalent susceptance. This term nulls at the resonant frequency determined by L and the series equivalent of C 1 and C. Design of the pi-network is flexible because the resonant frequency is determined by the signal frequency, and the C 1 -C ratio is determined by the desired reflected value of the load resistance, but the characteristic resistance of the resonant network is left undetermined. This may be set to achieve a bandwih objective. When bandwih is not critical, it is common to set Q to the range of 5-10 to keep the losses in the inductor manageable. 14 of 14

### Chapter 12. RL Circuits. Objectives

Chapter 12 RL Circuits Objectives Describe the relationship between current and voltage in an RL circuit Determine impedance and phase angle in a series RL circuit Analyze a series RL circuit Determine

### LAB #1: TIME AND FREQUENCY RESPONSES OF SERIES RLC CIRCUITS Updated July 19, 2003

SFSU - ENGR 3 ELECTRONICS LAB LAB #: TIME AND FREQUENCY RESPONSES OF SERIES RLC CIRCUITS Updated July 9, 3 Objective: To investigate the step, impulse, and frequency responses of series RLC circuits. To

### FREQUENCY RESPONSE AND PASSIVE FILTERS LABORATORY. We start with examples of a few filter circuits to illustrate the concept.

FREQUENCY RESPONSE AND PASSIVE FILTERS LABORATORY In this experiment we will analytically determine and measure the frequency response of networks containing resistors, AC source/sources, and energy storage

### Chapter 10. RC Circuits. Objectives

Chapter 10 RC Circuits Objectives Describe the relationship between current and voltage in an RC circuit Determine impedance and phase angle in a series RC circuit Analyze a series RC circuit Determine

### Chapter 21 Band-Pass Filters and Resonance

Chapter 21 Band-Pass Filters and Resonance In Chapter 20, we discussed low-pass and high-pass filters. The simplest such filters use RC components resistors and capacitors. It is also possible to use resistors

### Chapt ha e pt r e r 12 RL Circuits

Chapter 12 RL Circuits Sinusoidal Response of RL Circuits The inductor voltage leads the source voltage Inductance causes a phase shift between voltage and current that depends on the relative values of

### Filters and Waveform Shaping

Physics 333 Experiment #3 Fall 211 Filters and Waveform Shaping Purpose The aim of this experiment is to study the frequency filtering properties of passive (R, C, and L) circuits for sine waves, and the

### FILTER CIRCUITS. A filter is a circuit whose transfer function, that is the ratio of its output to its input, depends upon frequency.

FILTER CIRCUITS Introduction Circuits with a response that depends upon the frequency of the input voltage are known as filters. Filter circuits can be used to perform a number of important functions in

### 2.161 Signal Processing: Continuous and Discrete Fall 2008

MT OpenCourseWare http://ocw.mit.edu.6 Signal Processing: Continuous and Discrete Fall 00 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. MASSACHUSETTS

### Lab 8: Basic Filters: Low- Pass and High Pass

Lab 8: Basic Filters: Low- Pass and High Pass Names: 1.) 2.) 3.) Objectives: 1. Show students how circuits can have frequency- dependent resistance, and that many everyday signals are made up of many frequencies.

### Experiment V: The AC Circuit, Impedance, and Applications to High and Low Pass Filters

Experiment : The AC Circuit, Impedance, and Applications to High and Low Pass Filters I. eferences Halliday, esnick and Krane, Physics, ol. 2, 4th Ed., Chapters 33 Purcell, Electricity and Magnetism, Chapter

### Electrical Circuits (2)

Electrical Circuits () Lecture 4 Parallel Resonance and its Filters Dr.Eng. Basem ElHalawany Parallel Resonance Circuit Ideal Circuits It is usually called tank circuit Practical Circuits Complex Coniguration

### UNIVERSITY OF NORTH CAROLINA AT CHARLOTTE. Department of Electrical and Computer Engineering

UNIVERSITY OF NORTH CAROLINA AT CHARLOTTE Department of Electrical and Computer Engineering Experiment No. 9 - Resonance in Series and parallel RLC Networks Overview: An important consideration in the

### EXPERIMENT 5: SERIES AND PARALLEL RLC RESONATOR CIRCUITS

EXPERIMENT 5: SERIES AND PARALLEL RLC RESONATOR CIRCUITS Equipment List S 1 BK Precision 4011 or 4011A 5 MHz Function Generator OS BK 2120B Dual Channel Oscilloscope V 1 BK 388B Multimeter L 1 Leeds &

### Filter Considerations for the IBC

application note AN:202 Filter Considerations for the IBC Mike DeGaetano Application Engineering July 2013 Contents Page Introduction 1 IBC Attributes 1 Damping and 2 Converter Bandwidth Filtering 3 Filter

### Frequency response: Resonance, Bandwidth, Q factor

Frequency response: esonance, Bandwidth, Q factor esonance. Let s continue the exploration of the frequency response of circuits by investigating the series circuit shown on Figure. C + V - Figure The

### CHAPTER 6 Frequency Response, Bode Plots, and Resonance

ELECTRICAL CHAPTER 6 Frequency Response, Bode Plots, and Resonance 1. State the fundamental concepts of Fourier analysis. 2. Determine the output of a filter for a given input consisting of sinusoidal

### Chapter 13. RLC Circuits and Resonance. Objectives

Chapter 13 RLC Circuits and Resonance Objectives Determine the impedance of a series RLC circuit Analyze series RLC circuits Analyze a circuit for series resonance Analyze series resonant filters Analyze

### ECE215 Lecture 16 Date:

Lecture 16 Date: 20.10.2016 Bode Plot (contd.) Series and Parallel Resonance Example 1 Find the transfer function H(ω) with this Bode magnitude plot Example 2 Find the transfer function H(ω) with this

### ε: Voltage output of Signal Generator (also called the Source voltage or Applied

Experiment #10: LR & RC Circuits Frequency Response EQUIPMENT NEEDED Science Workshop Interface Power Amplifier (2) Voltage Sensor graph paper (optional) (3) Patch Cords Decade resistor, capacitor, and

### EXPERIMENT 6 - ACTIVE FILTERS

1.THEORY HACETTEPE UNIVERSITY DEPARTMENT OF ELECTRICAL AND ELECTRONICS ENGINEERING ELE-313 ELECTRONICS LABORATORY II EXPERIMENT 6 - ACTIVE FILTERS A filter is a circuit that has designed to pass a specified

### DC Circuits: Operational Amplifiers Hasan Demirel

DC Circuits: Operational Amplifiers Hasan Demirel Op Amps: Introduction Op Amp is short form of operational amplifier. An op amp is an electronic unit that behaves like a voltage controlled voltage source.

### April 8. Physics 272. Spring Prof. Philip von Doetinchem

Physics 272 April 8 Spring 2014 http://www.phys.hawaii.edu/~philipvd/pvd_14_spring_272_uhm.html Prof. Philip von Doetinchem philipvd@hawaii.edu Phys272 - Spring 14 - von Doetinchem - 218 L-C in parallel

### Frequency Response of Filters

School of Engineering Department of Electrical and Computer Engineering 332:224 Principles of Electrical Engineering II Laboratory Experiment 2 Frequency Response of Filters 1 Introduction Objectives To

### Chapter 3. Simulation of Non-Ideal Components in LTSpice

Chapter 3 Simulation of Non-Ideal Components in LTSpice 27 CHAPTER 3. SIMULATION OF NON-IDEAL COMPONENTS IN LTSPICE 3.1 Pre-Lab The answers to the following questions are due at the beginning of the lab.

### Chapter 12. RL Circuits ISU EE. C.Y. Lee

Chapter 12 RL Circuits Objectives Describe the relationship between current and voltage in an RL circuit Determine impedance and phase angle in a series RL circuit Analyze a series RL circuit Determine

### Eðlisfræði 2, vor 2007

[ Assignment View ] [ Pri Eðlisfræði 2, vor 2007 31. Alternating Current Circuits Assignment is due at 2:00am on Wednesday, March 21, 2007 Credit for problems submitted late will decrease to 0% after the

### EE 311: Electrical Engineering Junior Lab Active Filter Design (Sallen-Key Filter)

EE 311: Electrical Engineering Junior Lab Active Filter Design (Sallen-Key Filter) Objective The purpose of this experiment is to design a set of second-order Sallen-Key active filters and to investigate

### RLC Resonant Circuits

C esonant Circuits Andrew McHutchon April 20, 203 Capacitors and Inductors There is a lot of inconsistency when it comes to dealing with reactances of complex components. The format followed in this document

### Lesson 27. (1) Root Mean Square. The emf from an AC generator has the time dependence given by

Lesson 27 () Root Mean Square he emf from an AC generator has the time dependence given by ℇ = ℇ "#\$% where ℇ is the peak emf, is the angular frequency. he period is he mean square value of the emf is

### CIRCUITS LABORATORY EXPERIMENT 3. AC Circuit Analysis

CIRCUITS LABORATORY EXPERIMENT 3 AC Circuit Analysis 3.1 Introduction The steady-state behavior of circuits energized by sinusoidal sources is an important area of study for several reasons. First, the

### Lab 8: Basic Filters: Low- Pass and High Pass

Lab 8: Basic Filters: Low- Pass and High Pass Names: 1.) 2.) 3.) Beginning Challenge: Build the following circuit. Charge the capacitor by itself, and then discharge it through the inductor. Measure the

### Since any real component also has loss due to the resistive component, the average power dissipated is 2 2R

Quality factor, Q Reactive components such as capacitors and inductors are often described with a figure of merit called Q. While it can be defined in many ways, it s most fundamental description is: Q

### Analogue Filter Design

Analogue Filter Design Module: SEA Signals and Telecoms Lecturer: URL: http://www.personal.rdg.ac.uk/~stsgrimb/ email: j.b.grimbleby reading.ac.uk Number of Lectures: 5 Reference text: Design with Operational

### The Flyback Converter

The Flyback Converter Lecture notes ECEN4517! Derivation of the flyback converter: a transformer-isolated version of the buck-boost converter! Typical waveforms, and derivation of M(D) = V/! Flyback transformer

### INTRODUCTION SELF INDUCTANCE. Introduction. Self inductance. Mutual inductance. Transformer. RLC circuits. AC circuits

Chapter 13 INDUCTANCE Introduction Self inductance Mutual inductance Transformer RLC circuits AC circuits Magnetic energy Summary INTRODUCTION Faraday s important contribution was his discovery that achangingmagneticflux

### Chapter 10. RC Circuits ISU EE. C.Y. Lee

Chapter 10 RC Circuits Objectives Describe the relationship between current and voltage in an RC circuit Determine impedance and phase angle in a series RC circuit Analyze a series RC circuit Determine

### EE133 Winter 2002 Cookbook Filter Guide Welcome to the Cookbook Filter Guide!

Welcome to the! Don t have enough time to spice out that perfect filter before Aunt Thelma comes down for dinner? Well this handout is for you! The following pages detail a fast set of steps towards the

### Analysis of Dynamic Circuits in MATLAB

Transactions on Electrical Engineering, Vol. 4 (2015), No. 3 64 Analysis of Dynamic Circuits in MATLAB Iveta Tomčíková 1) 1) Technical University in Košice/Department of Theoretical and Industrial Electrical

### The Ideal Transformer. Description and Circuit Symbol

The Ideal Transformer Description and Circuit Symbol As with all the other circuit elements, there is a physical transformer commonly used in circuits whose behavior can be discussed in great detail. However,

### Chapter 17 11/13/2014

Chapter 17 Voltage / Current source conversions Mesh and Nodal analysis in an AC circuit Balance conditions and what elements are needed in a bridge network ECET 207 AC Circuit Analysis, PNC 2 1 Magnitude

### R f. V i. ET 438a Automatic Control Systems Technology Laboratory 4 Practical Differentiator Response

ET 438a Automatic Control Systems Technology Laboratory 4 Practical Differentiator Response Objective: Design a practical differentiator circuit using common OP AMP circuits. Test the frequency response

### Chapter 15. Active Filter Circuits

hapter 5 Active Filter ircuits 5.0 Introduction Filter is circuit that capable of passing signal from input to put that has frequency within a specified band and attenuating all others side the band. This

### Chapter 5: Analysis of Time-Domain Circuits

Chapter 5: Analysis of Time-Domain Circuits This chapter begins the analysis of circuits containing elements with the ability to store energy: capacitors and inductors. We have already defined each of

### See Horenstein 4.3 and 4.4

EE 462: Laboratory # 4 DC Power Supply Circuits Using Diodes by Drs. A.V. Radun and K.D. Donohue (2/14/07) Department of Electrical and Computer Engineering University of Kentucky Lexington, KY 40506 Updated

### RLC Circuits. OBJECTIVES To observe free and driven oscillations of an RLC circuit.

ircuits It doesn t matter how beautiful your theory is, it doesn t matter how smart you are. If it doesn t agree with experiment, it s wrong. ichard Feynman (1918-1988) OBJETIVES To observe free and driven

### Transistor Tuned Amplifiers

5 Transistor Tuned Amplifiers 389 Transistor Tuned Amplifiers 5. Tuned Amplifiers 5. Distinction between Tuned Amplifiers and other Amplifiers 5.3 Analysis of Parallel Tuned Circuit 5.4 Characteristics

### Lab Exercise: RLC CIRCUITS AND THE ELECTROCARDIOGRAM

and vary Lab Exercise: RLC CIRCUITS AND THE ELECTROCARDIOGRAM OBJECTIVES Explain how resistors, capacitors and inductors behave in an AC circuit. Explain how an electrocardiogram (EKG) works Explain what

### GENESYS S/FILTER. Eagleware Corporation. Copyright

GENESYS S/FILTER Copyright 1986-2000 Eagleware Corporation 635 Pinnacle Court Norcross, GA 30071 USA Phone: (678) 291-0995 FAX: (678) 291-0971 E-mail: eagleware@eagleware.com Internet: http://www.eagleware.com

### EXERCISES in ELECTRONICS and SEMICONDUCTOR ENGINEERING

Department of Electrical Drives and Power Electronics EXERCISES in ELECTRONICS and SEMICONDUCTOR ENGINEERING Valery Vodovozov and Zoja Raud http://learnelectronics.narod.ru Tallinn 2012 2 Contents Introduction...

### Lesson 3: RLC circuits & resonance

P. Piot, PHYS 375 Spring 008 esson 3: RC circuits & resonance nductor, nductance Comparison of nductance and Capacitance nductance in an AC signals R circuits C circuits: the electric pendulum RC series

### Bharathwaj Muthuswamy EE100 Active Filters

Bharathwaj Muthuswamy EE100 mbharat@cory.eecs.berkeley.edu 1. Introduction Active Filters In this chapter, we will deal with active filter circuits. Why even bother with active filters? Answer: Audio.

### Chapter 5. Basic Filters

Chapter 5 Basic Filters 39 CHAPTER 5. BASIC FILTERS 5.1 Pre-Lab The answers to the following questions are due at the beginning of the lab. If they are not done at the beginning of the lab, no points will

### Fourier Series Analysis

School of Engineering Department of Electrical and Computer Engineering 332:224 Principles of Electrical Engineering II aboratory Experiment 6 Fourier Series Analysis 1 Introduction Objectives The aim

### Chapter 13. RLC Circuits and Resonance

Chapter 13 RLC Circuits and Resonance Impedance of Series RLC Circuits A series RLC circuit contains both inductance and capacitance Since X L and X C have opposite effects on the circuit phase angle,

### In modern electronics, it is important to be able to separate a signal into different

Introduction In modern electronics, it is important to be able to separate a signal into different frequency regions. In analog electronics, four classes of filters exist to process an input signal: low-pass,

### Modern Definition of Terms

Filters In the operation of electronic systems and circuits, the basic function of a filter is to selectively pass, by frequency, desired signals and to suppress undesired signals. The amount of insertion

### 7.1 POWER IN AC CIRCUITS

C H A P T E R 7 AC POWER he aim of this chapter is to introduce the student to simple AC power calculations and to the generation and distribution of electric power. The chapter builds on the material

### EXPERIMENT 4:- MEASUREMENT OF REACTANCE OFFERED BY CAPACITOR IN DIFFERENT FREQUENCY FOR R-C CIRCUIT

Kathmandu University Department of Electrical and Electronics Engineering BASIC ELECTRICAL LAB (ENGG 103) EXPERIMENT 4:- MEASUREMENT OF REACTANCE OFFERED BY CAPACITOR IN DIFFERENT FREQUENCY FOR R-C CIRCUIT

### Circuits with inductors and alternating currents. Chapter 20 #45, 46, 47, 49

Circuits with inductors and alternating currents Chapter 20 #45, 46, 47, 49 RL circuits Ch. 20 (last section) Symbol for inductor looks like a spring. An inductor is a circuit element that has a large

### First and Second Order Filters

First and Second Order Filters These functions are useful for the design of simple filters or they can be cascaded to form high-order filter functions First Order Filters General first order bilinear transfer

### PIEZO FILTERS INTRODUCTION

For more than two decades, ceramic filter technology has been instrumental in the proliferation of solid state electronics. A view of the future reveals that even greater expectations will be placed on

### ENGR 210 Lab 11 Frequency Response of Passive RC Filters

ENGR 210 Lab 11 Response of Passive RC Filters The objective of this lab is to introduce you to the frequency-dependent nature of the impedance of a capacitor and the impact of that frequency dependence

### LCR Series Circuits. AC Theory. Introduction to LCR Series Circuits. Module 9. What you'll learn in Module 9. Module 9 Introduction

Module 9 AC Theory LCR Series Circuits Introduction to LCR Series Circuits What you'll learn in Module 9. Module 9 Introduction Introduction to LCR Series Circuits. Section 9.1 LCR Series Circuits. Amazing

### Electronic Components. Electronics. Resistors and Basic Circuit Laws. Basic Circuits. Basic Circuit. Voltage Dividers

Electronics most instruments work on either analog or digital signals we will discuss circuit basics parallel and series circuits voltage dividers filters high-pass, low-pass, band-pass filters the main

### Phasors. Phasors. by Prof. Dr. Osman SEVAİOĞLU Electrical and Electronics Engineering Department. ^ V cos (wt + θ) ^ V sin (wt + θ)

V cos (wt θ) V sin (wt θ) by Prof. Dr. Osman SEVAİOĞLU Electrical and Electronics Engineering Department EE 209 Fundamentals of Electrical and Electronics Engineering, Prof. Dr. O. SEVAİOĞLU, Page 1 Vector

### Lab #4 Capacitors and Inductors. Capacitor and Inductor Transient Response

Capacitor and Inductor Transient Response Capacitor Theory Like resistors, capacitors are also basic circuit elements. Capacitors come in a seemingly endless variety of shapes and sizes, and they can all

### Questions. Question 1

Question 1 Questions Explain why transformers are used extensively in long-distance power distribution systems. What advantage do they lend to a power system? file 02213 Question 2 Are the transformers

### Electrical Circuits I Lecture 1

Electrical Circuits I Lecture Course Contents Basic dc circuit elements, series and parallel Networks Ohm's law and Kirchoff's laws Nodal Analysis Mesh Analysis Source Transformation

### Understanding Adjustable Speed Drive Common Mode Problems and Effective Filter Solutions

Understanding Adjustable Speed Drive Common Mode Problems and Effective Filter Solutions September 22, 2014 Todd Shudarek, Principal Engineer MTE Corporation N83 W13330 Leon Road Menomonee Falls WI 53051

### Measuring Impedance and Frequency Response of Guitar Pickups

Measuring Impedance and Frequency Response of Guitar Pickups Peter D. Hiscocks Syscomp Electronic Design Limited phiscock@ee.ryerson.ca www.syscompdesign.com April 30, 2011 Introduction The CircuitGear

### CHAPTER 16 OSCILLATORS

CHAPTER 16 OSCILLATORS 16-1 THE OSCILLATOR - are electronic circuits that generate an output signal without the necessity of an input signal. - It produces a periodic waveform on its output with only the

### Resonant and cut-off frequencies Tuned network quality, bandwidth, and power levels Quality factor

Chapter 20 Resonant and cut-off frequencies Tuned network quality, bandwidth, and power levels Quality factor ECET 207 AC Circuit Analysis, PNC 2 1 20.1-20.7 A condition established by the application

### Reactance and Impedance

Reactance and Impedance Capacitance in AC Circuits Professor Andrew H. Andersen 1 Objectives Describe capacitive ac circuits Analyze inductive ac circuits Describe the relationship between current and

### Chapter 12 Driven RLC Circuits

hapter Driven ircuits. A Sources... -. A ircuits with a Source and One ircuit Element... -3.. Purely esistive oad... -3.. Purely Inductive oad... -6..3 Purely apacitive oad... -8.3 The Series ircuit...

### R-L-C Circuits and Resonant Circuits

P517/617 Lec4, P1 R-L-C Circuits and Resonant Circuits Consider the following RLC series circuit What's R? Simplest way to solve for is to use voltage divider equation in complex notation. X L X C in 0

### Laboratory #5: RF Filter Design

EEE 194 RF Laboratory Exercise 5 1 Laboratory #5: RF Filter Design I. OBJECTIVES A. Design a third order low-pass Chebyshev filter with a cutoff frequency of 330 MHz and 3 db ripple with equal terminations

### ALTERNATING CURRENTS

ALTERNATING CURRENTS VERY SHORT ANSWER QUESTIONS Q-1. What is the SI unit of? Q-2. What is the average value of alternating emf over one cycle? Q-3. Does capacitor allow ac to pass through it? Q-4. What

### UNIVERSITY of PENNSYLVANIA DEPARTMENT of ELECTRICAL and SYSTEMS ENGINEERING ESE206 - Electrical Circuits and Systems II Laboratory.

UNIVERSITY of PENNSYLVANIA DEPARTMENT of ELECTRICAL and SYSTEMS ENGINEERING ESE06 - Electrical Circuits and Systems II Laboratory. Objectives: Transformer Lab. Comparison of the ideal transformer versus

### Quality Factor, Bandwidth, and Harmonic Attenuation of Pi Networks

Bill Kaune, W7IE 6 Cedarview Dr, Port Townsend, WA 98363: wtkaune@ieee.org uality Factor, Bandwidth, and Harmonic Attenuation of Pi Networks The author looks at several definitions of, and makes some interesting

### Introduction to Series-Parallel DC Circuits. Online Resource for ETCH 213 Faculty: B. Allen

Introduction to Series-Parallel DC Circuits Series-parallel circuit A network or circuit that contains components that are connected in both series and parallel. Series-parallel resistive circuits Tracking

### First Order Transient Response

First Order Transient Response When non-linear elements such as inductors and capacitors are introduced into a circuit, the behaviour is not instantaneous as it would be with resistors. A change of state

### EE 221 AC Circuit Power Analysis. Instantaneous and average power RMS value Apparent power and power factor Complex power

EE 1 AC Circuit Power Analysis Instantaneous and average power RMS value Apparent power and power factor Complex power Instantaneous Power Product of time-domain voltage and time-domain current p(t) =

### An Introduction to the Mofied Nodal Analysis

An Introduction to the Mofied Nodal Analysis Michael Hanke May 30, 2006 1 Introduction Gilbert Strang provides an introduction to the analysis of electrical circuits in his book Introduction to Applied

### Understanding Poles and Zeros

MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 2.14 Analysis and Design of Feedback Control Systems Understanding Poles and Zeros 1 System Poles and Zeros The transfer function

### USING THE ANALOG DEVICES ACTIVE FILTER DESIGN TOOL

USING THE ANALOG DEVICES ACTIVE FILTER DESIGN TOOL INTRODUCTION The Analog Devices Active Filter Design Tool is designed to aid the engineer in designing all-pole active filters. The filter design process

### Let s examine the response of the circuit shown on Figure 1. The form of the source voltage Vs is shown on Figure 2. R. Figure 1.

Examples of Transient and RL Circuits. The Series RLC Circuit Impulse response of Circuit. Let s examine the response of the circuit shown on Figure 1. The form of the source voltage Vs is shown on Figure.

### University of Technology Laser & Optoelectronics Engineering Department Communication Engineering Lab.

OBJECT: To establish the pass-band characteristic. APPARTUS: 1- Signal function generator 2- Oscilloscope 3- Resisters,capacitors 4- A.V.O. meter. THEORY: Any combination of passive (R, L, and C) and/or

### Alternating-Current Circuits

hapter 1 Alternating-urrent ircuits 1.1 A Sources... 1-1. Simple A circuits... 1-3 1..1 Purely esistive load... 1-3 1.. Purely Inductive oad... 1-5 1..3 Purely apacitive oad... 1-7 1.3 The Series ircuit...

### Keysight Technologies Understanding the Fundamental Principles of Vector Network Analysis. Application Note

Keysight Technologies Understanding the Fundamental Principles of Vector Network Analysis Application Note Introduction Network analysis is the process by which designers and manufacturers measure the

### Coupled Electrical Oscillators Physics 3600 Advanced Physics Lab Summer 2010 Don Heiman, Northeastern University, 5/10/10

Coupled Electrical Oscillators Physics 3600 Advanced Physics Lab Summer 00 Don Heiman, Northeastern University, 5/0/0 I. Introduction The objectives of this experiment are: () explore the properties of

### CIRCUITS AND SYSTEMS LABORATORY EXERCISE 6 TRANSIENT STATES IN RLC CIRCUITS AT DC EXCITATION

CIRCUITS AND SYSTEMS LABORATORY EXERCISE 6 TRANSIENT STATES IN RLC CIRCUITS AT DC EXCITATION 1. DEVICES AND PANELS USED IN EXERCISE The following devices are to be used in this exercise: oscilloscope HP

### Positive Feedback and Oscillators

Physics 3330 Experiment #6 Fall 1999 Positive Feedback and Oscillators Purpose In this experiment we will study how spontaneous oscillations may be caused by positive feedback. You will construct an active

### Slide 1 / 26. Inductance. 2011 by Bryan Pflueger

Slide 1 / 26 Inductance 2011 by Bryan Pflueger Slide 2 / 26 Mutual Inductance If two coils of wire are placed near each other and have a current passing through them, they will each induce an emf on one

### Lab 4 Band Pass and Band Reject Filters

Lab 4 Band Pass and Band Reject Filters Introduction During this lab you will design and build three filters. First you will build a broad-band band-pass filter by cascading the high-pass and low-pass

### Introduction to RF Filter Design. RF Electronics Spring, 2016 Robert R. Krchnavek Rowan University

Introduction to RF Filter Design RF Electronics Spring, 2016 Robert R. Krchnavek Rowan University Objectives Understand the fundamental concepts and definitions for filters. Know how to design filters

### AN-837 APPLICATION NOTE

APPLICATION NOTE One Technology Way P.O. Box 916 Norwood, MA 262-916, U.S.A. Tel: 781.329.47 Fax: 781.461.3113 www.analog.com DDS-Based Clock Jitter Performance vs. DAC Reconstruction Filter Performance

### Laboratory 4: Feedback and Compensation

Laboratory 4: Feedback and Compensation To be performed during Week 9 (Oct. 20-24) and Week 10 (Oct. 27-31) Due Week 11 (Nov. 3-7) 1 Pre-Lab This Pre-Lab should be completed before attending your regular