This study material provides an in-depth understanding of RLC circuits and their reactants. It explains the behavior of resistance, inductance, and capacitance in a circuit with alternating current. The material includes theory, equations, phasor diagrams, and experimental results.
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2RLC CIRCUITS REACTANTS RLC CIRCUITS REACTANTS Introduction This experiment aims at familiarity with the components mentioned and how they behave in a circuit when put together, either parallel or connected in series, and the circuit fed by an alternating current. Theory The three basic passive components that are of focus are the resistance, inductance, and capacitance which have distinct phase relationships among themselves when the circuitry is of alternating supply that is sinusoidal in nature. When the resistor is alone in the circuit, the waveform shall be in-phase with the current. When there is only an inductor the voltage wave form is leading the current by 90O. When the circuit contain only capacitance, the wave representing voltage shall be lagging the current by 90O. The reactive value produced by the components mentioned above is what gives the phase differences. When the circuit is purely resistive, only has resistance, the reactance is zero, it will be negative when the circuit is capacitive and positive then the circuit is purely inductive. The table below shows a relationship of the RLC circuit as described, only that this time there shall the impedance, which the table shows how it relates to the other components as well (talked about in the next section) Component Resisto r (R) Reactance (X) Impedance (Z) ResistorR0Z=R =R<0 Inductor0ωL ZL= jωL = ωL<+90 Capacitor01/ωC ZC=1/jωC =1/ωC<-90 The impedance of the components The three elements can be combined together and analysed, where they can be connected in series or parallel, to form an RLC circuit. When analysing the RLC circuit, we shall need to take into consideration the inductive, capacitive reactance and the resistances, XL, XC, and R respectively. This will enable us to obtain the reactance of the circuit. Series RLC circuit are in a class of 2ndorder circuits since they store energy in two elements which are the capacitance and inductance, see figure 1.
3RLC CIRCUITS REACTANTS Figure1: RLC circuit connected in series There only one loop in the circuit of instantaneous current that flows through the loop, that means the current flowing is the same for every component of the circuit. The inductive and capacitive reactance of the circuit depends on the frequency at the supply, therefore the response happening in sinusoidal way varies with the frequency. The voltage drops across individual each element will be out of phase with one another. i(t)=Imaxsin(ωt)……………………………………..(1) Voltage across the resistor is written as VRshall be in phase with the current Voltage across the inductor or inductive load, VLwill be leading the current by 90O The voltage across the capacitor or capacitive load shall lag the current by 90O We can infer that VLand VCare 180Oout of phase and opposite each other. Figure2:Circuits showing the waveforms and the circuit and how they relate Considering all the parameters in place and considering the circuit given therein, we can have the following equations: For inductive reactance we have:XL=2πfL=ωL
4RLC CIRCUITS REACTANTS Reactance of the capacitor is given by:XC=1 2πfC=1 ωC IfXL>XCit is an inductive circuit IfXC>XLit becomes a capacitive circuit Reactance total for the circuit¿XC−XL∨XL−XC Impedance total forthe circuit shall beZ=√R2+XT 2=R+jX The above equation of the inductive reactance, has a condition that if the inductance or frequency is made big then the value of the inductance reactance will increase. The inductive reactance against frequency phasor diagram is shown below, which is a linear curve and direct proportionality. On the other hand, the frequency of the capacitive loads and the reactance are inversely proportional. Consider the following phasor diagrams that relates with the voltages across the components of the RLC series circuit.
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5RLC CIRCUITS REACTANTS Figure3: Phasor diagram for a series RLC circuit, where computation of voltages is concerned. From the laws of Pythagoras theorem, we note the following, in relation to the voltages and the mathematical laws. VS 2=VR 2+¿ VS 2=√¿¿ In RLC series circuit, the current is the same in terms of amplitude across all the components, this makes the voltage to be as follows mathematically. VR=IRsinωt+0O=IR……………………………..(iii) VL=IXLsinωt+90O=I(jωL)……………………………..(iv) VC=IXCsinωt−90O=I(1 jωL)……………………………..(v) When these values get substituted, we get the following: That isVR=IR;VL=IXL;VC=IXC………………………………………………(vi) VS 2=√(IR)2+¿¿ VS 2=I√R2+¿¿ That meansVS=IZ;whereZisrepresentedby(viii)
6RLC CIRCUITS REACTANTS The angle θ is the angle in the triangle between VSand supply current, which is equivalent to the angle that lies between the impedance and the resistance, Z and R respectively. the voltage source determines the positivity and the negativity of the angle, where it might lead or lag. Consider the equations below. cos∅=R Z;sin∅=XL−XC Z;tan∅=XL−XC R…………………………..(ix) Apparatus The following apparatus shall be used for this experiment, Network and bridges 100kΩ resistor 47μF Capacitor 100mH inductor Connection wires AC source or a DDS function generator The Set UP
7RLC CIRCUITS REACTANTS Results and analysis Procedure Theoretical Part Here, the calculations were done to determine the values and compared to the measured values at 50Hz frequency and recorded at the “measured values” section. Calculations, shown below, were done using the formulas (iii), (v), and (iv), and answers obtained were shown in the “Calculated values” part. Taking current =3.23mA, frequency =50Hz, Resistor=100K, Capacitor=47μF, and Inductor=100mH VR=IRsinωt+0O=(3.23X10−3)(1000)=3.23V VC=IXCsinωt−90O=(1 jωL)= (3.23X10−3 2πX50X47X10−6)=0.218V VL=IXLsinωt+90O=I(jωL)=0.00323X2πX50X0.1=0.101V Voltages across components Measured values (V) Calculated values (V) Observation VR3.3483.23Small difference observed VC5.1230.218Big difference observed VL0.1780.101Small difference observed The circuit was formed by connecting the wires on the board to form the circuit below. The AC source is set and voltages of VR, VCand VLthen we can compare these values with the theoretical values. Take the readings and fill the table below; the AC source varies from 30Hz to 70Hz and an interval of 10Hz
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8RLC CIRCUITS REACTANTS Figure4:Series connected RLC circuit Frequency (Hz) VR (V) VC (V) VL (V) Impedance (Z) (X103Ω) Current (I) (X10-3A) 302.46.1040.1238.732.29 402.9385.6120.1567.062.83 503.3485.1230.1786.193.23 603.6594.6720.1975.643.54 703.8954.2670.215.293.78 The change in voltage drop per each circuit as frequency changes VRand VLare seen to be on the increase with the increase in frequency, direct proportion, VCand the impedance decreases with the increase in frequency, which is inverse proportionality Conclusion The arrangement of an RLC circuit when supplied with an ac current, behave the way we have seen and noted. From the real-time experiment conducted here, we have seen how the voltage responds with the change in frequency, because we know that the change in frequency affects the capacitance and inductance and therefore the impedance, which at a constant current is changed. We can infer the following as well: the voltage across the resistor and inductor is directly proportional to the frequency, and current, yet inversely proportional to the impedance, the voltage across the capacitor is inversely proportional to the frequency and impedance, yet directly proportional to the current, impedance is inversely proportional to the frequency, the current is directly proportional to the frequency all these happens at a given time. The errors can be avoided by using a good voltmeter and an ammeter. It is further true that the resistivity of a cable can be known and considered in the calculation, to achieve the true values of impedance and other parameters.