Circuit Analysis using Complex Exponentials Physics 121 - Electricity and Magnetism Lecture 15E - AC Circuits & Resonance II No Y&F reference for slides on Complex analysis Circuit Analysis using Complex Exponentials Imaginary and Complex Numbers Complex Exponential Phasors and Rotations Phasors as Solutions of Steady State Oscillator Equations Phasor representation applied to current versus voltage in R. L. C. Series LCR Circuit using Complex Phasors Parallel LCR Circuit using Complex Phasors Transient Solution of Damped Oscillator using Complex Phasors

Summary: AC Series LCR Circuit Z Im Em F wDt VL-VC VR XL-XC R sketch shows XL > XC L C E vR vC vL VL = ImXL +90º (p/2) Lags VL by 90º XL=wdL L Inductor VC = ImXC -90º (-p/2) Leads VC by 90º XC=1/wdC C Capacitor VR = ImR 0º (0 rad) In phase with VR R Resistor Amplitude Relation Phase Angle Phase of Current Resistance or Reactance Symbol Circuit Element

Complex Exponentials in Circuit Analysis is “pure imaginary” Roots of a quadratic equation az2 + bz +c = 0 are complex if b2<4ac: Complex number in rectangular form: REVERSE SIGN OF IMAGINARY PART Re{z} and Im{z} are both real numbers Complex conjugate: Addition: EE’s use j instead of i (i is for current)

Representation using the Complex Plane Imaginary axis (y) Real axis (x) x y |z| q Polar form: “Argand diagram” Magnitude2: Argument: Picture also displays complex-valued functions

Complex Exponentials: Euler’s Formula Taylor’s Series definition of exponential function: Evaluate for u = iq: use i2n+1 = i(-1)2n Recognize series definitions of sine and cosine: Hence: Above is a complex number of magnitude 1, argument q: Special Properties:

Rules for Complex Quantities Euler’s Formula for magnitude r: Polar form of arbitrary complex z Complex conjugate of z Sum: Product: Magnitude of a product = product of individual magnitudes Argument of a product = sum of individual arguments Two complex entities are equal if and only if their amplitudes (magnitudes) are equal and their arguments (phases) are equal Periodicity 2p: DeMoivre’s Theorem: for integer n & k = 0,1….n-1

Phasors, Rotation Operators, Evolution Operators: Im Re q -f +f e+if rotates complex number z by angle +f in complex plane Time Evolution: Let f = w(t-t0) evolves z from t0 to t for tf > t0 A(t) is called a “time domain phasor” Re {A} = is the measurable instantaneous value of A(t) Time dependent exponentials like eiwt are sometimes called evolution operators Cancelling common factors of eiwt leaves a “frequency domain phasor” A (often simply called a “phasor”): Represent sinusoidally varying real quantity a(t) as a vector in the complex plane, rotating (counterclockwise) at angular frequency w.

Phasor Definitions, continued A phasor represents a sinusoidal, steady state signal whose amplitude Amax, phase f, and frequency w are time invariant. peak value of a(t), real frequency domain phasor A, complex, rotated by f from real axis time domain phasor, complex instantaneous value, time domain, real Phasor Transform P: forward - time to frequency domain inverse - frequency to time domain Advantage of phasor transform: For sinusoidal signals, differential equations (time domain) become algebraic equations (frequency domain) as common factors of eiwt cancel). Process: Replace real variables in time dependent analysis problem with variables written using complex exponentials. Cancel common factors of eiwt and solve remaining algebraic problem. Then return to real solution in time domain.

Time domain phasors are alternative solutions to sines and cosines in differential equations representing oscillations. Recall: a can be complex chain rule: First and Second Derivatives: set a = wt+f, amplitude Amax Example: Simple Harmonic Oscillator Trig Solutions: Complex Solutions:

Apply complex voltage & current (time domain phasors): Complex Exponential Representation applied to Passive Circuit Elements: Revisit AC Voltage vs. Current in L, C, and R Apply complex voltage & current (time domain phasors): Note: now using j = sqrt(-1) vR( t ) i(t) magnitudes: exponents: Resistor: Inductor: vL(t) L i(t) magnitudes: exponents: i(t) vC ( t ) C Capacitor: magnitudes: exponents:

Summary: Complex Exponential Representation of AC Voltage vs Summary: Complex Exponential Representation of AC Voltage vs. Current in L, C, and R & current: Applied voltage: time domain Phase angle factor ejF rotates E CCW from current phasor Passive circuit elements: Resistor: Inductor: Capacitor: VR& IRm in phase VL leads ILm by p/2 VC lags ICm by p/2 e0 =1 e+jp/2 = +j e-jp/2 = -j Phasor rotations of voltage drops relative to currents: Voltage drop phasors relative to currents: time domain Frequency domain (factor out time dependence) Define: Complex impedance z = E(t)/i(t) and |z| = Vmax/Imax Impedances of simple circuit elements (complex): Magnitudes of Impedances |z| = [zz*]1/2 (real)

Complex Impedance z Applied voltage: & current: time domain Definition: complex impedance z (or simply impedance) is the ratio of the (complex) voltage phasor to the (complex) current phasor (in time or frequency domain). Sketch shows positive Im{z}. is positive, implying that Em leads Im. Definition: Phase angle F measures rotation of the applied voltage referenced to the current in the branch. It is also the angle between z and the real axis. The rotation operator is ejF. Z F Im(z) Re(z) Re Im Em 1/z occurs when analyzing parallel branches:

Impedances of Series or Parallel Collections of Circuit Elements Impedances of individual passive circuit elements: The kth sub-circuit (arbitrary complexity) consists of R, L, and/or C basic elements: zk Assertion: Follow series or parallel resistor addition rules to compute equivalent impedances (complex) Series branch formula: zk z2 z1 Parallel sub-circuit formula: z1 z2 zk

Revisit series LCR circuit using complex phasors AC voltage: Current: time domain phasors R L C E vR vC vL Currents are the same everywhere in an essential branch, same phase, same magnitude. Kirchhoff Loop rule for series LRC circuit: Substitute the time domain voltage phasors for vR, vL, vC. Divide equation (1) by i(t) ( = Imejwt ) Magnitude of Z: Im Em F wDt VL VC VR Sketch shows F positive for VL>VC  XL>XC A t = 0 sketch would show phasors in frequency domain

Revisit the series LCR circuit, Continuation Impedance z (Equation 1) can also be found by summing the impedances of the 3 basic circuit elements in series LCR circuit (invoke Series Formula*) Z F XL-XC R Re Im z is not a phasor, as it is time independent. Previous “phasor diagrams” showed z rotating with Em. Sketch shows F >0 for XL > XC  VL > VC. F is positive when Im{z} is positive Phase angle F for the circuit: Power factor: * The equivalent (complex) impedance for circuit elements in series (arbitrarily many) is the sum of the individual (complex) impedances. Resonance: As before, z is real for XL= XC (w2 = 1/LC). |z| is minimized. Current amplitude Im is maximized at resonance Phase angle in terms of admittance 1/z:

Parallel LCR circuit using complex phasors vC vL R L C E vR iR iL iC i a b All steady state voltages and currents oscillate at driving frequency wD AC voltage: Instantaneous Current: Kirchhoff Loop Rule (time domain): Instantaneous voltages across parallel branches have the same magnitude and phase: 2 essential nodes “a” & “b” 4 essential branches, Not all independent Common voltage phase, all branches: Common voltage peak, all branches: Use voltage as reference instead of current Kichhoff Current Rule (node a or b): Current Amplitudes in each parallel branch (reference now to voltage drop):

Parallel LCR circuit, Continuation #1 IRm Im Em, VR, VL, VC F wDt ICm Re ILm ICm- ILm Instantaneous currents in each branch lead, lag, or are in phase with the (reference) voltages: in phase with E(t) lags E(t) by p/2 leads E(t) by p/2 Current amplitude addition rule is Pythagorean Sketch shows F < 0 (applied voltage lags current) for ICm > ILm  XL>XC Substitute currents into junction rule equation (1): Cancel ejwt factor and multiply by e-jF (current phasor – frequency domain)

Parallel LCR circuit, continuation #2 Recall, impedance “admittance” Substitute, then divide (Eq. 2.1) above by Em Note that: Admittance 1/z in Equation 3 is also the sum of the reciprocal impedances of the 3 basic circuit elements in parallel LCR circuit (invoke Parallel Formula) Find |1/z|: multiply 1/z by complex conjugate (1/z)* and take square root: Represent z in terms of 1/z:

Parallel LCR circuit, Continuation #3 Sketch shows F negative for 1/XC > 1/XL  IC > IL  XL > XC 1/z F 1/XC-1/XL 1/R E Im Phase angle F: For XL > XC: 1/XL – 1/XC < 0 -Tan (F) and F are positive in Series LCR circuit (see above), Voltage Em leads current Im - Tan(F) and F are negative in Parallel LCR circuit Current Im leads applied voltage Em Converse for XC > XL F = 0 at resonance (XL = XC) in both Series and Parallel circuits

Parallel LCR circuit, Continuation #4 Resonance: Minimizes as function of frequency Minimum of 1/|z| when XL = XC, i.e. when w2 = 1/LC Same resonant frequency as series LCR, but current is minimized instead of maximized at resonance At resonance, the current amplitudes ILm and ICm in the L & C branches are equal, but are 180o apart in phase. These cancel at all times at nodes a and b of the circuit.

Using complex exponentials to solve a differential equation Revisit Damped Oscillator: After “step response” to switch at ‘a’ saturates, turn switch to ‘b’. Decaying natural oscillations start when damping is weak. a L C E + b R R dissipates potential energy not a steady state system solutions not simple sinusoids second order equation for Q(t) – see Lect. 13 Trial solution… … but if w is real, solution oscillates forever wx and wy assumed real  Assume complex frequency Derivatives: Substitute into Equation (1): Cancel common factors of Q: “characteristic equation” Phasor-like trial solution (2) turned differential equation into polynomial equation

Using complex exponentials to solve a differential equation, #2 Equation (3) becomes 2 separate equations for real and imaginary terms Expand: Re{Eq 3}  Substitute wy: Im{Eq 3}  Shifted natural frequency wx is real for underdamping imaginary for overdamping For real wx (underdamping): damping oscillation For critical damping, wx = 0: No oscillations, decay only:

Using complex exponentials to solve a differential equation, #3 For overdamping: wx becomes imaginary. Equations 4.1 & 4.2 invalid Return to Eq. 3.0. Assume frequency is pure imaginary ( no oscillation) Eq. 3.0 becomes: quadratic, real coefficients Solution: two roots both pure imaginary Both roots lead to decay w/o oscillation + root: w+ implies damping faster than e-Rt/2L - root: w- implies damping slower than e-Rt/2L but not growth Most general solution: linear combination of Q+ and Q-, each of form of Eq. 2.0 Recall definition, hyperbolic cosine: Correctly reduces to critically damped Eq. 8.0