Principles & Applications Electricity Principles & Applications Sixth Edition Richard J. Fowler Chapter 13 R, C, and L Circuits ©2003 Glencoe/McGraw-Hill

INTRODUCTION RC, RL, and RCL Circuits Resonance (fr) Quality (Q) Bandwidth (BW) Filters

Facts About R, C, and L Circuits Impedance (Z) is the combined opposition of R and X. The ohm is the base unit of impedance. Except at resonance, Z is always larger than R or X in a series circuit . Except at resonance, Z is always smaller than R plus X in a series circuit. Except at resonance, Z always causes phase shift.

More Facts About R, C, and L Circuits The current phasor is the reference for series circuits. For all impedance circuits, Z = VT / IT and fr = 1 / (6.28 Ö LC ). The voltage phasor is the reference for parallel circuits. A given L and C are resonant at only one frequency. At a given f, BW is inversely proportional to Q.

Graphical and Mathematical Addition of Phasors 3 4 5 C = A2 + B2 = 32 + 42 = 5 Two right angle phasors can be added graphically by completing the rectangle and bisecting it diagonally. The bisecting line is the phasor resulting from the addition. Mathematical addition can be done with this equation. Substituting for A and B, and solving the equation yields 5 which is the value of the resultant phasor (C).

Series RC Circuit VR VC VT IT V 50 V V 40 V V 30 V VR + VC does not equal VT because these voltages are out-of-phase. These voltages must be added using phasor addition. Solution: VT = Ö VR2 + VC2 = Ö 302 + 402 = Ö 2500 = 50 V

Series RC Impedance R XC Z XC = 60 W R = 80 W V 150 V A 1.5 A Z does not equal R + XC because R and XC are 90° out-of-phase. Determine Z using I and V. Z = V / I = 150 V / 1.5 A = 100 W Or, using phasor addition: Z = Ö R2 + XC2 = Ö 802 + 602 = Ö 10000 = 100 W

Parallel RC Circuit A 0.5 A IR IC IT VT A 0.4 A A 0.3 A IR + IC does not equal IT because these currents are out-of-phase. These currents must be added using phasor addition. Solution: IT = Ö IR2 + IC2 = Ö 0.42 + 0.32 = Ö 0.25 = 0.5 A

RC-Circuits Quiz VT = ____ when VR = 20 V and VC = 15 V in a series circuit. 25 V IT ____ VT in a series RC circuit. leads Z = ____ when R = 400 W and XC = 300 W in a series circuit. 500 W In a parallel RC circuit, IT ____ VT. leads When IR = 0.5A, IC = 0.8A, and VT = 30V, Z of a parallel RC circuit will be ____ W. 33.7

Series RL Circuit VR VL VT IT V 100 V V 80 V V 60 V VR + VL does not equal VT because these voltages are out-of-phase. These voltages must be added using phasor addition. Solution: VT = Ö VR2 + VL2 = Ö 602 + 802 = Ö10000 = 100 V

Series RL Impedance R XL Z XL = 60 W R = 80 W V 150 V A 1.5 A Z does not equal R + XL because R and XL are 90° out-of-phase. Determine Z using I and V. Z = V / I = 150 V / 1.5 A = 100 W Or, using phasor addition: Z = Ö R2 + XL2 = Ö 802 + 602 = Ö 10000 = 100 W

Parallel RL Circuit A 1.5 A IR IL IT VT A 0.9 A A 1.2 A IR + IL does not equal IT because these currents are out-of-phase. These currents must be added using phasor addition. Solution: IT = Ö IR2 + IL2 = Ö 0.92 + 1.22 = Ö 2.25 = 1.5 A

RL-Circuits Quiz VT = ____ when VR = 15 V and VL = 12 V 19.2 V in a series circuit. 19.2 V In a series RL circuit, IT ____ VT . lags In a series circuit, Z = ____ when R = 600 W and XL = 800 W. 1000 W In a parallel RL circuit, IT ____ VT. lags Z of a parallel RL circuit will be ____ W when IR = 0.7A, IL = 0.4A, and VT = 25 V. 38.5

Series RCL Circuits L R C V 15 V VR VL VT VC VX V 38 V V 26 V V 9 V Notice that either (or both) VC and VL can be greater than VT . Also, VT must be greater than VX or VR. VL + VC + VR does not equal VT because these voltages are out-of-phase. VT can be calculated using phasor addition: VT = Ö VR2 + (VL - VC)2 = Ö 92 + 122 = Ö 225 = 15 V

Series RCL Impedance A 0.6 A R XL Z XC X XL = 50 W R = 15 W XC = 30 W V 15 V Notice that either (or both) XC and XL can be greater than Z. Also, Z must be greater than X or R. Z does not equal R +XC + XL because R and X are 90° out-of-phase. Determine Z using I and V. Z = V / I = 15 V / 0.6A = 25 W Or, using phasor addition: Z = Ö R2 + (XL -XC)2 = Ö 152 + 202 = Ö 625 = 25 W

Parallel RCL Circuits VT IR IC IT IL IX A 0.89 A A 0.6 A A 0.8 A A Notice that either (or both) IC and IL can be greater than IT . Also, IT must be greater than IX or IR . IL + IC + IR does not equal IT because these currents are out-of-phase. IT can be calculated using phasor addition: IT = Ö IR2 + (IC - IL)2 = Ö 0.82 + 0.42 = Ö 0.8 = 0.894 A

RCL-Circuit Quiz The current will ____ the voltage in a series RCL circuit in which R = 20 W, XC = 45 W, and XL = 65 W. lag The current will ____ the voltage in a parallel RCL circuit in which R = 30 W, XC = 50 W, and XL = 75 W. lead In a series RCL circuit, VT is always ____ than VX . greater In a series RCL circuit, if XL is ____ than XC , then I leads V. less In a parallel RCL circuit, the ____ current can be less than the ____ current or the _____ current. total capacitor inductor Z = ____ for a parallel RCL circuit in which R = 20 W, XC = 30 W, and XL = 60 W. 19 W

Series Resonance Demonstration 100 W light bulb Wooden wedge for adjusting the air gap Variable inductor ( 1 to 4 H ) with multiple taps 6 μF ac capacitor The next slide shows the circuit as it goes from capacitive, to resonant, to inductive. Watch the brilliancy of the bulb.

Click on the image to run the display. Notice how rapidly L changes as the air gap approaches zero. To rerun the display, click again on the image.

Resonant Circuits XC3 fr Reactance XL3 XL1 fr2 XL2 XC1 XC2 Frequency Resonance occurs when XL equals XC. There is only one resonant frequency for each LC combination. However, an infinite number of LC combinations have the same fr. L1 and C3 are resonant at fr2.

Effects of Q on the BW of Resonant Circuits Frequency Impedance or Current 0.707 BW 0.707 BW Response curve and BW for a high-Q circuit. The curve plots I for series circuits and Z for parallel circuits. A low-Q circuit has a flatter curve and a wider bandwidth. A high-Q produces a narrow BW and good selectivity.

Low-Pass Filters XC decreases as f increases. Frequency Voutput Multiple- frequency input Output voltage XC decreases as f increases. Thus, less V develops across C as f increases. Multiple- frequency input Output voltage Frequency Voutput XL increases as f increases. Thus, more V develops across L as f increases.

High-Pass Filters XC decreases as f increases. Multiple- frequency input Output voltage Frequency Voutput XC decreases as f increases. Thus, less V develops across C as f increases. Multiple- frequency input Output voltage Frequency Voutput XL increases as f increases. Thus, more V develops across L as f increases.

Resonance and Filters Quiz A given value of inductance and capacitance produces ____ value(s) of resonant frequency(ies). one A circuit is ____ when the inductive reactance is equal to the capacitive reactance. resonant The ____ the Q , the wider the BW of the circuit. lower The ____ the bandwidth, the more selective the circuit. narrower An RC filter with the output taken across the capacitor is a ____ pass filter. low- An RL filter with the output taken across the inductor is a ____ pass filter. high-

REVIEW RC, RL, and RCL Circuits Resonance Quality Bandwidth Filters