TEACHING RLC PARALLEL CIRCUITS IN HIGH SCHOOL

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Volume 8, Number 4, 2015TEACHING RLC PARALLEL CIRCUITSIN HIGH-SCHOOL PHYSICS CLASSAlpár SimonAbstract: This paper will try to give an alternative treatment of the subject "parallel RLC circuits"and "resonance in parallel RLC circuits" from the Physics curricula for the XIth grade fromRomanian high-schools, with an emphasis on practical type circuits and their possible applications,and intends to be an aid for both Physics teachers and students eager to learn and understand more.Key words: alternating current, RLC circuits, resonance, teaching1. IntroductionBoth Electricity and Magnetism has been known as basic subjects in Physics education, at all levels,because of several reasons: (a) both are the main source of knowledge about the structure, propertiesand applications of matter, (b) because of their practical applications have a great relevance in oureveryday lives, under every aspect of it (social, cultural, personal, technological, etc.). Therefore, inseveral studies [1-7] regarding learning difficulties of Physics, subjects like steady state or dynamicelectric circuits were used to measure the level of the problem (helding misconceptions,misunderstanding concepts, erroneous reasoning, conceptual difficulties, etc.). As expected, there areseveral studies of alternative methods to help students to overcome those difficulties. A fair survey ofthe literature [8-15] suggest two type of methodes: (a) a traditional one, with emphasis onexperimental activities (laboratory) and (b) a modern one, with emphasis on computational resourcesusing modeling and simulations.Alternating current (AC) and related phenomena, physical quantities and applications are a veryimportant part of the Romanian high-school Physics curricula for both Xth and XIth grades [16-17]. Inthe Xth grade it takes the students to an exciting journey from definition and generation, through circuitelements behavior and AC energy/power notions, to applications like transformers, electric motors orhome appliances. Students from the XIth grade are taken to the "next level". They learn about the RLCcircuits, electromagnetic oscillations and resonance, and some practical applications of oscillatingcircuits.All these AC related knowledge could be a considerable challenge for the students due to two reasons:(a) the physical quantities have a very different dynamic behavior and properties, as compared to whatthey know until than (time and frequency dependence, periodicity, reverse in direction, phase, lead orlag relationship between voltage and current), (b) the math's that applies is quite difficult(trigonometry, operation with time dependent quantities, Fresnel type phasor diagram, differentialequations, complex numbers, etc.).One of the major problem with the Physics textbooks, designed and written according to the abovecited curricula, is that they are using ideal, theoretical models and concepts that are, sometimes veryfar or not related at all with reality or the practical aspects of the subject. This is the special case of theparallel RLC circuit in the XIth grade curricula and accredited Physics textbooks [18-21].This paper will try to give an alternative treatment of the subject "parallel RLC circuits" and"resonance in parallel RLC circuits" with an emphasis on practical type circuits and their possibleapplications.Received December 2015.

34Alpár Simon2. Basic aspects of the RLC circuits in XIth grade Physics textbookAccording to the mentioned Physics textbooks [18-20] an RLC circuit is an oscillating electric circuitconsisting of a resistor (R), an inductor (L) and a capacitor (C) connected in series or in parallel (seeFigure 1 a and b).(a)(b)Figure 1. Ideal RLC series (a) and parallel (b) circuitsThe name of the circuit is derived from the initials of the constituent passive components of the circuit,connected is that particular order. Obviously, if they would be connected in an other sequence, thecircuit name will not be different, as it is expected!Oscillating means that such a circuit is able to produce a periodic, oscillating signal by the periodicaltransfer of the stored energy between the two reservoirs (L and C), the resistance (R) being responsiblefor the dumping, the loss of some energy during the back and forth transformation via Joule heatdissipation, leading to the exponential decay of these oscillations.The most important facts about the two circuits, stated in the textbook, are synthesized comparativelyin Table 1.Table 1. Summary of properties of RLC circuits based on XIth grade Physics textbook [19]TypeSERIESPropertyImpedancePhase angleResonance type1 Z s R2 L C tan s 1 CR L PARALLEL2Zp 11 1 C 2R L 1 tan p R C L . of voltages and. of currents andimpedance is minimizedimpedance is maximizedResonance frequencyfs Quality factorQs 12 L C1L R C2fp 12 L Cnot given3. Some critical aspects of the RLC circuits in XIth grade Physics textbookWhen analyzing the information given about the parallel RLC circuit one can see that, it issignificantly less than that given for the series circuit (which is also incomplete and deficientsometimes).Acta Didactica Napocensia, ISSN 2065-1430

Teaching RLC parallel circuits in high-school Physics class35There is no information about the half-power frequencies and bandwidths. A very superficialdefinition of quality factor Q is given and only for the series circuit. Both AC powers (active, reactiveand apparent) and power factor are defined and given for just the series case.The chapter ends with just two possible applications of AC circuits, i.e. transformers and electricmotors, presented and treated very briefly.All the components were considered to be ideal, but both inductor and capacitor usually have a lossresistance - therefore it is not even necessary to have an extra resistor, the dumping being assured bythese loss resistances.Such a real parallel RLC circuit is depicted in Figure 2, where RL and RC are the loss resistances of theinductor and the capacitor, respectively.Figure 2. Two branch RLC parallel circuitStudying such a parallel circuit is considerably more difficult than those depicted in Figure 1, even ifin the case of the ideal circuits it is customary to say that the parallel circuit is "dual" of the series one[22] - the current and the voltage exchange roles, the parallel circuit has a current gain instead of thevoltage gain found for the series one, the impedance will be maximized for the parallel oscillator atresonance rather than minimized, as it is for the series one.We may suspect that it will be somewhat similar for the real parallel circuit, but this is not so obviousat the first sight and has to be proven later. All this uncertainty is mainly due to the fact that, in thecase of Figure 2, each branch of the circuit will have its own phase angle and they cannot be combinesimply, like in for ideal ones.When studying AC circuits, the following three methods are available: a) Analytical method; b)Fresnel phasor (vector) method; c) Complex number method. Each one has its own advantages, butdrawbacks too, mainly due to the not so easy maths.The analysis of the two branch real parallel circuit can be found rarely in the literature, authorsprivilege the easier ideal models. Despite of this, some excellent treatment of the realistic RLC circuitsare available: a very short and synthetic, but useful presentation [23] or a more detailed one, withseveral calculus examples [24].4. Detailed analysis of the practical RLC parallel circuitIn order to give a real aid for the curious and enquiring high-school students, let's presume that thecapacitor is ideal (RC 0). This is a presumption very close to reality, the dielectric found in capacitorare almost lossless when used under working conditions, and will lead us to a much approachable andtreatable circuit called practical RLC parallel circuit (see Figure 3).Figure 3. Practical RLC parallel circuitLet us consider the practical parallel circuit presented in Figure 4, where the circuit is connected to asignal generator providing a time varying input signal described by the expression:u(t ) U 0 sin( t )Volume 8 Number 4, 2015

36Alpár Simonwhere U0 is the peak value of the voltage (unit: V) and represents the angular frequency (unit: rad/s)defined using the physical frequency f (number of cycles per second, unit: Hz) like being 2 f .Figure 4. Circuit diagram for the idealized LRC parallel circuitFor such a configuration, the instantaneous voltage across the branches will be the same and the timevarying analytical expressions for the current in the main branch of the circuit and in the two parallelbranches become, respectively:i(t ) I 0 sin( t )iL (t ) I L 0 sin( t L )iC (t ) I C 0 sin( t C )where the indexes "L" and "C" are depicting the branches containing the inductor and the capacitor,respectively, index "0" indicates the peak values of the currents and the Greek letter (phi) is standingfor the phase angle appeared between currents and voltage, the " " or " " signs are denoting theleading or lagging relationship between the current and voltage. The value of the phase angle in thebranch of the capacitor is C / 2 (or 90o) because it was considered to be ideal. The phase angle inthe inductors branch will depend on both loss resistance and inductance, L tan-1( L/R).It will be useful to make a short detour here: in Reference [25] a very good suggestion is made formemorizing the current/voltage relationships in the case of capacitors and inductors - the mnemonicCIVIL is introduced.Considering the positions of the letters in the word CIVIL we will have: C - I - V (in the case of thecapacitor C, the current I will LEAD the voltage V) and V (repeated) - I - L (voltage V LEAD thecurrent I, in the case of the inductor L), respectively.In order to complete the analysis of such a circuit it will be necessary to calculate the currents in eachbranch (main and secondary), work out the phase relationship between the main current and the supplyvoltage, find the impedance of the circuit and the resonance frequency, and figure out the time(frequency) dependent behaviour of the circuit.Beacuse of the necessity of solving differential equations, an exhaustive analytical treatment of thesubject exceeds the Romanian high-school maths curricula. Anyway, in order to find out the currents,some analytical expressions has to be written down.For the inductors branch, we will have the Kirchhoff's voltage law (KVL) for the instantaneousvoltages:u(t ) U 0 sin( t ) uR (t ) uL (t ) R iL (t ) L diL (t )dtdiL (t ) I L 0 cos( t L ) I L 0 sin( t L )dt2Acta Didactica Napocensia, ISSN 2065-1430

Teaching RLC parallel circuits in high-school Physics class37 U 0 sin( t ) R I 0 sin( t L ) L I L 0 sin( t L )2For the capacitors branch, we have for the instantaneous value of the electric charge on the capacitorplates:QC (t ) C u(t ) C U 0 sin( t )The current through this branch will become:iC (t ) dQC (t )du(t ) C C U 0 sin( t ) I C 0 sin( t )dtdt22The current in the main branch of the circuit will be given by the Kirchhoff's current law (KCL):i(t ) iC (t ) iL (t ) I 0 sin( t ) I L 0 sin( t L ) I C 0 sin( t )2These currents can be represented with the phasor diagram shown in Figure 5.Figure 5. Phasor diagram for the idealized LRC parallel circuit(a) with capacitive behavior (currents leads voltage), (b) with inductive behavior (currents lags voltage)Solving such diagrams using only geometry and trigonometry is not such an easy task.Let's take a look on Figure 5a. The peak value of the current in the main branch will be:I 02 ( I C 0 I L 0 sin L ) 2 ( I L 0 cos L ) 2I 02 I C2 0 I L20 sin 2 L 2 I C 0 I L0 sin L I L0 cos 2 LI 02 I C2 0 I L20 2 I C 0 I L 0 sin LThis result lead us to the impedance and later, to the resonant frequency of the circuit.In AC circuits the complex ratio of voltage to current is called impedance. This somehow extends theconcept of resistance (known from direct current part of the curricula, Xth grade). Basicly, it is ageneralized, extended resistance which has both magnitude and phase.Thus for the impedance one can write:Volume 8 Number 4, 2015

38Alpár Simon U0 Z pwith sin L 2 U 02(C ) U 02 (C ) 2 2 U 02 sin L22 22R (L )R (L ) L R2 (L )2.Substituting and simplifying, we will find:2 U0 U 02C L U 02 (C ) 2 2 U 02 22 Z R (L )R 2 (L )2R 2 (L )2 p 2 U0 U 02C L 22 U 02 (C ) 2 2 U 0 2 Z R 2 (L )2R (L )2 p 2 1 1C L 2 (C ) 2 2 Z R 2 (L )2R 2 (L )2 p Zp 2R2 (L )2( R C ) 2 ( 2 L C 1) 2Thus, the impedance of the practical parallel RLC circuit is frequency dependent and given by therelationship:Z p ( ) R2 (L )2( R C ) 2 ( 2 L C 1) 2Resonance is a very important phenomenon in physics occuring in all sorts of systems. In the case ofoscillating circuits, it occurs when the phase angle between the current in the main branch of thecircuit and voltage across the circuit will be equal to zero.From Figure 5a one can see that:tan I C 0 I L 0 sin LI L 0 cos LSubstituting and simplifying, we will find:(C ) [ R 2 ( L ) 2 ] ( L )tan RThe resonance condition ( tan 0 ) will give for the resonant frequency of the practical RLC circuitthe following expression:11 R fp 2 L C L 2One can see that, this value is slightly smaller, due to the loss resistance of the coil, than the resonancefrequencyActa Didactica Napocensia, ISSN 2065-1430

Teaching RLC parallel circuits in high-school Physics classfs 3912 L Cof the series RLC circuit obtained using the same components. Combining the two expressions for thefrequencies, one can write:R2 Cf p fs 1 fsLThe value of the impedance of the practical parallel RLC circuit at the resonance frequency fp, calleddynamic resistance, and its expression will be:Z p ( ) 2 f p LR CAs no one expects, at least after reading and learning from the XIth grade textbook, this impedance willbe near to its maximum value, but will not be quite that maximum! The frequency at which themaximum impedance will occur is defined by an other, slightly higher frequency that the resonanceone, as demonstrated in Figure 6. This frequency is determined by differentiating (calculus) thegeneral equation for the parallel impedance with respect to frequency and then determining thefrequency at which the resulting equation is equal to zero. The math's is quite extensive andcumbersome for the XIth grade, and will not be included here, but Physics teacher should deal with iteasily. However, the expression for this frequency is the following:1 R2 Cfm fs 1 4LFigure 6. Example of frequency dependence of the impedance of a practical RLC parallel circuitAn other very important quantity, the quality factor (Q) of the resonant system is a measure of how"sharp" or narrow the frequency dependence of the impedance Z p Z p ( ) is.Volume 8 Number 4, 2015

40Alpár SimonAccordind to one possible definition, the quality factor Q represents 2 multiplied by the ratiobetween the maximum energy stored and the total energy lost per cycle, at resonance.For the maximum energy stored in the circuit we have:Emax storedU 02 2 Cand the for the total energy lost due to the resistance in circuit, per cycle will find:Etotal lostU 0211 2 R I L0 R2 R 2 (L )22 ffrespectively.Thus, for the quality factor Q, after substitution and simplifications, one can find:Q 2 Emax storedEtotallostU 02R 2 (L )22 C (C ) U 021R 2 R22 R (L )f2At resonance p 2 f p 1 R the quality factor for the practical parallel RLCL C L circuit will become:Qp L 1R C2With the values given in Figure 6 one can see that the quality factor of such a circuit will be Qp 16.11.The sharpness is a very important property because it depends upon how quickly the system loses thestored energy due to resistance in the circuit. When the curve is sharp or more narrowly peaked aroundthe resonance frequency, one can say that the circuit has higher selectivity. This means that, thepractical parallel RLC circuit can be used as a bandpass filter, a circuit which will let through signalswith frequencies very close to the resonant frequency, but rejecting frequencies very different (loweror even higher) to fp.The influence of the value of the resistance R on the sharpness is depicted in Figure 7.Acta Didactica Napocensia, ISSN 2065-1430

Teaching RLC parallel circuits in high-school Physics class41Figure 7. Influence of loss resistance on the quality factor of a practical RLC parallel circuitThe bandwidth (BW), the width of the resonant power curve at half maximum, depends upon Q:211 R fp fp2 L C L RBW Qp2 LL 12R CWith the values given in Figure 6 we will get BW 8 kHz 2.55 kHz .This means that, around the resonant frequency (fp 41.015 kHz), there will be a passing window forsuch a frequency range (from 39.74 kHz to 42.29 kHz), the signals being attenuated less with lessthan 3dB. The signal with frequencies outside this range will be drastically attenuated and not allowedto pass.5. The easy (or not so hard) way to deal with the practical RLC parallel circuitIf the treatment of the ideal RLC parallel circuit is still preferred, there is a compromise, a bridgetowards the practical RLC parallel circuit: the series to parallel transformation. This technique willgive the expressions of the elements of a parallel circuit as function of the elements of the series one.This subject can be treated easily with the complex number formalism, but such an approach is outsidethe high-school math curricula. Let us consider the two circuits depicted in Figure 8. Figure 8. Series to parallel equivalent circuitsVolume 8 Number 4, 2015

42Alpár SimonThe two circuit will be equivalent, if both their impedances and quality factors are equal.For the series circuit we have Q LRand Z R 2 L 2Rxrelationships will become Qx and Z x Lxand for the parallel one, theRx Lx , respectively.22Rx Lx 22Using the equivalence conditions Q Qx and Z Z x , respectively, for the elements of the parallelcircuitweform Rx willfind Rx R 1 Q 2 andLx L 1 Q2Q2orinamoredetailedR 2 ( L) 2R 2 ( L) 2and Lx RLThus, the practical RLC circuit, formed by the parallel connection of a series RL branch with acapacitor C, could be easily transfromed this way in a parallel RxLxC circuit. One can see that both Rxand Lx are frequency dependent. This is the, not so obvious, reason why at resonance the impedance isnot exactly maximum.6. Applications of the parallel RLC circuitGenerally speaking, resonant RLC circuits are used in a variety of configurations in communicationsystems for coupling, filtering, tuning, generation of harmonic oscillations etc.When listening to a radio, one should know that radio signals are broadcasted via electromagneticwaves through atmosphere towards the receving antennas, where small voltages are induced. From aquite wide range of frequencies only one or a narrow band must be extracted, transmitted to thereceiver and amplified in order to be listened by us. Figure 9 shows some typical arrangements forantenna coupling (a), tuned amplification (b) and frequency selection (c), respectively.Both coupling to theantenna and tuning tothe desired frequencyare realised via parallelresonant circuit formedby a real inductor and avariable capacitor.(a)(b)Figure 9. Tuning and selection with parallel RLC circuitsActa Didactica Napocensia, ISSN 2065-1430(c)

Teaching RLC parallel circuits in high-school Physics class43In TV receivers, the electronics m

Received December 2015. Volume 8, Number 4, 2015 TEACHING RLC PARALLEL CIRCUITS IN HIGH-SCHOOL PHYSICS CLASS Alpár Simon Abstract: This paper will try to give an alternative treatment of the subject "parallel RLC circuits" and "resonance in parallel RLC circuits" from the Physics curricula for the XIth grade from Romanian high-schools,

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