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Circuits and Analog Electronics Ch4 Sinusoidal Steady State Analysis 4.1 Characteristics of Sinusoidal 4.2 Phasors 4.3 Phasor Relationships for R, L and C 4.4 Impedance 4.5 Parallel and Series Resonance 4.6 Examples for Sinusoidal Circuits Analysis 4.7 Magnetically Coupled Circuits References: Hayt-Ch7; Gao-Ch3; Ch4 Sinusoidal Steady State Analysis • Any steady state voltage or current in a linear circuit with a sinusoidal source is a sinusoid – All steady state voltages and currents have the same frequency as the source • In order to find a steady state voltage or current, all we need to know is its magnitude and its phase relative to the source (we already know its frequency) • We do not have to find this differential equation from the circuit, nor do we have to solve it • Instead, we use the concepts of phasors and complex impedances • Phasors and complex impedances convert problems involving differential equations into circuit analysis problems Focus on steady state; Focus on sinusoids. Ch4 Sinusoidal Steady State Analysis 4.1 Characteristics of Sinusoidal Key Words: Period: T , Frequency: f , Radian frequency Phase angle Amplitude: Vm Im Ch4 Sinusoidal Steady State Analysis 4.1 Characteristics of Sinusoidal vt Vm sin t I1 I1 I1 I1 I1 I1 + U1 - U R1 5 R1 5 i + v、i R _ IS E I1 I 0 I1 I1 I1 I I1 t t2 R1 5 R1 5 i - + U1 - U t1 IS + R E I1 Both the polarity and magnitude of voltage are changing. Ch4 Sinusoidal Steady State Analysis 4.1 Characteristics of Sinusoidal Period: T — Time necessary to go through one cycle. (s) Frequency: f — Cycles per second. (Hz) f = 1/T Radian frequency(Angular frequency): = 2f = 2/T (rad/s) Amplitude: Vm Im i = Imsint, v =Vmsint v、i Vm、Im 0 2 t Ch4 Sinusoidal Steady State Analysis 4.1 Characteristics of Sinusoidal Effective Roof Mean Square (RMS) Value of a Periodic Waveform — is equal to the value of the direct current which is flowing through an R-ohm resistor. It delivers the same average power to the resistor as the periodic current does. 1 T T 0 i 2 Rdt I 2 R Effective Value of a Periodic Waveform I eff I eff 1 T T 0 I m2 T I sin tdt 2 m 2 Veff 1 T T 0 T 0 1 T 1 cos 2 t dt 2 v 2dt Vm 2 T 0 i 2dt 1 2 T I Im m T 2 2 Ch4 Sinusoidal Steady State Analysis 4.1 Characteristics of Sinusoidal Phase (angle) i I m sin t Phase angle 08 6 4 2 i0 I m sin 0 -2 0 <0 -4 -6 -8 0.01 0.02 0.03 0.04 0.05 Ch4 Sinusoidal Steady State Analysis 4.1 Characteristics of Sinusoidal Phase difference i I m sin( t 2 ) v Vm sin( t 1 ) v i t 1 (t 2 ) 1 2 1 2 0 — v(t) leads i(t) by (1 - 2), or i(t) lags v(t) by (1 - 2) 1 2 0 — v(t) lags i(t) by (2 - 1), or i(t) leads v(t) by (2 - 1) 1 2 0 1 2 v、i In phase. v、i v 1 2 2 Out of phase。 v、i v v i i i t t t Ch4 Sinusoidal Steady State Analysis 4.1 Characteristics of Sinusoidal Review The sinusoidal means whose phases are compared must: ① Be written as sine waves or cosine waves. ② With positive amplitudes. ③ Have the same frequency. 360°—— does not change anything. 90° —— change between sin & cos. 180°—— change between + & 2 sin cos cos 3 2 cos sin 2 Ch4 Sinusoidal Steady State Analysis 4.1 Characteristics of Sinusoidal Phase difference P4.1, v1 220 2 sin 314t 30 Find v2 220 2 cos314t 30 ? v2 220 2 cos314t 30 220 2 sin 314t 30 90 220 2 sin 314t 120 1 2 30 120 150 v2 220 2 cos314t 30 220 2 cos314t 30 180 220 2 cos 360 314t 210 2 sin 314t 60 220 2 sin 314t 150 90 220 1 2 30 60 30 Ch4 Sinusoidal Steady State Analysis 4.1 Characteristics of Sinusoidal Phase difference Vm sin t 3 P4.2, v、i v i • -/3 • /3 • I m sin t 3 t Ch4 Sinusoidal Steady State Analysis 4.2 Phasors A sinusoidal voltage/current at a given frequency , is characterized by only two parameters :amplitude an phase Key Words: Complex Numbers Rotating Vector Phasors Ch4 Sinusoidal Steady State Analysis 4.2 Phasors E.g. voltage response v t Vm cos t Re v t Time domain Complex form: v t Vm e Angular frequency ω is known in the circuit. Phasor form: j t Frequency domain A sinusoidal v/i Complex transform Phasor transform By knowing angular frequency ω rads/s. Ch4 Sinusoidal Steady State Analysis 4.2 Phasors Rotating Vector i1 I m1 sint 1 i 2 I m2 sint 2 i i1 i2 I m sint y i i t Im Im x i(t1) A complex coordinates number: I m e j t t1 t I m cos t jI m sin t Real value: i t I m sin t I max I m e j t Ch4 Sinusoidal Steady State Analysis 4.2 Phasors Rotating Vector y v Vm sin( t ) Vm 0 x Ch4 Sinusoidal Steady State Analysis 4.2 Phasors Complex Numbers A a jb — Rectangular Coordinates imaginary axis A A cos j sin b A A e j— Polar Coordinates real axis a conversion: A a jb A A e j A e j a jb e j 90 cos 90 j sin 90 0 j j A a2 b2 arctg a A cos b A sin b a Ch4 Sinusoidal Steady State Analysis 4.2 Phasors Complex Numbers Arithmetic With Complex Numbers Addition: A = a + jb, B = c + jd, A + B = (a + c) + j(b + d) Imaginary Axis A+B B A Real Axis Ch4 Sinusoidal Steady State Analysis 4.2 Phasors Complex Numbers Arithmetic With Complex Numbers Subtraction : A = a + jb, B = c + jd, A - B = (a - c) + j(b - d) Imaginary Axis B A A-B Real Axis Ch4 Sinusoidal Steady State Analysis 4.2 Phasors Complex Numbers Arithmetic With Complex Numbers Multiplication : A = Am A, B = Bm B A B = (Am Bm) (A B) Division: A = Am A , B = Bm B A / B = (Am / Bm) (A B) P4.3, sint 60 sint 30 i1 I m1 sint 1 100sin t 45 i2 I m2 2 Find:i i1 i2 Ch4 Sinusoidal Steady State Analysis 4.2 Phasors Phasors A phasor is a complex number that represents the magnitude and phase of a sinusoid: im cost I I m Phasor Diagrams • A phasor diagram is just a graph of several phasors on the complex plane (using real and imaginary axes). • A phasor diagram helps to visualize the relationships between currents and voltages. Ch4 Sinusoidal Steady State Analysis 4.2 Phasors Complex Exponentials A A e j Ae jt A e j ( t ) A cos(t ) j A sin( t ) Re{ Ae jt } | A | cos(t ) A real-valued sinusoid is the real part of a complex exponential. Complex exponentials make solving for AC steady state an algebraic problem. Ch4 Sinusoidal Steady State Analysis 4.3 Phasor Relationships for R, L and C Key Words: I-V Relationship for R, L and C, Power conversion Ch4 Sinusoidal Steady State Analysis 4.3 Phasor Relationships for R, L and C Resistor v~i relationship for a resistor + S i Suppose R v _ v Vm sin t v Vm i sin t I m sin t R R V Relationship between RMS: I R v、i v Wave and Phasor diagrams: i t V I R I V Ch4 Sinusoidal Steady State Analysis 4.3 Phasor Relationships for R, L and C Resistor Time domain frequency domain With a resistor θ﹦φ, v(t) and i(t) are in phase . Ch4 Sinusoidal Steady State Analysis 4.3 Phasor Relationships for R, L and C Resistor Power i + • Transient Power R 2 p vi Vm sin t I m sin t I mVm sin t v _ I mVm 1 cos 2t IV IV cos 2t 2 p0 • Average Power v、i v i P=IV t 1 P T 1 T 0 pdt T 0VI 1 cos 2t dt VI T V2 P IV I R R 2 Ch4 Sinusoidal Steady State Analysis 4.3 Phasor Relationships for R, L and C Resistor P4.4 , v 311sin 314t , R=10,Find i and P。 Vm 311 V 220V 2 2 I V 220 22 A R 10 i 22 2 sin 314t P IV 220 22 4840W Ch4 Sinusoidal Steady State Analysis 4.3 Phasor Relationships for R, L and C Inductor v~i relationship v v AB di L dt Suppose i I m sin t vL di d I m sin t L I mL cos t dt dt I mL sin t 90 Vm sin t 90 1 t 1 0 1 t 1 t i vdt L vdt L 0 vdt i0 0 vdt L L Ch4 Sinusoidal Steady State Analysis 4.3 Phasor Relationships for R, L and C Inductor v~i relationship di v L I mL sin t 90 Vm sin t 90 dt Vm I mL Relationship between RMS: V IL V X L L 2fL I L XL f For DC,f = 0,XL = 0. v(t) leads i(t) by 90º, or i(t) lags v(t) by 90º Ch4 Sinusoidal Steady State Analysis 4.3 Phasor Relationships for R, L and C Inductor v ~ i relationship i(t) = Im ejt di I m jLe jt jLi(t ) dt Represent v(t) and i(t) as phasors: V jLI I V V jL jX L • The derivative in the relationship between v(t) and i(t) becomes a multiplication by j in the relationship between V and I. • The time-domain differential equation has become the algebraic equation in the frequency-domain. • Phasors allow us to express current-voltage relationships for inductors and capacitors in a way such as we express the current-voltage relationship for a resistor. v (t ) L Ch4 Sinusoidal Steady State Analysis 4.3 Phasor Relationships for R, L and C Inductor v ~ i relationship Wave and Phasor diagrams: V jIX L v、i v i V eL t I Ch4 Sinusoidal Steady State Analysis 4.3 Phasor Relationships for R, L and C Inductor Power p vi Vm sin t 90 I m sin t Vm I m cos t sin t Vm I m sin 2t VI sin 2t 2 t i 1 Energy stored:W vidt Lidi Li 2 0 0 2 1 Wmax LI m2 LI 2 2 Average Power P 1 T T 0 1 T pdt VI sin 2tdt 0 T 0 V2 Reactive Power Q IV I X L XL 2 (Var) P + + - - t v、i v i t Ch4 Sinusoidal Steady State Analysis 4.3 Phasor Relationships for R, L and C Inductor P4.5,L = 10mH,v = 100sint,Find iL when f = 50Hz and 50kHz. X L 2fL 2 50 10 10 3 3.14 I 50 V 100 / 2 22.5 A XL 3.14 iL t 22.5 2 sin t 90 A X L 2fL 2 50 10 3 10 10 3 3140 V 100 / 2 I 50k 22.5mA XL 3.14 iL t 22.5 2 sin t 90 mA Ch4 Sinusoidal Steady State Analysis 4.3 Phasor Relationships for R, L and C Capacitor + v _ v ~ i relationship dq dv i C dt dt i Suppose: v Vm sin t C I m CVm i CVm cos t CVm sin t 90 I m sin t 90 1 t 1 0 1 t 1 t v idt idt idt v0 idt c c c 0 c 0 Relationship between RMS: I CV V V 1 XC C 1 1 XC C 2fC 1 XC For DC,f = 0, XC f i(t) leads v(t) by 90º, or v(t) lags i(t) by 90º Ch4 Sinusoidal Steady State Analysis 4.3 Phasor Relationships for R, L and C Capacitor + v _ v ~ i relationship v(t) = Vm ejt i C dv(t ) dVme j t i (t ) C C jCVme j t dt dt V Represent v(t) and i(t) as phasors: I = jωCV = jX C • The derivative in the relationship between v(t) and i(t) becomes a multiplication by j in the relationship between V and I. • The time-domain differential equation has become the algebraic equation in the frequency-domain. • Phasors allow us to express current-voltage relationships for inductors and capacitors much like we express the current-voltage relationship for a resistor. Ch4 Sinusoidal Steady State Analysis 4.3 Phasor Relationships for R, L and C Capacitor v ~ i relationship Wave and Phasor diagrams: V jI X C v、i I i v t V Ch4 Sinusoidal Steady State Analysis 4.3 Phasor Relationships for R, L and C Capacitor Power p vi Vm sin t I m sin t 90 Energy stored: Vm I m sin 2t VI sin 2t 2 P v dv 1 W vidt v C dt Cvdv Cv 2 0 0 0 dt 2 1 Wmax CVm2 CV 2 2 t v + + - - t v、i Average Power: P=0 i 2 V Reactive Power Q IV I X C (Var) XC 2 v t Ch4 Sinusoidal Steady State Analysis 4.3 Phasor Relationships for R, L and C Capacitor P4.7,Suppose C=20F,AC source v=100sint,Find XC and I for f = 50Hz, 50kHz。 f 50Hz X c I V V m 1.38A Xc 2Xc f 50KHz X c I 1 1 159 C 2fC V Xc 1 1 0.159 C 2fC Vm 1380A 2Xc Ch4 Sinusoidal Steady State Analysis 4.3 Phasor Relationships for R, L and C Review (v-I relationship) Time domain R L C v Ri Frequency domain V R I , v and i are in phase. vL L di dt V jL I vC C dv dt 1 1 V I , XC , v lags i by 90°. jC C , X L L , v leads i by 90°. Ch4 Sinusoidal Steady State Analysis 4.3 Phasor Relationships for R, L and C Summary R: XR R 0 L: X L L 2fL f v i C: XC 1 1 1 c 2fc f 2 v i 2 V IX Frequency characteristics of an Ideal Inductor and Capacitor: A capacitor is an open circuit to DC currents; A Inducter is a short circuit to DC currents. Ch4 Sinusoidal Steady State Analysis 4.4 Impedance Key Words: complex currents and voltages. Impedance Phasor Diagrams Ch4 Sinusoidal Steady State Analysis 4.4 Impedance Complex voltage, Complex current, Complex Impedance • AC steady-state analysis using phasors allows us to express the relationship between current and voltage using a formula that looks likes Ohm’s law: Z is called impedance. V IZ measured in ohms () V Vm e jv Vm v I I m e ji I mi V Vm j (v i ) Z e Z e j Z I I m Ch4 Sinusoidal Steady State Analysis 4.4 Impedance Complex Impedance V Vm j (v i ) Z e Z e j Z I I m • Complex impedance describes the relationship between the voltage across an element (expressed as a phasor) and the current through the element (expressed as a phasor) • Impedance is a complex number and is not a phasor (why?). • Impedance depends on frequency Ch4 Sinusoidal Steady State Analysis 4.4 Impedance Complex Impedance Resistor——The impedance is R ZR = R = 0( = 0); or ZR = R 0 Capacitor——The impedance is 1/jC 1 j2 j Zc e jx c C C =-/2 ( v i ) 2 or ZC 1 90 C Inductor——The impedance is jL Z L Le j 2 =/2 jL jxL ( v i 2 or Z L L90 ) Ch4 Sinusoidal Steady State Analysis 4.4 Impedance Complex Impedance I1 US I1 I1 I1 I1 Impedance in series/parallel can be combined as resistors. I + I1 I1 Z1 Z2 I1 U U _ + U1 - Zn R1 5 R1 5 + IS n Z Z 1 Z 2 ... Z n Z k k 1 Voltage divider: Zi Vi V n Zk k 1 U Z1 Zn Z2 _ US _ I + n 1 1 1 1 1 ... Z Z1 Z 2 Z n k 1 Z k Current divider: I1 I Z2 Z1 Z 2 I2 I Z1 Z1 Z 2 Ch4 Sinusoidal Steady State Analysis 4.4 Impedance Complex Impedance P4.8, + Z1 I1 Z2 V _ I Z I I Z 2 1 Z Z2 V V Z Z 2 I 1 1 ZZ1 Z 2 Z1 ZZ 2 1 1 Z1 Z2 Z Z V 2 I ZZ1 Z 2 Z1 ZZ 2 Ch4 Sinusoidal Steady State Analysis 4.4 Impedance Complex Impedance Phasors and complex impedance allow us to use Ohm’s law with complex numbers to compute current from voltage and voltage from current P4.9 10V 0 + - 20k 1F + - VC = 377 Find VC • How do we find VC? • First compute impedances for resistor and capacitor: ZR = 20k = 20k 0 ZC = 1/j (377 *1F) = 2.65k -90 Ch4 Sinusoidal Steady State Analysis 4.4 Impedance Complex Impedance P4.9 + 10V 0 - 20k 0 20k + 1F = 377 VC Find VC - Now use the voltage divider to find VC: 2.65k 90 VC 10V0 ( ) 2.65k 90 20k0 10V 0 + - + VC - 2.65k -90 2.65 90 20.17 7.54 1.31V 82.46 VC 10V 0 Ch4 Sinusoidal Steady State Analysis 4.4 Impedance Complex Impedance Impedance allows us to use the same solution techniques for AC steady state as we use for DC steady state. • All the analysis techniques we have learned for the linear circuits are applicable to compute phasors – KCL & KVL – node analysis / loop analysis – superposition – Thevenin equivalents / Norton equivalents – source exchange • The only difference is that now complex numbers are used. Ch4 Sinusoidal Steady State Analysis 4.4 Impedance Kirchhoff’s Laws KCL and KVL hold as well in phasor domain. n KCL: ik 0 k 1 n Ik ik- Transient current of the #k branch 0 k 1 n KVL: v k 1 k n V k 1 k 0 0 vk- Transient voltage of the #k branch Ch4 Sinusoidal Steady State Analysis 4.4 Impedance Admittance • I = YV, Y is called admittance, the reciprocal of impedance, measured in siemens (S) • Resistor: – The admittance is 1/R • Inductor: – The admittance is 1/jL • Capacitor: – The admittance is j C Ch4 Sinusoidal Steady State Analysis 4.4 Impedance Phasor Diagrams • A phasor diagram is just a graph of several phasors on the complex plane (using real and imaginary axes). • A phasor diagram helps to visualize the relationships between currents and voltages. I = 2mA 40, VR = 2V 40 VC = 5.31V -50, V = 5.67V -29.37 2mA 40 + 1F V 1k – + Imaginary Axis VC – + – VR VR Real Axis V VC Ch4 Sinusoidal Steady State Analysis 4.5 Parallel and Series Resonance Key Words: RLC Circuit, Series Resonance Parallel Resonance Ch4 Sinusoidal Steady State Analysis 4.5 Parallel and Series Resonance Series RLC Circuit (2nd Order RLC Circuit ) v vR vL vC vR v Phasor vL V VR2 (VL VC ) 2 vC ( IR ) 2 ( IX L IX C ) 2 VL I R 2 ( X L X C )2 V I VC V VR VL VC VR I R2 X 2 IZ Z R X 2 2 ( X X L X C) 1 2 R (L ) c 2 Ch4 Sinusoidal Steady State Analysis 4.5 Parallel and Series Resonance Series RLC Circuit (2nd Order RLC Circuit ) V VR2 (VL VC )2 IZ Z X = XL-XC 2 R 2 (L Phase difference: VX VL VC = arctg XL VR XL>XC >0,v leads i by ——Inductance Circuit XL<XC XL=XC 1 2 ) c VL - VC = arctg VR R V Z R X 2 <0,v lags i by ——Capacitance Circuit =0,v and i in phase——Resistors Circuit XC R Ch4 Sinusoidal Steady State Analysis 4.5 Parallel and Series Resonance Series RLC Circuit (2nd Order RLC Circuit ) vR V VR VL VC IR jIX L jIX C I( R j( X L X C )] I( R jX ) IZ v vL vC V Z R j( X L X C ) I Z R jX Z Z R2 ( X L X C )2 arctg X L XC R v i Ch4 Sinusoidal Steady State Analysis 4.5 Parallel and Series Resonance Series RLC Circuit (2nd Order RLC Circuit ) P4.9, R. L. C Series Circuit,R = 30,L = 127mH,C = 40F,Source v 220 2 sin( 314t 20o ) , Find 1) XL、XC、Z;2) I and i;3) VR and vR; VL and vL; VC and vC; 4) Phasor Diagrams vR v vL vC P4.10,Computing I by (complex numbers) Phasors Ch4 Sinusoidal Steady State Analysis 4.5 Parallel and Series Resonance Series Resonance (2nd Order RLC Circuit ) V VR VL VC IR jIX L jIX C VL VC X L XC arctg arctg VR R 1 When X L X C , L VL VC C Resonance condition 0 1 1 or f 0 LC 2 LC Resonance frequency VR V and 0 ——Series Resonance VL X X L 2 fL VR V I VC XC f0 1 2 f C f Ch4 Sinusoidal Steady State Analysis 4.5 Parallel and Series Resonance Series Resonance (2nd Order RLC Circuit ) Resonance condition: • X L XC ( 1 L) C Z 0 R 2 ( X L X C )2 R I 0 VL VC V V Z0 R Zmin;when V=constant, I=Imax=I0。 •When X L X C R , I 0 X L I 0 X C I 0 R •Quality factor Q, Q VL VC X L X C V V R R VL VC V Ch4 Sinusoidal Steady State Analysis 4.5 Parallel and Series Resonance Series Resonance (2nd Order RLC Circuit ) Ch4 Sinusoidal Steady State Analysis 4.5 Parallel and Series Resonance Series Resonance (2nd Order RLC Circuit ) Ch4 Sinusoidal Steady State Analysis 4.5 Parallel and Series Resonance Series Resonance (2nd Order RLC Circuit ) Ch4 Sinusoidal Steady State Analysis 4.5 Parallel and Series Resonance Series Resonance (2nd Order RLC Circuit ) Ch4 Sinusoidal Steady State Analysis 4.5 Parallel and Series Resonance Series Resonance (2nd Order RLC Circuit ) Ch4 Sinusoidal Steady State Analysis 4.5 Parallel and Series Resonance Series Resonance (2nd Order RLC Circuit ) Ch4 Sinusoidal Steady State Analysis 4.5 Parallel and Series Resonance Parallel RLC Circuit I V IL IC 1 1 1 jC R jL j / C R jL R jL jC R jL R jL R L 2 j ( C ) 2 2 2 2 2 R L R L L R ) 0 , When (C 2 Y0 2 R 2 L2 R 2 L2 Y V In phase with I Parallel Resonance Parallel Resonance frequency In generally R X L Zmax Imin: 0 0 1 1 LC CR 2 1 L ( f0 1 ) 2 LC LC R R R RC I I 0 VY0 V 2 V V V 1 2 L R 02 L2 L 2 2 R L R LC C Ch4 Sinusoidal Steady State Analysis 4.5 Parallel and Series Resonance Parallel RLC Circuit IL V I V IL IC 1 V C j j V R j0 L 0 L L C IC j0CV j V L | IL || IC || I0 | 0 •Quality factor Q, I C I L YL YC Q I 0 I 0 Y0 Y0 IL jQI0 IC jQI0 Q 0 L R 1 0 RC Z . Ch4 Sinusoidal Steady State Analysis 4.5 Parallel and Series Resonance Parallel RLC Circuit R1 3, X L 4, R2 8, X C 6 P4.10, Find i1、 i2、 i i i2 v i1 v 220 2 sin 314t Ch4 Sinusoidal Steady State Analysis 4.5 Parallel and Series Resonance Parallel RLC Circuit Review For sinusoidal circuit, Series :v v1 v2 Parallel : i i1 i2 V V1 V2 I I1 I 2 Two Simple Methods: Phasor Diagrams and Complex Numbers ? Ch4 Sinusoidal Steady State Analysis 4.6 Examples for Sinusoidal Circuits Analysis Key Words: Bypass Capacitor RC Phase Difference Low-Pass and High-Pass Filter Ch4 Sinusoidal Steady State Analysis 4.6 Examples for Sinusoidal Circuits Analysis Bypass Capacitor P4.11, Let i 3 10 3 2 sin t 3 10 3 2 sin 2ft 3 10 3 2 sin 1000 t f = 500Hz,Determine VAB before the C is connected . And VAB after parallel C = 30F Before C is connected i VAB IR 3 103 500 1.5(V ) After C is connected v XC 1 1 10() 2fc 1000 30 10 6 1 1 Z R jX C 1 0.2 10 j 10 88.85 VAB I C Z 3 103 10 30(mV ) Ch4 Sinusoidal Steady State Analysis 4.6 Examples for Sinusoidal Circuits Analysis RC Phase Difference P4.12, + vi _ f = 300Hz, R = 100。 If vo - vi= /4,C =? R=100 + Vi IR jX C vi vi Vo I jX C vo vo 1 5.3110 4 XC vo C C C vo 90 Vo jX C _ 5.3110 6 Vi R jX C vi arctg C 6 5.3110 vo vi arctg 2 C 4 6 5.3110 arctg C 4 5.31106 0.0411 C C 1.29 10 4 F Ch4 Sinusoidal Steady State Analysis 4.6 Examples for Sinusoidal Circuits Analysis Low-Pass and High-Pass Filter + R=200 + RC---- High-Pass Filter vi _ vo C XC _ VR R 1 VC C 1 f VR X C RVC P4.13, The voltage sources are vi=240+100sin2100t(V), R=200, C=50F, Determine VAC and VDC in output voltage vo. VDC = 240V V AC V 32 XC 100 16(V ) Z 200 XC 1 1 32 6 2fc 2 100 50 10 Z R 2 X C2 2002 322 200 Ch4 Sinusoidal Steady State Analysis 4.6 Examples for Sinusoidal Circuits Analysis Low-Pass and High-Pass Filter 260V 250V 240V 230V 220V 50ms V(2) 55ms 60ms 65ms 70ms 75ms Time 80ms 85ms 90ms 95ms 100ms Ch4 Sinusoidal Steady State Analysis 4.6 Examples for Sinusoidal Circuits Analysis Low-Pass and High-Pass Filter 400V 300V 200V 100V 50ms V(2) 55ms V(1) 60ms 65ms 70ms 75ms Time 80ms 85ms 90ms 95ms 100ms Ch4 Sinusoidal Steady State Analysis SEL>> 4.6 Examples for Sinusoidal Circuits Analysis 400V 300V 200V 100V 50ms V(2) 55ms V(1) 60ms 65ms 70ms 75ms 80ms 85ms 90ms 95ms 100ms Time 300V 300V 200V 200V 100V 100V 0V 0Hz 0.2KHz V(1) 0.4KHz 0.6KHz 0.8KHz 0V 0Hz 1.0KHz 0.2KHz 1.2KHz V(2) 0.4KHz 1.4KHz 0.6KHz 1.6KHz 0.8KHz 2.0KHz1.0KHz 1.8KHz Ch4 Sinusoidal Steady State Analysis 4.6 Examples for Sinusoidal Circuits Analysis Low-Pass and High-Pass Filter 1.0V 0.5V 0V 1.0Hz V(2) 3.0Hz 10Hz 30Hz 100Hz 300Hz Frequency 1.0KHz 3.0KHz 10KHz 30KHz 100KHz Ch4 Sinusoidal Steady State Analysis SEL>> 4.6 Examples for Sinusoidal Circuits Analysis 1.0V 0.5V 0V 0s V(2) 50ms V(1) 100ms 150ms 200ms 250ms 300ms 350ms 400ms 450ms 500ms 550ms Time 800mV 400mV SEL>> 0V V(2) 1.0V 0.5V 0V 0Hz 50Hz V(1) 100Hz 150Hz 200Hz 250Hz 300Hz 350Hz 400Hz 450Hz 500Hz 600ms Ch4 Sinusoidal Steady State Analysis SEL>> 4.6 Examples for Sinusoidal Circuits Analysis 1.0V 0.5V 0V 0Hz 800mV 50Hz 100Hz 150Hz 200Hz 250Hz 300Hz 350Hz 400Hz 450Hz 300Hz 350Hz 400Hz 450Hz 500Hz V(2) Frequency 400mV SEL>> 0V V(2) 1.0V 0.5V 0V 0Hz 50Hz V(1) 100Hz 150Hz 200Hz 250Hz 500Hz Ch4 Sinusoidal Steady State Analysis 4.6 Examples for Sinusoidal Circuits Analysis Complex Numbers Analysis P4.14, Find I1 I2 I3 V in the circuit of the following fig. 2 i3 v1=120sint i1 i2 v2 Ch4 Sinusoidal Steady State Analysis 4.6 Examples for Sinusoidal Circuits Analysis Complex Numbers Analysis P4.15, Let Vm = 100V. Use Thevenin’s theorem to find ICD v v Ch4 Sinusoidal Steady State Analysis 4.7 Magnetically Coupled Circuits Key Words: Self- inductance and Mutual inductance Magnetically Coupled Circuits and v ~ i relationship Dot convention Ideal transformer Ch4 Sinusoidal Steady State Analysis 4.7 Magnetically Coupled Circuits Coupled Circuits and v~i relationship Magnetic flux: 1 = f(i1) (1 = N11) The flux is proportional to the current in linear inductor: 1(t) = L1i1(t) L is a lumped element abstraction for the coil. i1 + v1 - i1 v1( t ) d 1 di1 L1 dt dt v1 Voltage be proportional to the time rate of change of the magnetic field. Ch4 Sinusoidal Steady State Analysis 4.7 Magnetically Coupled Circuits Coupled Circuits and v~i relationship 1(t ) L1i1(t ) M12i2(t ) M12 M 21 M i1 + v1 - i2 + v2 - 2(t ) M 21i1(t ) L2i2(t ) d 1 di di L1 1 M 2 dt dt dt d 2 di di v2 M 1 L2 2 dt dt dt v1 ——Ideal Coupled Circuits’ v ~ i relationship L1、L2、M represent Ideal Coupled Inductor d 1 di di v1 L1 1 M 2 dt dt dt Self- inductance voltage Mutual- inductance voltage Ch4 Sinusoidal Steady State Analysis 4.7 Magnetically Coupled Circuits Coupled Circuits and v ~ i relationship + v1 - + i2 + v2 - i1 i2 + v2 i1 v1 - - 1(t ) L1i1(t ) Mi2(t ) 1(t ) L1i1(t ) Mi2(t ) 2(t ) Mi1(t ) L2i2(t ) 2(t ) Mi1(t ) L2i2(t ) v1 d 1 di di L1 1 M 2 dt dt dt d 2 di1 di2 v2 dt M dt L2 dt v1 v2 d 1 di di L1 1 M 2 dt dt dt d 2 di1 di2 dt M dt L2 dt i2 i1 Ch4 Sinusoidal Steady State Analysis 4.7 Magnetically Coupled Circuits v1 • • Dot convention i2 + v2 - + v1 - + 1 i v1 - i1 v1 d 1 di di L1 1 M 2 dt dt dt d 2 di1 di2 v2 dt M dt L2 dt v1 i 2 + v2 - 1 i • i2 • v2 v1 v2 d 1 di di L1 1 M 2 dt dt dt d 2 di1 di2 dt M dt L2 dt A current entering the dotted terminal of one coil produces an open circuit voltage with a positive voltage reference at the dotted terminal of the second coil. Inversely , current leaving of the dotted terminal of one coil produces a negative voltage reference at the dotted terminal of the other end. v2 Ch4 Sinusoidal Steady State Analysis 4.7 Magnetically Coupled Circuits Question:The terminal is dotted,how can we get v ~ i equations to coupled inductor? i2 i1 • • i2 i1 v1 v2 v1 u 2 of the i and Suppose direction M with Dot convention! • v2 di is• consistent dt Steps to determine the coupled circuit voltage 1. For self inductance voltage +i / i-i / i+ 2. For mutual inductance voltage + vs - vs + vm - vm Ch4 Sinusoidal Steady State Analysis 4.7 Magnetically Coupled Circuits P4.16,For the circuit shown in following figures, determine v1and v2. i1 v1 • L1 M i2 i1 • v L2 u2 2 v1 • L1 M i2 i1 L2 v • u2 2 v1 M L1 • i2 - • L2 uv2 2 + dv1 di M 2 dt dt di dv v2 L2 2 M 1 dt dt v1 L1 dv1 di M 2 dt dt di di v2 L2 2 M 1 dt dt v1 L1 di1 di M 2 dt dt di di v2 L2 2 M 1 dt dt v1 L1 Ch4 Sinusoidal Steady State Analysis 4.7 Magnetically Coupled Circuits Coupled Circuits and v ~ i relationship 1(t ) L1i1(t ) M12i2(t ) M12 M 21 M i1 + v1 - i2 + v2 - 2(t ) M 21i1(t ) L2i2(t ) d 1 di di L1 1 M 2 dt dt dt d 2 di di v2 M 1 L2 2 dt dt dt v1 ——Ideal Coupled Circuits’s v~i relationship For sinusoidal circuit, V1 jL1I1 jMI2 V2 jMI1 jL2 I2 Ch4 Sinusoidal Steady State Analysis 4.7 Magnetically Coupled Circuits Ideal transformer n=N1/N2 (k = 1, L = , M = ) • v1 v1( t ) nv2 ( t ) 1 i1( t ) i2 ( t ) n i2 i1 N1 • N2 u2 v2 ZL v1i1 v2i2 0 For ideal transformer, p=0 Z1 V1 nV2 Impedance Transformation Z1 n2Z L I1 1 I 2 n