Ch4 Sinusoidal Steady State Analysis 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 Readings: Gao-Ch3; Hayt-Ch7

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 v、i t t1 t2

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): =2f =2 /T (rad/s） Amplitude: Vm Im i=Imsint ， v=Vmsint v、i t  2 Vm、Im

Ch4 Sinusoidal Steady State Analysis 4.1 Characteristics of Sinusoidal Effective(RMS) Value of a Periodic Waveform——is equal to the value of the direct current which, flowing through an R-ohm resistor, delivers the same average power to the resistor as does the periodic current. Effective Value of a Periodic Waveform

Ch4 Sinusoidal Steady State Analysis 4.1 Characteristics of Sinusoidal Phase(angle) Phase angle 0 ①如果正弦波的起始最小值发生在时间起点之前，则为正值。 ②如果正弦波的起始最小值发生在时间起点之后，则为负值。 <0

Ch4 Sinusoidal Steady State Analysis 4.1 Characteristics of Sinusoidal Phase difference ——v(t) leads i(t) by (1- 2), or i(t) lags v(t) by (1- 2) ——v(t) lags i(t) by (2- 1), or i(t) leads v(t) by (2- 1) v、i t v i Out of phase。 t v、i v i v、i t v i In phase.

Ch4 Sinusoidal Steady State Analysis 4.1 Characteristics of Sinusoidal Phase difference P4.1, Find

Ch4 Sinusoidal Steady State Analysis 4.1 Characteristics of Sinusoidal Phase difference P4.2, v、i t v i -/3 /3 •  P4.2, v、i波形如图，问，v、i初相各为多少？若将时间起点右移/3，则v、i初相有何改变？改变否？若时间起点右移，则v、i初相有何改变？ 改变否？若将时间起点左移/3 ，则v、i初相有何改变？ 改变否？

Ch4 Sinusoidal Steady State Analysis 4.2 Phasors Key Words: Complex Numbers, Rotating Vector Phasors

Ch4 Sinusoidal Steady State Analysis 4.2 Phasors Rotating Vector Im  t x y i t  Im i P4.2, v、i波形如图，问，v、i初相各为多少？若将时间起点右移/3，则v、i初相有何改变？改变否？若时间起点右移，则v、i初相有何改变？ 改变否？若将时间起点左移/3 ，则v、i初相有何改变？ 改变否？ t1 i(t1)

Ch4 Sinusoidal Steady State Analysis 4.2 Phasors Rotating Vector Vm x y   P4.2, v、i波形如图，问，v、i初相各为多少？若将时间起点右移/3，则v、i初相有何改变？改变否？若时间起点右移，则v、i初相有何改变？ 改变否？若将时间起点左移/3 ，则v、i初相有何改变？ 改变否？

Ch4 Sinusoidal Steady State Analysis 4.2 Phasors Complex Numbers ——Rectangular Coordinates  |A| a b real axis imaginary axis ——Polar Coordinates P4.2, v、i波形如图，问，v、i初相各为多少？若将时间起点右移/3，则v、i初相有何改变？改变否？若时间起点右移，则v、i初相有何改变？ 改变否？若将时间起点左移/3 ，则v、i初相有何改变？ 改变否？ j——旋转90的算子 conversion：

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) Real Axis Imaginary Axis A B A + B P4.2, v、i波形如图，问，v、i初相各为多少？若将时间起点右移/3，则v、i初相有何改变？改变否？若时间起点右移，则v、i初相有何改变？ 改变否？若将时间起点左移/3 ，则v、i初相有何改变？ 改变否？

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) Real Axis Imaginary Axis A B A - B P4.2, v、i波形如图，问，v、i初相各为多少？若将时间起点右移/3，则v、i初相有何改变？改变否？若时间起点右移，则v、i初相有何改变？ 改变否？若将时间起点左移/3 ，则v、i初相有何改变？ 改变否？

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) P4.2, v、i波形如图，问，v、i初相各为多少？若将时间起点右移/3，则v、i初相有何改变？改变否？若时间起点右移，则v、i初相有何改变？ 改变否？若将时间起点左移/3 ，则v、i初相有何改变？ 改变否？ Find： P4.3,

Ch4 Sinusoidal Steady State Analysis 4.2 Phasors Phasors A phasor is a complex number that represents the magnitude and phase of a sinusoid: Phasor Diagrams P4.2, v、i波形如图，问，v、i初相各为多少？若将时间起点右移/3，则v、i初相有何改变？改变否？若时间起点右移，则v、i初相有何改变？ 改变否？若将时间起点左移/3 ，则v、i初相有何改变？ 改变否？ 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 real-valued sinusoid is the real part of a complex exponential. Complex exponentials make solving for AC steady state an algebraic problem. P4.2, v、i波形如图，问，v、i初相各为多少？若将时间起点右移/3，则v、i初相有何改变？改变否？若时间起点右移，则v、i初相有何改变？ 改变否？若将时间起点左移/3 ，则v、i初相有何改变？ 改变否？

1.正弦量的相量表示法 Ch3正弦交流电路  3.2正弦量的相量表示法 3）相量 ——表示正弦量的复数 相量在复平面上的几何表示称为相量图． 讨论： ①只有正弦周期量才能用相量表示，相量不能表示非正弦周期量． ②只有同频率的正弦量才能画在同一相量图上，不同频率的正弦量不能画在一个相量图上。 ③相量只能代表正弦信号，不能将其与正弦信号画等号． ④正弦量用相量表示后，它们运算就变换为复数的运算． ⑤表示正弦量的相量有两种形式：相量图、复数式

1.正弦量的相量表示法 Ch3正弦交流电路  3.2正弦量的相量表示法 正弦量的四种表示方法： 三角函数式 波形图 旋转矢量 相量（复数） 　相量（复数） 例3：已知， 求：

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 Suppose Relationship between RMS: v、i t v i Wave and Phasor diagrams：

Ch4 Sinusoidal Steady State Analysis 4.3 Phasor Relationships for R, L and C Resistor Power Transient Power p0 v、i t v i Average Power P=IV

Ch4 Sinusoidal Steady State Analysis 4.3 Phasor Relationships for R, L and C Resistor P4.4 ， , R=10，Find i and P。

Ch4 Sinusoidal Steady State Analysis 4.3 Phasor Relationships for R, L and C Inductor v~i relationship Suppose

Ch4 Sinusoidal Steady State Analysis 4.3 Phasor Relationships for R, L and C Inductor v~i relationship Relationship between RMS: 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 ejwt Represent v(t) and i(t) as phasors: 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 diffierential 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 Inductor v~i relationship Wave and Phasor diagrams： v、i t v i eL

Ch4 Sinusoidal Steady State Analysis 4.3 Phasor Relationships for R, L and C Inductor Power P t Energy stored: v、i t v i + - Average Power Reactive Power （Var）

Ch4 Sinusoidal Steady State Analysis 4.3 Phasor Relationships for R, L and C Inductor P4.5，L=10mH，v=100sint，Find iL when f=50Hz and 50kHz.

Ch4 Sinusoidal Steady State Analysis 4.3 Phasor Relationships for R, L and C Inductor P4.6，L=0.5H，iL is shown in fig。Find eLand v - I i (mA) t (ms) 2 4 6 - I eL、v (V) t (ms) 0.2 0.4 2 4 6 -0.4 -0.2 eL v

Ch4 Sinusoidal Steady State Analysis 4.3 Phasor Relationships for R, L and C Capacitor v~i relationship Suppose: Relationship between RMS: For DC，f=0， XC 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~i relationship v(t) = Vm ejt Represent v(t) and i(t) as phasors: 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 diffierential 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、i t v i

Ch4 Sinusoidal Steady State Analysis 4.3 Phasor Relationships for R, L and C Capacitor Power P t Energy stored: v、i t v i + - Average Power: P=0 Reactive Power （Var）

Ch4 Sinusoidal Steady State Analysis 4.3 Phasor Relationships for R, L and C Capacitor P4.7，Suppose C=20F，AC source v=100sint，Find XC and I for f=50Hz, 50kHz。

Ch4 Sinusoidal Steady State Analysis 4.3 Phasor Relationships for R, L and C Summary R： L： C： Frequency characteristics of an Ideal Inducter 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. measured in ohms ()

Ch4 Sinusoidal Steady State Analysis 4.4 Impedance Complex Impedance 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 ZR=R =0 （＝0); or ZR=R0 Resistor——The impedance is R =-/2 or Capacitor——The impedance is 1/jwC =/2 or Inductor——The impedance is jwL

Ch4 Sinusoidal Steady State Analysis 4.4 Impedance Complex Impedance Impedance in series/parallel can be combined as resistors. Voltage divider: Current divider:

Ch4 Sinusoidal Steady State Analysis 4.4 Impedance Complex Impedance P4.8,

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 20kW + - 1mF 10V  0 VC w = 377 Find VC P4.9 How do we find VC? First compute impedances for resistor and capacitor: ZR = 20kW= 20kW  0 ZC = 1/j (377 *1mF) = 2.65kW  -90

Ch4 Sinusoidal Steady State Analysis 4.4 Impedance Complex Impedance 20kW + - 1mF 10V  0 VC w = 377 Find VC P4.9 Now use the voltage divider to find VC: 20kW  0 + - 2.65kW  -90 10V  0 VC

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/Notron 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. KCL: ik- Transient current of the #k branche KVL： vk- Transient voltage of the #k branche

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/jL 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 – 1mF VC + 1kW VR V Real Axis Imaginary Axis VR VC V

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 Phasor 

Ch4 Sinusoidal Steady State Analysis 4.5 Parallel and Series Resonance Series RLC Circuit (2nd Order RLC Circuit ) Z X=XL-XC R  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

Ch4 Sinusoidal Steady State Analysis 4.5 Parallel and Series Resonance Series RLC Circuit (2nd Order RLC Circuit ) v vR vL vC

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=40F，Source , Find 1）XL、XC、Z；2）I and i；3）VR and vR； VL and vL； VC and vC； 4） Phasor Diagrams v vR vL vC P4.10，Computing by (complex numbers) Phasors

Ch4 Sinusoidal Steady State Analysis 4.5 Parallel and Series Resonance Series Resonance (2nd Order RLC Circuit ) Resonance condition Resonance frequency and ——Series Resonance f0 f X

Ch4 Sinusoidal Steady State Analysis 4.5 Parallel and Series Resonance Series Resonance (2nd Order RLC Circuit ) Resonance condition: Zmin；when V=constant, I=Imax=I0。 When ， Quality Factor Q,

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   当 时 In phase with the Parallel Resonance Parallel Resonance frequency In generally Zmax Imin:

Ch4 Sinusoidal Steady State Analysis 4.5 Parallel and Series Resonance Parallel RLC Circuit   Z。 Quality Factor Q,

Ch4 Sinusoidal Steady State Analysis 4.5 Parallel and Series Resonance Parallel RLC Circuit P4.10, Find i1、 i2、 i v  i i1   i2

Ch4 Sinusoidal Steady State Analysis 4.5 Parallel and Series Resonance Parallel RLC Circuit Review For sinusoidal circuit， Series ： ？ Parallel : 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 f＝500Hz，Determine VAB before the C have been not connected . And VAB=? after parallel C= 30F Before C connected v i After C connected

Ch4 Sinusoidal Steady State Analysis 4.6 Examples for Sinusoidal Circuits Analysis RC Phase Difference P4.12, f=300Hz, R=100。 If vo-vi=/4，C=？

Ch4 Sinusoidal Steady State Analysis 4.6 Examples for Sinusoidal Circuits Analysis Low-Pass and High-Pass Filter RC---- High-Pass Filter P4.13, The voltage sources are vi=240+100sin2100t(V), R＝200， C＝50F， Determine VAC and VDC in output voltage vo. VDC=240V

Ch4 Sinusoidal Steady State Analysis 4.6 Examples for Sinusoidal Circuits Analysis Low-Pass and High-Pass Filter

Ch4 Sinusoidal Steady State Analysis 4.6 Examples for Sinusoidal Circuits Analysis Low-Pass and High-Pass Filter

Ch4 Sinusoidal Steady State Analysis 4.6 Examples for Sinusoidal Circuits Analysis

Ch4 Sinusoidal Steady State Analysis 4.6 Examples for Sinusoidal Circuits Analysis Low-Pass and High-Pass Filter

Ch4 Sinusoidal Steady State Analysis 4.6 Examples for Sinusoidal Circuits Analysis

Ch4 Sinusoidal Steady State Analysis 4.6 Examples for Sinusoidal Circuits Analysis

Ch4 Sinusoidal Steady State Analysis 4.6 Examples for Sinusoidal Circuits Analysis Complex Numbers Analysis P4.14, Find in the circuit of the following fig. v1=120sint v2  i3  i1  i2

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 v v