SUMMARY The goal of Chapter 35 has been to understand and apply basic techniques of AC circuit analysis. IMPORTANT CONCEPTS Basic circuit elements AC circuits are driven by an emf Element i and v Resistance/ reactance I and V Power Resistor Capacitor Inductor In phase i leads v by 90° i lags v by 90° R is fixed XC 5 1/vC XL 5 vL V 5 IR V 5 IXL V 5 IXC Vrms Irms 0 0 E 5 E0 cos vt that oscillates with angular frequency v 5 2pf. Phasors can be used to represent the oscillating emf, current, and voltage. The length of the phasor is the peak value E0. E0 For many purposes, especially calculating power, the root-mean-square (rms) quantities vt Vrms 5 V/ !2 E The horizontal projection is the instantaneous value E. Irms 5 I/ !2 Erms 5 E/ !2 are equivalent to the corresponding DC quantities. KEY SKILLS Phasor diagrams Kirchhoff’s laws VR • Start with a phasor (v or i) common to two or more circuit elements. Loop law The sum of the potential differences around a loop is zero. E0 I vt • The sum of instantaneous quantities is vector addition. • Use the Pythagorean theorem to relate peak quantities. Junction law The sum of currents entering a junction equals the sum leaving the junction. VC Instantaneous and peak quantities For an RC circuit, shown here, vR 1 vC 5 E Instantaneous quantities v and i generally obey different relationships than peak quantities V and I. VR2 1 VC2 5 E02 APPLICATIONS Series RLC circuits RC filter circuits R VC 5 E0 XC /"R2 1 XC2 C E R I 5 E0/Z where Z is the impedance VC VC S E0 as v S 0 E L Z 5 "R2 1 (XL 2 XC )2 VR 5 IR A low-pass filter transmits low frequencies and blocks high frequencies. C VR 5 E0 R/"R2 1 XC2 E R VR VR S E0 as v S ` A high-pass filter transmits high frequencies and blocks low frequencies. Copyright ©2004 Pearson Education, Inc., publishing as Addison Wesley VL 5 IXL VC 5 IXC C When v 5 v0 5 1/ !LC (the resonance frequency), the current in the circuit is a maximum Imax 5 E0/R. In general, the current i lags behind E by the phase angle f 5 tan21 ( (XL 2 XC )/R). The power supplied by the emf is Psource 5 IrmsErms cos f, where cos f is called the power factor. The power lost in a resistor is PR 5 IrmsVrms 5 (Irms )2 R.