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Mohammed Waris Senan

- Any steady state voltage or current in a linear circuit with a sinusoidal source is a sinusoid with the same frequency as the source. Phasors and complex impedances allow conversion of differential equations to circuit analysis by representing magnitude and phase of sinusoids. - For a resistor, the voltage and current are in phase. In the phasor domain, the resistor phasor relationship is V=IR. In the time domain, the average power dissipated is proportional to the product of RMS current and voltage.

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 Characteristics of Sinusoidal  Phasors  Phasor Relationships for R, L and C  Impedance  Parallel and Series Resonance  Examples for Sinusoidal Circuits Analysis Single Phase AC
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
Characteristics of Sinusoids Outline: 1. Time Period: T 2. Frequency: f (Hertz) 3. Angular Frequency:  (rad/sec) 4. Phase angle: Φ 5. Amplitude: Vm Im
Characteristics of Sinusoids :   tVv mt sin iI1 I1 I1 I1 I1 I1 R1 R1 R 5 5 + _ IS  E I1 U1 + - U I  iI1 I1 I1 I1 R1 R1 R 5 5 - + IS  E I1 U1 + - U I  v ,i tt1 t20 Both the polarity and magnitude of voltage are changing.
Radian frequency(Angular frequency):  = 2f = 2/T (rad/s） Time Period: T — Time necessary to go through one cycle. (s) Frequency: f — Cycles per second. (Hz) f = 1/T Amplitude: Vm Im i = Imsint， v =Vmsint v ,i t 20 Vm , Im Characteristics of Sinusoids :
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. RIRdti T T 2 0 21  Effective Value of a Periodic Waveform  T eff dti T I 0 21 22 1 2 2cos1 sin 1 2 0 2 0 22 m m T m T meff IT I T dt t T I tdtI T I       2 1 0 2 m T eff V dtv T V   Characteristics of Sinusoids :
Phase (angle)    tIi m sin   sin0 mIi  Phase angle -8 -6 -4 -2 0 2 4 6 8 0 0.01 0.02 0.03 0.04 0.05 <0 0 Characteristics of Sinusoids :
)sin( 1  tVv m )sin( 2  tIi m Phase difference 2121 )(   ttiv 021   — v(t) leads i(t) by (1 - 2), or i(t) lags v(t) by (1 - 2) 2 21    v, i t v i   21 Out of phase t v, i v i v, i t v i 021   In phase 021   — v(t) lags i(t) by (2 - 1), or i(t) leads v(t) by (2 - 1) Characteristics of Sinusoids :
Review The sinusoidal waves 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                                 Characteristics of Sinusoids :
Phase difference   30314sin22201  tv     9030314sin222030314cos22202  ttv   120314sin2220  t  1501203021     30314cos22202  tv   30314cos22202  tv   18030314cos2220  t    210314360cos2220  t   90150314sin2220  t   60314sin2220  t  30603021   Find ?   30314cos22202  tvIf Characteristics of Sinusoids :
Phase difference v, i t v i -/3 /3 • ••         3 sin  tVm        3 sin  tIm Characteristics of Sinusoids :
Outline: 1. Complex Numbers 2. Rotating Vector 3. Phasors A sinusoidal voltage/current at a given frequency, is characterized by only two parameters : amplitude and phase A phasor is a Complex Number which represents magnitude and phase of a sinusoid Phasors
e.g. voltage response A sinusoidal v/i Complex transform Phasor transform By knowing angular frequency ω rads/s. Time domain Frequency domain   eR v t Complex form:    cosmv t V t   Phasor form:    j t mv t V e    Angular frequency ω is known in the circuit.  || mVV  || mVV Phasors
Rotating Vector    tIti m sin)( i Im t1 i t Im  t x y              max cos sin sin j t m m m j t m m I e I t jI t i t I t I I e                    A complex coordinates number: Real value: i(t1) Imag Phasors
Rotating Vector Vm x y 0  )sin(   tVv m Phasors
Complex Numbers jbaA  — Rectangular Coordinates   sincos jAA  j eAA  — Polar Coordinates j eAAjbaA  conversion： 22 baA  a b arctg jbaeA j  cosAa  sinAb   a b Real axis Imaginary axis jjje j  090sin90cos90  Phasors
Complex Numbers Arithmetic With Complex Numbers Addition: A = a + jb, B = c + jd, A + B = (a + c) + j(b + d) Real Axis Imaginary Axis AB A + B Phasors
Complex Numbers Arithmetic With Complex Numbers Subtraction : A = a + jb, B = c + jd, A - B = (a - c) + j(b - d) Real Axis Imaginary Axis AB A - B 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) Phasors
Phasors A phasor is a complex number that represents the magnitude and phase of a sinusoid:   tim cos  mI 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. Phasors
)sin()cos()(    tAjtAeAAe tjtj )cos(||}Re{   tAAe tj Complex Exponentials j eAA   A real-valued sinusoid is the real part of a complex exponential.  Complex exponentials make solving for AC steady state an algebraic problem. Phasors
Phasor Relationships for R, L and C Outline: I-V Relationship for R, L and C, Power conversion
Phasor Relationships for R, L and C  v~i relationship for a resistor _ v i R   + S    tIt R V R v i m m  sinsin  tVv m sin Relationship between RMS: R V I  Wave and Phasor diagrams： v、i t v i  I  V R V I    Resistor Suppose
 Time domain Frequency domainResistor With a resistor θ﹦ϕ, v(t) and i(t) are in phase . )cos()( )cos()(     wtIti wtVtv m m IRV RIV eRIeV eRIeV mm j m j m wtj m wtj m         )()( Phasor Relationships for R, L and C
 PowerResistor _ v i R +   P  0 tItVvip mm  sinsin  tVI mm 2 sin  t VI mm 2cos1 2  tIVIV 2cos v, i t v i P=IV  T pdt T P 0 1    T VIdttVI T 0 2cos1 1  R V RIIVP 2 2  • Average Power • Transient Power Note: I and V are RMS values. Phasor Relationships for R, L and C
Resistor , R=10，Find i and Ptv 314sin311  V V V m 220 2 311 2   A R V I 22 10 220  ti 314sin222  WIVP 484022220  Phasor Relationships for R, L and C
 v~i relationshipInductor dt di Lvv AB    tLI dt tId L dt di Lv m m   cos sin    90sin  tLIm    90sin  tVm     t vdt L i 1    t vdt L vdt L 0 0 11  t vdt L i 0 0 1 tIi m sinSuppose Phasor Relationships for R, L and C
 v~i relationshipInductor   90sin  tLIm  dt di Lv    90sin  tVm  LIV mm  Relationship between RMS: LIV  L V I   fLLXL  2   For DC，f = 0，XL = 0. fXL  v(t) leads i(t) by 90º, or i(t) lags v(t) by 90º Phasor Relationships for R, L and C
 v ~ i relationshipInductor v, i t v i eL V I LXIjV  Wave and Phasor diagrams： Phasor Relationships for R, L and C
 PowerInductor vip    tItV mm  sin90sin   ttIV mm  sincos  t IV mm 2sin 2  tVI 2sin P t v, i t v i ++ --22 max 2 1 LILIW m  2 00 2 1 LiLidividtW it  Energy stored:    T T tdtVI T pdt T P 0 0 02sin 11 Average Power Reactive Power L L X V XIIVQ 2 2  （Var） Phasor Relationships for R, L and C
Inductor L = 10mH，v = 100sint，Find iL when f = 50Hz and 50kHz.    14.310105022 3 fLX L      Atti A X V I L L  90sin25.22 5.22 14.3 2/100 50       31401010105022 33 fLX L      mAtti mA X V I L L k  90sin25.22 5.22 14.3 2/100 50    Phasor Relationships for R, L and C
 v ~ i relationshipCapacitor _ v i +   C dt dv C dt dq i  tVv m sinSuppose:   90sincos  tCVtCVi mm    90sin  tIm       t tt idt c vidt c idt c idt c v 0 0 0 0 1111 i(t) leads v(t) by 90º, or v(t) lags i(t) by 90º Relationship between RMS: CX V C V CVI    1   fCC XC  2 11  For DC，f = 0， XC   f XC 1  mm CVI  Phasor Relationships for R, L and C
_ v i +   C tj m tj m eCVj dt edV C dt tdv Cti    )( )( v(t) = Vm ejt Represent v(t) and i(t) as phasors: CjX V VCωjI ==   • The derivative in the relationship between v(t) and i(t) becomes a multiplication by in the relationship between and . • 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.  v ~ i relationshipCapacitor V I wC j  Phasor Relationships for R, L and C
 v ~ i relationshipCapacitor v, i t v i I V CXIjV   Wave and Phasor diagrams： Phasor Relationships for R, L and C
PowerCapacitor Average Power: P = 0 Reactive Power C C X V XIIVQ 2 2  （Var）   90sinsin  tItVvip mm  tVIt IV mm  2sin2sin 2  P t v, i t v i ++ -- Energy stored:    t vv CvCvdvdt dt dv CvvidtW 0 0 2 0 2 1 22 max 2 1 CVCVW m  Phasor Relationships for R, L and C
Capacitor Suppose C=20F，AC source v=100sint，Find XC and I for f = 50Hz, 50kHz。  159 2 11 Hz50 fCC Xf c  A44.0 2  c m c X V X V I   159.0 2 11 KHz50 fCC Xf c  A440 2  c m c X V X V I Phasor Relationships for R, L and C
Review (v – i Relationship) Time domain Frequency domain iRv  IRV   I Cj V    1 ILjV    dt di LvL  dt dv CiC  C XC  1  LXL , , , v and i are in phase. , v leads i by 90°. , v lags i by 90°. R C L Phasor Relationships for R, L and C
Summary:  R： RX R  0 L： ffLLXL   2 2    iv C： ffcc XC 1 2 11   2    iv  IXV   Frequency characteristics of an Ideal Inductor and Capacitor: A capacitor is an open circuit to DC currents; A Inductor is a short circuit to DC currents. Phasor Relationships for R, L and C
Impedance (Z) Outline: Complex currents and voltages. Impedance Phasor Diagrams
• 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: ZIV  Complex voltage， Complex current， Complex Impedance vm j m VeVV v   im j m IeII i      ZeZe I V I V Z jj m m iv )(   ‘Z’ is called impedance measured in ohms () Impedance (Z)
Complex Impedance    ZeZe I V I V Z jj m m iv )(    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. Impedance (Z)
Complex Impedance ZR = R  ＝ 0; or ZR = R  0 Resistor——The impedance is R c j c jX C j e C Z       21 ) 2 (    iv or  90 1  C ZC  Capacitor——The impedance is 1/jωC L j L jXLjLeZ    2 ) 2 (    iv or  90 LZL  Inductor——The impedance is jωL Impedance (Z)
Complex Impedance Impedance in series/parallel can be combined as resistors. _ U U Z1 +  Z2 Zn   I    n k kn ZZZZZ 1 21 ... _ US In I1 I1 I1 I1 I1 I1     R1 R1 Zn 5 5 5 5 + + _ US IS  U1 + - U  I  Z2Z1     n k kn ZZZZZ 121 11 ... 111 21 1 2 21 2 1 ZZ Z II ZZ Z II      Current divider:   n k k i i Z Z VV 1  Voltage divider: Impedance (Z)
Complex Impedance _ +   V  I   1IZ1 Z2 Z    2121 2 2121 2 1 2 1 1 2 2 1 11 ZZZZZZ ZV I ZZZZZZ ZZV ZZ Z V I ZZ Z II                      Impedance (Z)
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 20kW + - 1mF10V  0 VC + - w = 377 Find VC • How do we find VC? • First compute impedances for resistor and capacitor: ZR = 20kW = 20kW  0 ZC = 1/j (377 *1mF) = 2.65kW  -90 Impedance (Z)
Complex Impedance 20kW + - 1mF10V  0 VC + - w = 377 Find VC 20kW  0 + - 2.65kW  - 90 10V  0 VC + - Now use the voltage divider to find VC:     46.82V31.1 54.717.20 9065.2 010VCV ) 0209065.2 9065.2 (010       kk k VVC Impedance (Z)
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. Complex Impedance Impedance (Z)
Kirchhoff’s Laws KCL and KVL hold as well in phasor domain. KVL： 0 1  n k kv vk- Transient voltage of the #k branch 0 1  n k kV KCL: 0 1   n k ki 0 1   n k kI ik- Transient current of the #k branch Impedance (Z)
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/jL • Capacitor: – The admittance is jC Impedance (Z)
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. 2mA  40 – 1mF VC + – 1kW VR + + – V I = 2mA  40, VR = 2V  40 VC = 5.31V  -50, V = 5.67V  -29.37 Real Axis Imaginary Axis VR VC V Impedance (Z)
Parallel and Series Resonance Outline: RLC Circuit, Series Resonance Parallel Resonance
v vR vL vC CLR vvvv  CLR VVVV  Phasor  I V LV CV RV IZ XRI XXRI IXIXIR VVVV CL CL CLR      22 22 22 22 )( )()( )( ）CL XXX ( 22 XRZ  22 ) 1 ( c LR    (2nd Order RLC Circuit )Series RLC Circuit Parallel and Series Resonance :
22 XRZ  22 ) 1 ( c LR   IZVVVV CLR  22 )( Z X = XL-XC R   V RV CLX VVV   R XX V VV CL R CL      1 1 tan - tan Phase difference: XL>XC   >0，v leads i by  — Inductance Circuit XL<XC   <0，v lags i by  — Capacitance Circuit XL=XC   =0，v and i in phase — Resistors Circuit Series RLC Circuit Parallel and Series Resonance :
CLR VVVV   CL XIjXIjRI   ZIjXRIXXjRI CL   )()](( )( CL XXjR I V Z     ZjXRZ 22 )( CL XXRZ  R XX CL   1 tan iv   v vR vL vC Series RLC Circuit Parallel and Series Resonance :
Series Resonance (2nd Order RLC Circuit ) CLR VVVV   CL XIjXIjRI   R XX arctg V VV arctg CL R CL     CLCL VVL C XXWhen    1 , VVR  0and —— Series Resonance Resonance condition I LV CV VVR   LC for LC   2 11 00  f0 f X Cf XC 2 1  fLXL 2 Resonant frequency Parallel and Series Resonance :
Series Resonance R V Z V IRXXRZ CL  0 0 22 0 )(• Zmin；when V = constant, I = Imax= I0 RXX CL  RIXIXI CL 000  VVV CL  • Quality factor Q, R X R X V V V V Q CLCL  CLCL VVL C XX  ) 1 (   Resonance condition: When, Parallel and Series Resonance :
Parallel RLC Circuit V  I  LI  CI    )( 1 / 11 222222 LR L Cj LR R Cj LjRLjR LjR Cj LjRCjLjR Y                         Parallel Resonance Parallel Resonance frequency L CR LC 2 0 1 1  LXR In generally ) 2 1 ( 0 LC f   LC 1 0  0)( 222    LR L C   When 2220 LR R Y  , In phase withV I V L RC C L R R V L LC R R V LR R VVYII        222 22 0 200 1 Zmax Imin: Parallel and Series Resonance :
Parallel RLC Circuit V  I  LI  CI V L C j L V j LjR VIL       00 1  V L C jVCjIC   0 0|||||| 0  III CL  Z  . RCR L Q 0 0 1   0IjQIL   0IjQIC   •Quality factor Q, 0000 Y Y Y Y I I I I Q CLLC  Parallel and Series Resonance :
Parallel RLC Circuit Review For sinusoidal circuit， Series ： 21 vvv  21 VVV  21 iii  21 III  ？ Two Simple Methods: Phasor Diagrams and Complex Numbers Parallel : Parallel and Series Resonance :

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Ac single phase

  • 1.  Characteristics of Sinusoidal  Phasors  Phasor Relationships for R, L and C  Impedance  Parallel and Series Resonance  Examples for Sinusoidal Circuits Analysis Single Phase AC
  • 2. 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
  • 3. Characteristics of Sinusoids Outline: 1. Time Period: T 2. Frequency: f (Hertz) 3. Angular Frequency:  (rad/sec) 4. Phase angle: Φ 5. Amplitude: Vm Im
  • 4. Characteristics of Sinusoids :   tVv mt sin iI1 I1 I1 I1 I1 I1 R1 R1 R 5 5 + _ IS  E I1 U1 + - U I  iI1 I1 I1 I1 R1 R1 R 5 5 - + IS  E I1 U1 + - U I  v ,i tt1 t20 Both the polarity and magnitude of voltage are changing.
  • 5. Radian frequency(Angular frequency):  = 2f = 2/T (rad/s） Time Period: T — Time necessary to go through one cycle. (s) Frequency: f — Cycles per second. (Hz) f = 1/T Amplitude: Vm Im i = Imsint， v =Vmsint v ,i t 20 Vm , Im Characteristics of Sinusoids :
  • 6. 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. RIRdti T T 2 0 21  Effective Value of a Periodic Waveform  T eff dti T I 0 21 22 1 2 2cos1 sin 1 2 0 2 0 22 m m T m T meff IT I T dt t T I tdtI T I       2 1 0 2 m T eff V dtv T V   Characteristics of Sinusoids :
  • 7. Phase (angle)    tIi m sin   sin0 mIi  Phase angle -8 -6 -4 -2 0 2 4 6 8 0 0.01 0.02 0.03 0.04 0.05 <0 0 Characteristics of Sinusoids :
  • 8. )sin( 1  tVv m )sin( 2  tIi m Phase difference 2121 )(   ttiv 021   — v(t) leads i(t) by (1 - 2), or i(t) lags v(t) by (1 - 2) 2 21    v, i t v i   21 Out of phase t v, i v i v, i t v i 021   In phase 021   — v(t) lags i(t) by (2 - 1), or i(t) leads v(t) by (2 - 1) Characteristics of Sinusoids :
  • 9. Review The sinusoidal waves 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                                 Characteristics of Sinusoids :
  • 10. Phase difference   30314sin22201  tv     9030314sin222030314cos22202  ttv   120314sin2220  t  1501203021     30314cos22202  tv   30314cos22202  tv   18030314cos2220  t    210314360cos2220  t   90150314sin2220  t   60314sin2220  t  30603021   Find ?   30314cos22202  tvIf Characteristics of Sinusoids :
  • 11. Phase difference v, i t v i -/3 /3 • ••         3 sin  tVm        3 sin  tIm Characteristics of Sinusoids :
  • 12. Outline: 1. Complex Numbers 2. Rotating Vector 3. Phasors A sinusoidal voltage/current at a given frequency, is characterized by only two parameters : amplitude and phase A phasor is a Complex Number which represents magnitude and phase of a sinusoid Phasors
  • 13. e.g. voltage response A sinusoidal v/i Complex transform Phasor transform By knowing angular frequency ω rads/s. Time domain Frequency domain   eR v t Complex form:    cosmv t V t   Phasor form:    j t mv t V e    Angular frequency ω is known in the circuit.  || mVV  || mVV Phasors
  • 14. Rotating Vector    tIti m sin)( i Im t1 i t Im  t x y              max cos sin sin j t m m m j t m m I e I t jI t i t I t I I e                    A complex coordinates number: Real value: i(t1) Imag Phasors
  • 16. Complex Numbers jbaA  — Rectangular Coordinates   sincos jAA  j eAA  — Polar Coordinates j eAAjbaA  conversion： 22 baA  a b arctg jbaeA j  cosAa  sinAb   a b Real axis Imaginary axis jjje j  090sin90cos90  Phasors
  • 17. Complex Numbers Arithmetic With Complex Numbers Addition: A = a + jb, B = c + jd, A + B = (a + c) + j(b + d) Real Axis Imaginary Axis AB A + B Phasors
  • 18. Complex Numbers Arithmetic With Complex Numbers Subtraction : A = a + jb, B = c + jd, A - B = (a - c) + j(b - d) Real Axis Imaginary Axis AB A - B Phasors
  • 19. 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) Phasors
  • 20. Phasors A phasor is a complex number that represents the magnitude and phase of a sinusoid:   tim cos  mI 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. Phasors
  • 21. )sin()cos()(    tAjtAeAAe tjtj )cos(||}Re{   tAAe tj Complex Exponentials j eAA   A real-valued sinusoid is the real part of a complex exponential.  Complex exponentials make solving for AC steady state an algebraic problem. Phasors
  • 22. Phasor Relationships for R, L and C Outline: I-V Relationship for R, L and C, Power conversion
  • 23. Phasor Relationships for R, L and C  v~i relationship for a resistor _ v i R   + S    tIt R V R v i m m  sinsin  tVv m sin Relationship between RMS: R V I  Wave and Phasor diagrams： v、i t v i  I  V R V I    Resistor Suppose
  • 24.  Time domain Frequency domainResistor With a resistor θ﹦ϕ, v(t) and i(t) are in phase . )cos()( )cos()(     wtIti wtVtv m m IRV RIV eRIeV eRIeV mm j m j m wtj m wtj m         )()( Phasor Relationships for R, L and C
  • 25.  PowerResistor _ v i R +   P  0 tItVvip mm  sinsin  tVI mm 2 sin  t VI mm 2cos1 2  tIVIV 2cos v, i t v i P=IV  T pdt T P 0 1    T VIdttVI T 0 2cos1 1  R V RIIVP 2 2  • Average Power • Transient Power Note: I and V are RMS values. Phasor Relationships for R, L and C
  • 26. Resistor , R=10，Find i and Ptv 314sin311  V V V m 220 2 311 2   A R V I 22 10 220  ti 314sin222  WIVP 484022220  Phasor Relationships for R, L and C
  • 27.  v~i relationshipInductor dt di Lvv AB    tLI dt tId L dt di Lv m m   cos sin    90sin  tLIm    90sin  tVm     t vdt L i 1    t vdt L vdt L 0 0 11  t vdt L i 0 0 1 tIi m sinSuppose Phasor Relationships for R, L and C
  • 28.  v~i relationshipInductor   90sin  tLIm  dt di Lv    90sin  tVm  LIV mm  Relationship between RMS: LIV  L V I   fLLXL  2   For DC，f = 0，XL = 0. fXL  v(t) leads i(t) by 90º, or i(t) lags v(t) by 90º Phasor Relationships for R, L and C
  • 29.  v ~ i relationshipInductor v, i t v i eL V I LXIjV  Wave and Phasor diagrams： Phasor Relationships for R, L and C
  • 30.  PowerInductor vip    tItV mm  sin90sin   ttIV mm  sincos  t IV mm 2sin 2  tVI 2sin P t v, i t v i ++ --22 max 2 1 LILIW m  2 00 2 1 LiLidividtW it  Energy stored:    T T tdtVI T pdt T P 0 0 02sin 11 Average Power Reactive Power L L X V XIIVQ 2 2  （Var） Phasor Relationships for R, L and C
  • 31. Inductor L = 10mH，v = 100sint，Find iL when f = 50Hz and 50kHz.    14.310105022 3 fLX L      Atti A X V I L L  90sin25.22 5.22 14.3 2/100 50       31401010105022 33 fLX L      mAtti mA X V I L L k  90sin25.22 5.22 14.3 2/100 50    Phasor Relationships for R, L and C
  • 32.  v ~ i relationshipCapacitor _ v i +   C dt dv C dt dq i  tVv m sinSuppose:   90sincos  tCVtCVi mm    90sin  tIm       t tt idt c vidt c idt c idt c v 0 0 0 0 1111 i(t) leads v(t) by 90º, or v(t) lags i(t) by 90º Relationship between RMS: CX V C V CVI    1   fCC XC  2 11  For DC，f = 0， XC   f XC 1  mm CVI  Phasor Relationships for R, L and C
  • 33. _ v i +   C tj m tj m eCVj dt edV C dt tdv Cti    )( )( v(t) = Vm ejt Represent v(t) and i(t) as phasors: CjX V VCωjI ==   • The derivative in the relationship between v(t) and i(t) becomes a multiplication by in the relationship between and . • 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.  v ~ i relationshipCapacitor V I wC j  Phasor Relationships for R, L and C
  • 34.  v ~ i relationshipCapacitor v, i t v i I V CXIjV   Wave and Phasor diagrams： Phasor Relationships for R, L and C
  • 35. PowerCapacitor Average Power: P = 0 Reactive Power C C X V XIIVQ 2 2  （Var）   90sinsin  tItVvip mm  tVIt IV mm  2sin2sin 2  P t v, i t v i ++ -- Energy stored:    t vv CvCvdvdt dt dv CvvidtW 0 0 2 0 2 1 22 max 2 1 CVCVW m  Phasor Relationships for R, L and C
  • 36. Capacitor Suppose C=20F，AC source v=100sint，Find XC and I for f = 50Hz, 50kHz。  159 2 11 Hz50 fCC Xf c  A44.0 2  c m c X V X V I   159.0 2 11 KHz50 fCC Xf c  A440 2  c m c X V X V I Phasor Relationships for R, L and C
  • 37. Review (v – i Relationship) Time domain Frequency domain iRv  IRV   I Cj V    1 ILjV    dt di LvL  dt dv CiC  C XC  1  LXL , , , v and i are in phase. , v leads i by 90°. , v lags i by 90°. R C L Phasor Relationships for R, L and C
  • 38. Summary:  R： RX R  0 L： ffLLXL   2 2    iv C： ffcc XC 1 2 11   2    iv  IXV   Frequency characteristics of an Ideal Inductor and Capacitor: A capacitor is an open circuit to DC currents; A Inductor is a short circuit to DC currents. Phasor Relationships for R, L and C
  • 39. Impedance (Z) Outline: Complex currents and voltages. Impedance Phasor Diagrams
  • 40. • 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: ZIV  Complex voltage， Complex current， Complex Impedance vm j m VeVV v   im j m IeII i      ZeZe I V I V Z jj m m iv )(   ‘Z’ is called impedance measured in ohms () Impedance (Z)
  • 41. Complex Impedance    ZeZe I V I V Z jj m m iv )(    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. Impedance (Z)
  • 42. Complex Impedance ZR = R  ＝ 0; or ZR = R  0 Resistor——The impedance is R c j c jX C j e C Z       21 ) 2 (    iv or  90 1  C ZC  Capacitor——The impedance is 1/jωC L j L jXLjLeZ    2 ) 2 (    iv or  90 LZL  Inductor——The impedance is jωL Impedance (Z)
  • 43. Complex Impedance Impedance in series/parallel can be combined as resistors. _ U U Z1 +  Z2 Zn   I    n k kn ZZZZZ 1 21 ... _ US In I1 I1 I1 I1 I1 I1     R1 R1 Zn 5 5 5 5 + + _ US IS  U1 + - U  I  Z2Z1     n k kn ZZZZZ 121 11 ... 111 21 1 2 21 2 1 ZZ Z II ZZ Z II      Current divider:   n k k i i Z Z VV 1  Voltage divider: Impedance (Z)
  • 44. Complex Impedance _ +   V  I   1IZ1 Z2 Z    2121 2 2121 2 1 2 1 1 2 2 1 11 ZZZZZZ ZV I ZZZZZZ ZZV ZZ Z V I ZZ Z II                      Impedance (Z)
  • 45. 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 20kW + - 1mF10V  0 VC + - w = 377 Find VC • How do we find VC? • First compute impedances for resistor and capacitor: ZR = 20kW = 20kW  0 ZC = 1/j (377 *1mF) = 2.65kW  -90 Impedance (Z)
  • 46. Complex Impedance 20kW + - 1mF10V  0 VC + - w = 377 Find VC 20kW  0 + - 2.65kW  - 90 10V  0 VC + - Now use the voltage divider to find VC:     46.82V31.1 54.717.20 9065.2 010VCV ) 0209065.2 9065.2 (010       kk k VVC Impedance (Z)
  • 47. 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. Complex Impedance Impedance (Z)
  • 48. Kirchhoff’s Laws KCL and KVL hold as well in phasor domain. KVL： 0 1  n k kv vk- Transient voltage of the #k branch 0 1  n k kV KCL: 0 1   n k ki 0 1   n k kI ik- Transient current of the #k branch Impedance (Z)
  • 49. 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/jL • Capacitor: – The admittance is jC Impedance (Z)
  • 50. 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. 2mA  40 – 1mF VC + – 1kW VR + + – V I = 2mA  40, VR = 2V  40 VC = 5.31V  -50, V = 5.67V  -29.37 Real Axis Imaginary Axis VR VC V Impedance (Z)
  • 51. Parallel and Series Resonance Outline: RLC Circuit, Series Resonance Parallel Resonance
  • 52. v vR vL vC CLR vvvv  CLR VVVV  Phasor  I V LV CV RV IZ XRI XXRI IXIXIR VVVV CL CL CLR      22 22 22 22 )( )()( )( ）CL XXX ( 22 XRZ  22 ) 1 ( c LR    (2nd Order RLC Circuit )Series RLC Circuit Parallel and Series Resonance :
  • 53. 22 XRZ  22 ) 1 ( c LR   IZVVVV CLR  22 )( Z X = XL-XC R   V RV CLX VVV   R XX V VV CL R CL      1 1 tan - tan Phase difference: XL>XC   >0，v leads i by  — Inductance Circuit XL<XC   <0，v lags i by  — Capacitance Circuit XL=XC   =0，v and i in phase — Resistors Circuit Series RLC Circuit Parallel and Series Resonance :
  • 54. CLR VVVV   CL XIjXIjRI   ZIjXRIXXjRI CL   )()](( )( CL XXjR I V Z     ZjXRZ 22 )( CL XXRZ  R XX CL   1 tan iv   v vR vL vC Series RLC Circuit Parallel and Series Resonance :
  • 55. Series Resonance (2nd Order RLC Circuit ) CLR VVVV   CL XIjXIjRI   R XX arctg V VV arctg CL R CL     CLCL VVL C XXWhen    1 , VVR  0and —— Series Resonance Resonance condition I LV CV VVR   LC for LC   2 11 00  f0 f X Cf XC 2 1  fLXL 2 Resonant frequency Parallel and Series Resonance :
  • 56. Series Resonance R V Z V IRXXRZ CL  0 0 22 0 )(• Zmin；when V = constant, I = Imax= I0 RXX CL  RIXIXI CL 000  VVV CL  • Quality factor Q, R X R X V V V V Q CLCL  CLCL VVL C XX  ) 1 (   Resonance condition: When, Parallel and Series Resonance :
  • 57. Parallel RLC Circuit V  I  LI  CI    )( 1 / 11 222222 LR L Cj LR R Cj LjRLjR LjR Cj LjRCjLjR Y                         Parallel Resonance Parallel Resonance frequency L CR LC 2 0 1 1  LXR In generally ) 2 1 ( 0 LC f   LC 1 0  0)( 222    LR L C   When 2220 LR R Y  , In phase withV I V L RC C L R R V L LC R R V LR R VVYII        222 22 0 200 1 Zmax Imin: Parallel and Series Resonance :
  • 58. Parallel RLC Circuit V  I  LI  CI V L C j L V j LjR VIL       00 1  V L C jVCjIC   0 0|||||| 0  III CL  Z  . RCR L Q 0 0 1   0IjQIL   0IjQIC   •Quality factor Q, 0000 Y Y Y Y I I I I Q CLLC  Parallel and Series Resonance :
  • 59. Parallel RLC Circuit Review For sinusoidal circuit， Series ： 21 vvv  21 VVV  21 iii  21 III  ？ Two Simple Methods: Phasor Diagrams and Complex Numbers Parallel : Parallel and Series Resonance :