Chapter 9 Objectives

Chapter 9 Engr8 Circuit Analysis Dr Curtis Nelson Chapter 9 Objectives Understand the concept of a phasor; Be able to transform a circuit with a sinusoidal source into the frequency domain using phasor concepts; Know how to use the following circuit analysis techniques to solve a circuit in the frequency domain: Ohm s Law; Kirchhoff s laws; Series and parallel simplifications; Voltage and current division; Node-voltage method; Mesh-current method; Thévenin and Norton equivalents; Maximum power theorem. Engr8 - Chapter 9, Nilsson 0e

Properties of a Sinusoidal Waveform The general form of sinusoidal wave is where: v(t) = V m sin(wt +q) V m is the amplitude in peak voltage; ω is the angular frequency in radian/second, also πf; q is the phase shift in degrees or radians. Frequency Review volts period 0.8 0.6 0.4 0. 0-0. T = f -0.4-0.6-0.8-0 3 4 5 6 7 8 9 0 sec Period 6.8 seconds, Frequency = 0.59 Hz Engr8 - Chapter 9, Nilsson 0e

Amplitude Review volts 0.8 0.6 0.4 0. 0-0. -0.4-0.6-0.8-0 3 4 5 6 7 8 9 0 Peak: Blue volt, Red 0.8 volts Peak-to-Peak: Blue volts, Red.6 volts Average: 0 volts sec Phase Shift Review Period=6.8 0.8 0.6 0.4 0. 0-0. -0.4 y y blue red = sin( t) = 0.8sin( t + ) -0.6-0.8-0 3 4 5 6 7 8 9 0 Red leads Blue by 57.3 degrees ( radian) f = 6.8 360! = 57.3! Engr8 - Chapter 9, Nilsson 0e 3

More on Phase The red wave [V M sin(ωt + θ)] leads the wave in green by θ; The green wave [V M sin(ωt)] lags the wave in red by θ; The units of θ and ωt must be consistent. Basic AC Circuit Components AC Voltage and Current Sources (active elements) Resistors (R) Inductors (L) passive elements Capacitors (C) Inductors and capacitors have limited energy storage capability. Engr8 - Chapter 9, Nilsson 0e 4

AC Voltage and Current Sources Voltage Sources Current Sources + AC AC + - AC AC - AC + - 0sin(πt + π/4) Amplitude = 0V peak ω = π so F = Hz Phase shift = 45 Sinusoidal Steady State (SSS) Analysis SSS is important for circuits containing capacitors and inductors because these elements provide little value in circuits with only DC sources; Sinusoidal means that source excitations have the form V S cos(ωt + q) or V S sin(ωt + θ); Since V S sin(ωt + θ) can be written as V S cos(ωt + θ - p/), we will use V S cos(ωt + θ) as the form for our general source excitation; Steady state means that all transient behavior in the circuit has decayed to zero. Engr8 - Chapter 9, Nilsson 0e 5

Sinusoidal Steady State Response The SSS response of a circuit to a sinusoidal input is also a sinusoidal signal with the same frequency but with possibly different amplitude and phase shift. v(t) cos wave v(t) cos wave i(t) cos wave AC + - Ω 5cos(3t+π/3) 3H v L (t) cos wave Review of Complex Numbers Complex numbers can be viewed as vectors where the X-axis represents the real part and the Y-axis represents the imaginary part. There are two common ways to represent complex numbers: Rectangular form: 4 + j3 Polar form: 5 37 o jω 3 4 σ Engr8 - Chapter 9, Nilsson 0e 6

Complex Number Forms Rectangular form: Polar form: a + jb ρ θ r = a + b æ q = arctanç è a = r cosq b = r sinq b a ö ø Complex Math Rectangular Form p = a + jb q = c + jd Addition and subtraction x = p + q = (a + c) + j(b + d) y = p q = (a - c) + j(b - d) Example p = 3 + j4 q = - j x = p + q = (3 + ) + j(4 - ) = 4 + j y = p q = (3 ) + j(4 (-)) = + j6 Engr8 - Chapter 9, Nilsson 0e 7

Complex Math Rectangular Form p = a + jb q = c + jd Multiplication (easier in polar form) x = p q = ac + jad + jbc + j bd = (ac bd) + j(ad + bc) Example p = 3 + j4 q = - j x = p q = [(3)() - (4)(-)] + j[(3)(-) + (4)()] = j Complex Math Rectangular Form p = a + jb q = c + jd Division (easier in polar form) x Example p a + jb æ = = ç q c + jd è ( a + jb)( c - jd) ( c + jd)( c - jd) ö æ = ç ø è ( ac + bd ) + j( bc - ad ) ö ø = c + d p = 3 + j4 q = - j ((3)() + (4)(-)) + j( (4)() - (3)(-)) p -5+ j0 x = = = = -+ j q + (-) 5 Engr8 - Chapter 9, Nilsson 0e 8

Euler s Identity Euler s identity states that e jθ = cos(θ) + jsin(θ) A complex number can then be written as: r = a + jb = ρcos(θ) + jρsin(θ) = ρ[cos(θ) + jsin(θ)] = ρe jθ Using shorthand notation, we write this as: ρe jθ ρ θ Complex Math Polar Form x = a + jb = p = m e ( q ) j j re q = rðq q = m e j( q ) r = a + b q = tan - æ b ö ç è a ø Addition and subtraction - too hard in polar so convert to rectangular coordinates. Multiplication z = p q = m m e ( q +q ) j Example p = 6e æ p ö jç è 6 ø æ p ö jç è ø q = e z = p q = ( 6)( ) e æ p p ö jç + è 6 ø = e æ p ö jç è 3 ø p = 6Ð30 q = Ð90 z = p q = Ð0 Engr8 - Chapter 9, Nilsson 0e 9

Complex Math Polar Form x = a + jb = p = m e ( q ) j j re q = rðq q = m e j( q ) r = a + b q = tan - æ b ö ç è a ø Division m z = p q = m Example p = 6e æ p ö jç è 6 ø z = p q = e e ( q - ) j q q = e æ p ö jç è ø 6 æ p p ö æ p ö jç - jç - 6 3 è ø è ø = 3e = 3Ð - 60! More on Sinusoids Suppose you connect a function generator to any circuit containing resistors, inductors, and capacitors. If the function generator is set to produce a sinusoidal waveform, then every voltage drop and every current in the circuit will also be a sinusoid of the same frequency. Only the amplitudes and phase angles will (may) change. The same thing is not necessarily true for waveforms of other shapes like triangle or square waveforms. Fortunately, it turns out that sinusoids are not only the easiest waveforms to work with mathematically, they're also the most useful and occur quite frequently in realworld applications. Engr8 - Chapter 9, Nilsson 0e 0

Phasors A phasor is a vector that represents an AC electrical quantity such as a voltage waveform or a current waveform; The phasor's length represents the peak value of the voltage or current; The phasor's angle represents the phase angle of the voltage or current; Phasors are used to represent the relationship between two or more waveforms with the same frequency. More on Phasors Phasors are complex numbers used to represent sinusoids of a fixed frequency; Their primary purpose is to simplify the analysis of circuits involving sinusoidal excitation by providing an algebraic alternative to differential equations; A typical phasor current is represented as I = I M f For example, i(t) = 5cos(wt + 45º) has the phasor representation I = 5 45º A phasor voltage is written as V = V M f For example, v(t) = -5sin(wt + 30º) = 5cos(ωt + 0º) has the phasor representation V = 5 0º Engr8 - Chapter 9, Nilsson 0e

Phasor Equations The diagram at the right shows two phasors labeled v and v ; Phasor v is drawn at an angle of 0 and has a length of 0 units; Phasor v is drawn at an angle of 45 and is half as long as v ; In terms of the equations for sinusoidal waveforms, this diagram is a pictorial representation of the equations: v = 0 cos(ωt) v = 5 cos(ωt + 45 ) The equations above and the diagram convey the same information. Example Problems Useful trigonometric relationships: sin(wt) = cos(wt - 90º) -sin(wt) = cos(wt + 90º) cos(wt) = sin(wt + 90º) -cos(wt) = sin(wt - 90º) Express each of the following currents as a phasor:. sin(400t + 0º)A. (-7sin800t 3cos800t)A 3. 4cos(00t 30º) 5cos(00t + 0º)A. sin(400t + 0º)A = cos(400t + 0º)A = 0ºA. (-7sin800t 3cos800t)A = 7 90º - 3 0º = (0 +7j) + (-3 + 0j) = -3 + 7j = 7.66 3.ºA 3. 4cos(00t 30º) 5cos(00t + 0º) = 4-30º -5 0º = (3.464 -j) (4.70 +.7j) = -.35 3.7j = 3.9-08ºA Engr8 - Chapter 9, Nilsson 0e

Phasor Relationships for R, L, and C Now that we have defined phasor relationships for sinusoidal forcing functions, we need to define phasor relationships for the three basic circuit elements. The phasor relationship between voltage and current in a circuit is still defined by Ohm s law with resistance replaced by impedance, a frequency dependent form of resistance denoted as Z(jω). In terms of phasors, Ohm s law still applies so V = IZ(jω) where V is a phasor voltage, I is a phasor current, and Z(jω) is the impedance of the circuit element. - Note that Z is a real number for resistance and a complex number for capacitance and inductance. Since phasors are functions of frequency (ω), we often refer to them as being in the frequency domain. Phasors: The Resistor In the frequency domain, Ohm s Law takes the same form: Engr8 - Chapter 9, Nilsson 0e 3

Phasor Relationship for Inductors i(t) Finding the impedance (Z) of an inductor: di( t) v( t) = L dt i( t) = ò v( t) dt = ò Asin wtdt L L A A æ - coswt ö = ò sin wtdt = ç L L è w ø A A p = (-coswt) = (sinwt - ) wl wl AC + - Asin(ωt) Impedance of jωl Phase shift of -90º L Phasors: The Inductor By dividing the phasor voltage by the phasor current, we derive an expression for the phasor impedance of an inductor shown in the figure below. Differentiation in time becomes multiplication in phasor form: (calculus becomes algebra). Engr8 - Chapter 9, Nilsson 0e 4

Phasor Relationship for Capacitors i(t) dv( t) d( Asin wt) i( t) = C = C dt dt = AwC(coswt) A = æ ç è wc ö ø sin( wt + p ) AC + - Asin(ωt) C Impedance of /jωc Phase shift of +90º Phasors: The Capacitor Differentiation in time becomes multiplication in phasor form: (calculus becomes algebra again). Engr8 - Chapter 9, Nilsson 0e 5

Summary: Phasor Voltage/Current Relationships Time Domain Frequency Domain Calculus (real numbers) Algebra (complex numbers) Impedance We define impedance as Z = V/I or V = IZ Z R =R Z L =jωl Z C =/jωc Impedance is a complex number (with units of ohms): The real part of Z(jω) is called resistance; The imaginary part of Z(jω) is called reactance. Impedances in series or parallel can be combined using the same resistor rules that you learned in Chapter 3. Engr8 - Chapter 9, Nilsson 0e 6

Impedance Example Find the equivalent impedance, in polar form, for the circuit below if ω = 0.333 rad/sec. 3H Ω Z EQ = R + jwl = + j 3 = + j = Ð45 3! Example: Equivalent Impedance Find the impedance of the network at ω = 5 rad/s. Answer: 4.55 + j4.99 Ω Engr8 - Chapter 9, Nilsson 0e 7

Circuit Analysis Using Phasors Techniques that can be used in circuit analysis with phasors: Ohm s law; Kirchhoff s voltage law (KVL); Kirchhoff s current law (KCL); Source transformations; Nodal analysis; Mesh analysis; Thévenin's theorem; Norton s theorem; Maximum power theorem. Circuit Analysis Procedure Using Phasors Change the voltage/current sources into phasor form; Change R, L, and C values into phasor impedances; R L C R jωl /jωc Use normal DC circuit analysis techniques but the values of voltage, current, and impedance can be complex numbers; Change back to the time-domain form if required. Engr8 - Chapter 9, Nilsson 0e 8

Example Problem 9.55 - Nilsson 9E Use the node-voltage method to find V O. Answer: V O = 38.078 j8.v = 88.43-4.88º V Mesh Analysis Example Find the currents i (t) and i (t). i (t) =.4 cos(0 3 t + 9.7 ) A i (t) =.77 cos(0 3 t + 56.3 ) A Engr8 - Chapter 9, Nilsson 0e 9

Nodal Analysis Example Find the phasor voltages V and V. Answer: V = - j V and V = - + j4 V Example Problem 9.64 - Nilsson 9E Use the mesh current method to find the steady-state expression for v o if v g = 30cos(0,000t)V. Answer: v o = 56.57cos(0,000t 45º)V Engr8 - Chapter 9, Nilsson 0e 0

Example Problem Find v (t). v (t) = 34.36cos(ωt + 3.63º)V Example Problem Find v x (t) in the circuit below if v s = 0cos000t V and v s = 0sin000t V. v x (t) = 70.7cos(000t 45º) V Engr8 - Chapter 9, Nilsson 0e

Example Problem Find v X (t). v X (t) =.3cos(00t 75.96º)V Example Problem Compute the power dissipated by the Ω resistor. P Ω = 6.5mW Engr8 - Chapter 9, Nilsson 0e

Thévenin Example Thévenin s theorem also applies to phasors; use it to find V OC and Z TH in the circuit below. Answer: V oc = 6 j3 V Z TH = 6 + j Ω Example Problem 9.44 - Nilsson 9E Find the Thévenin equivalent circuit at terminals ab for v g = 47.49cos(000t + 45º ) V. V TH = 350V = 350 0º V Z TH = 00 + j00ω = 4.4 45º Ω Engr8 - Chapter 9, Nilsson 0e 3

Example Problem Find the Thévenin equivalent circuit at terminals ab. V TH = -50 + j50 = 58. 08.43º V Z TH = j50ω Example Problem 9.45 - Nilsson 0 th Use source transformations to find the Thévenin equivalent circuit with respect to terminals a and b. V TH = 8 + j6 V, R TH = 00 j00ω V TH = 8.97 8.43º V, R TH = 3.6-6.56ºΩ Engr8 - Chapter 9, Nilsson 0e 4

Chapter 9 Summary From the study of this chapter, you should: Understand phasor concepts; Be able to transform a circuit with a sinusoidal source into the frequency domain using phasor concepts; Know how to use the following circuit analysis techniques to solve a circuit in the frequency domain: Ohm s Law; Kirchhoff s laws; Series and parallel simplifications; Voltage and current division; Node-voltage method; Mesh-current method; Thévenin and Norton equivalents; Maximum power theorem. Engr8 - Chapter 9, Nilsson 0e 5