2.
Objectives• Describe the relationship between currentand voltage in an RL circuit• Determine impedance and phase angle in aseries RL circuit• Analyze a series RL circuit• Determine impedance and phase angle in aparallel RL circuit
3.
Objectives• Analyze a parallel RL circuit• Analyze series-parallel RL circuits• Determine power in RL circuits
4.
Sinusoidal Response of RLCircuits• The inductor voltage leads the source voltage• Inductance causes a phase shift between voltageand current that depends on the relative values ofthe resistance and the inductive reactance
5.
Impedance and Phase Angle ofSeries RL Circuits• Impedance of any RL circuit is the total oppositionto sinusoidal current and its unit is the ohm• The phase angle is the phase difference betweenthe total current and the source voltage• The impedance of a series RL circuit isdetermined by the resistance (R) and the inductivereactance (XL)
6.
The Impedance Triangle• In ac analysis, both R and XL are treated a phasorquantities, with XL appearing at a +90° angle with respectto R∀ θ is the phase angle between applied voltage and current
7.
The Impedance Triangle• The impedance magnitude of the series RLcircuit in terms of resistance and reactance:Z = √R2+ X2L– The magnitude of the impedance (Z) isexpressed in ohms• The phase angle is:θ = tan-1(XL/R)
8.
Ohm’s Law• Application of Ohm’s Law to series RLcircuits involves the use of the phasorquantities Z, V, and IV = IZI = V/ZZ = V/I
9.
Relationships of the Current andVoltages in a Series RL Circuit• Resistor voltage is inphase with the current• Inductor voltage leadsthe current by 90°• There is a phasedifference of 90°between the resistorvoltage, VR, and theinductor voltage, VL
10.
Kirchhoff’s Voltage Law• From KVL, the sum of the voltage drops mustequal the applied voltage• The magnitude of the source voltage is:Vs = √V2R + V2L• The phase angle between resistor voltage andsource voltage is:θ = tan-1(VL/VR)
11.
Variation of Impedance andPhase Angle with Frequency• Inductive reactance varies directly with frequency• Z is directly dependent on frequency• Phase angle θ also varies directly with frequency
12.
Impedance and Phase Angle ofParallel RL Circuits• The magnitude of the total impedance of atwo-component parallel RL circuit is:Z = RXL/ √R2+ X2L• The phase angle between the appliedvoltage and the total current is:θ = tan-1(R/XL)
13.
Conductance (G), Susceptance (B),and Admittance (Y)• Conductance is: G = 1/R• Inductive Susceptance is: BL = 1/XL• Admittance is: Y = 1/Z• Total admittance is the phasor sum of conductanceand the inductive susceptance:Y = √G2+ B2LThe unit for G, BL an Y is siemens (S)
14.
Analysis of Parallel RL Circuits• The total current, Itot , divides at the junction intothe two branch currents, IR and IL• Vs, VR , and VL are all in phase and of the samemagnitude
15.
Kirchhoff’s Current Law• The current through the inductor lags the voltageand the resistor current by 90°• By Kirchhoff’s Current Law, the total current isthe phasor sum of the two branch currents:Itot = √I2R + I2L• Phase angle: θ = tan-1(IL/IR)
16.
Series Parallel RL Circuits• A first approach to analyzing circuits withcombinations of both series and parallel Rand L elements is to:– Find the series equivalent resistance (R(eq)) andinductive reactance (XL(eq)) for the parallelportion of the circuit– Add the resistances to get the total resistanceand add the reactances to get the total reactance– Determine the total impedance
17.
Series Parallel RL Circuits• A second approach to analyzing circuitswith combinations of both series andparallel R and L elements is to:– Calculate the magnitudes of inductive reactance(XL)– Determine the impedance of each branch– Calculate each branch current in polar form– Use Ohm’s law to get element voltages
18.
Power in RL Circuits• When there is both resistance and inductance,some of the energy is alternately stored andreturned by the inductance and some is dissipatedby the resistance• The amount of energy converted to heat isdetermined by the relative values of the resistanceand the inductive reactance• The Power in the inductor is reactive power:Pr = I2XL
19.
Power Triangle for RL Circuits• The apparent power (Pa) is the resultant of theaverage power (Ptrue) and the reactive power (PR)• Recall Power Factor: PF = cos θ
20.
Significance of the Power Factor• Many practical loads have inductance as aresult of their particular function, and it isessential for their proper operation• Examples are: transformers, electric motorsand speakers• A higher power factor is an advantage indelivering power more efficiently to a load
21.
Power Factor Correction• Power factor of an inductive load can beincreased by the addition of a capacitor inparallel– The capacitor compensates for the the phase lagof the total current by creating a capacitivecomponent of current that is 180° out of phasewith the inductive component– This has a canceling effect and reduces thephase angle (and power factor) as well as thetotal current, as illustrated on next slide
23.
RL Circuit as a Low-Pass Filter• An inductor acts as a short to dc• As the frequency is increased, so does theinductive reactance– As inductive reactance increases, the outputvoltage across the resistor decreases– A series RL circuit, where output is takenacross the resistor, finds application as a low-pass filter
24.
RL Circuit as a High-Pass Filter• For the case when output voltage ismeasured across the inductor– At dc, the inductor acts a short, so the outputvoltage is zero– As frequency increases, so does inductivereactance, resulting in more voltage beingdropped across the inductor– The result is a high-pass filter
25.
Summary• When a sinusoidal voltage is applied to an RLcircuit, the current and all the voltage drops arealso sine waves• Total current in an RL circuit always lags thesource voltage• The resistor voltage is always in phase with thecurrent• In an ideal inductor, the voltage always leads thecurrent by 90°
26.
Summary• In an RL circuit, the impedance is determined byboth the resistance and the inductive reactancecombined• Impedance is expressed in units of ohms• The impedance of an RL circuit varies directlywith frequency• The phase angle (θ) if a series RL circuit variesdirectly with frequency
27.
Summary• You can determine the impedance of a circuit bymeasuring the source voltage and the total currentand then applying Ohm’s law• In an RL circuit, part of the power is resistive andpart reactive• The phasor combination of resistive power andreactive power is called apparent power• The power factor indicates how much of theapparent power is true power
28.
Summary• A power factor of 1 indicates a purely resistivecircuit, and a power factor of 0 indicates a purelyreactive circuit• In an RL lag network, the output voltage lags theinput voltage in phase• In an RL lead network, the output voltage leadsthe input voltage in phase• A filter passes certain frequencies and rejectsothers
Be the first to comment