AC Circuits Basics: Capacitors & Inductors Analysis Guide

1. Energy UnitAC CircuitsSchool of Electrical, Electronic and Computer Engineering AC Circuits Study: G. Rizzoni, J. Kearnes, 6th Ed.Principles and Applications of Electrical Engineering Chapter 4: AC Network Analysis

2. Key Concepts: Fundamentals and Techniques (1) • What is a Capacitor? • A capacitor (denoted by C)is a passive element designed to store energy in its electric field • Its current-voltage relationship is described by a differential equation: i= C (dv/dt). The current through a capacitor is proportional to the rate of change in voltage across the capacitor • No current through capacitor if voltage is static (acts like open-circuit)  Need an alternating or sinusoidal voltage across capacitor • What is an Inductor? • An inductor (denoted by L) is a passive element designed to store energy in its magnetic field. • Its voltage–current relationship is described by a differential equation: v = L (di/dt). The voltage across an inductor is proportional to the rate of change in current through the inductor • No voltage across the inductor if current is static (acts like short-circuit)  Need an alternating or sinusoidal current through inductor

3. Key Concepts: Fundamentals and Techniques (1) Key Concepts: Fundamentals and Techniques (1) • Phasor Analysis / Impedance Model / AC Circuits • How to analyse circuits excited by harmonic / sinusoidal excitation using complex numbers  phasors • Converting between time-domain and phasor-domain • How L and C components behave (the impedance model) • Using standard analysis (KCL, KVL, impedance model) for the steady-state response of circuits excited by sinusoids, but with complex equations • Impedance Combination • How to combine multiple series-connected/parallel-connected impedances • Implementing voltage-division/current-division in series/parallel ac circuits • Wye-delta and delta-to-wye transformations for complex impedances • The wye-delta and delta-to-wye transformations that we applied to resistive circuits are also valid for impedances.

4. Capacitors and Inductors • An electrical element is defined by its relation between v and i. This is called a constitutive relation. In general, we write • For a resistor, v = i * R • The constitutive relation of a resistor has no dependence upon time. v = f(i) or i = g(v) i + v–

5. Capacitors Capacitors • What is a Capacitor? • A capacitor is a passive element designed to store energy in its electric field. • A capacitor consists of two conducting plates separated by an insulator. • When a voltage source is connected to the capacitor, the source deposits a positive charge q on one plate and a negative charge on the other. • Circuit symbol for capacitors

6. Capacitors • Current-voltage relationship of capacitor i = C where the proportional constant C is capacitance(i unit is Farad or F) This relationship states that • currentthrough a capacitor is proportional to the rate of change in voltage across the capacitor, and • no current through capacitor if voltage is static (after initial phase)(acts like open-circuit)  Need an alternating or sinusoidal voltage across capacitor  AC analysis. • Voltage-current relationship of capacitor v = * t0tidt + v(t0) • The energy stored in the capacitor is w = C * v2

7. Capacitors Capacitor DC Response Source: Rizzoni, Kearns, Electrical Engineering

8. Inductors Capacitor DC Response Source: Hyperphysics GSU

9. Capacitors Capacitors Ceq = C1 + C2 + … + CN 1/Ceq = 1/C1 + 1/C2 + … + 1/CN • Series and Parallel Capacitors • The equivalent capacitance of N parallel-connected capacitors is the sum of the individual capacitances. • The equivalent capacitance of Nseries-connected capacitors is the reciprocal of the sum of the reciprocals of the individual capacitances.

10. Inductors Inductors • What is an Inductor? • An inductor is a passive element designed to store energy in its magneticfield. • An inductor consists of a coil of conducting wire. • Inductance (L) of an inductor is the property that an inductor exhibits oppositionto the change of current flowing through it, measured in henrys (H). • Circuit symbol for inductor

11. Inductors Inductors • voltage-current relationship of inductor v = L * where the proportional constant L is inductance(unit is Henry or H). This relationship states that • voltage across the inductor is directly proportional to the time rate of change of the current, and • If iL(t) = IL (does not change with time) then vL = 0 and L acts like a short-circuit. For L to do anything useful iL(t) needs to change with time, one such case is when iL(t)is alternating sinusoidally AC analysis • current-voltage relationship of inductor i = -∞ t v(t) dt • The power delivered to the inductor is: p = v * i = (L * di/dt) * i • The energy stored in the capacitor is: w = L * i2

12. Inductors Inductor DC Response Source: allaboutcircuits.com

13. Inductors Inductor DC Response Source: Hyperphysics GSU

14. Inductors Inductors Leq = L1 + L2 + … +LN 1/Leq = 1/L1 + 1/L2 + … + 1/LN • Series and Parallel Inductors • The equivalent inductance of series-connected inductors is the sum of the individual inductances. • The equivalent inductance of parallel inductors is the reciprocal of the sum of the reciprocals of the individual inductances.

15. Inductors Review Source: Rizzoni, Kearns, Electrical Engineering

16. Inductors Waveforms Source: Rizzoni, Kearns, Electrical Engineering

17. +1 0 –1 Inductors Phasor Consider a sinusoidal function A* ei(ωt+θ) amplitude frequency initial phase (all constant) Re(A * ei(ωt+θ)) = A * cos(ωt+θ) Im(A* ei(ωt+θ)) = A * sin(ωt+θ) Phasormeans staticcomplex vector (t=0): P = A* eiθ Notation: = A ∠θ angle time Source: Wikipedia, augmented by TB

18. Sinusoidal Analysis and Phasors Consider v(t) = Vm * cos(wt +f) Then v(t) = Re (Vm* ej(wt +f)) = Re (Vm * ejf* ejwt) phasorV = Vm * ejf= Vm∠fof sinusoid v(t) with frequency w = Re (V * ejwt ) }

19. Sinusoidal Analysis and Phasors Some handy relationships: Better: θ = atan2(X, R) Phasors are complex numbers  study Polar/Cartesian representations of complex numbers and how to add/multiply

20. Phasors and Impedance Models We define impedanceas: Z = V / I and admittanceas: Y = I / V We will write i(t) = Im * cos(wt+F) in phasor form I = Im∠F Similarly v(t) = Vm* cos(wt +F) in phasor form V = Vm∠F Node to Datum analysisand Parallel/Series reductions can be applied to circuits where resistance/conductance is replaced by impedance/admittance.These will be complex equations!

21. Phasors and Impedance Models Resistor v(t) = R * i(t) = R * Im* cos(wt + ϕ) The phasor form for the voltage is then (t=0): V = R * Im∠ϕ = R * I Hence we have: Z = R = R ∠0o and Y = (1/R) ∠0o

22. Phasor Model for Resistor Resistor Phasor diagram for the resistor; Iand V are in phase. Voltage-current relations for a resistor in (a) time domain, (b) frequency domain

23. Phasor Model for Inductor Remember: i(t) = Im* cos(wt + ϕ) (t=0) j ∠𝛷 90°

24. Phasor Model for Inductor Phasor diagram for the inductor; I lags V Voltage-current relations for an inductor in (a) time domain, (b) frequency domain

25. Phasor Model for Capacitor Remember: v(t) = Vm * cos(wt +F) j ∠𝛷 90°

26. Phasor Model for Capacitor Phasor diagram for the capacitor; I leads V Voltage-current relations for a capacitor in (a) time domain, (b) frequency domain

27. Voltage-Current over Time ResistorCapacitorInductor t t t supply voltage V = vR ≈ iR Supply voltage V = Vm * cos(wt) Voltage over element Current through element Note: For C and L there will generally also be a phase shift between supply voltage and element voltage/current

28. Phasor Relationships for Circuit Elements Summary of voltage-current/current-voltage relationships for resistor, inductor and capacitor V = R * I V = jωL * I I = jωC * V V = * I

29. Impedance and Admittance • The impedanceZ of a circuit is the ratio of the phasor voltage Vto the phasor current I, measured in ohms (W): Z = • The impedances of resistors, inductors, and capacitors are:V / I = R, V / I = jwL, V / I = • As a complex quantity, the impedance may be expressed in rectangular form as Z = R + jXwhere R is the resistance and X is the reactance. • The impedance may also be expressed in polar form asZ = |Z| ∠θ • Thus, we could obtain:|Z| = √(R2 + X2) , θ = tan-1(X / R) andR = |Z| * cos(θ), X = |Z| * sin(θ)

30. Impedance and Admittance • The admittance Yof an element (or a circuit) is the ratio of the phasor current through it to the phasor voltage across it. Equivalently, the admittance is the reciprocal of impedance, measured in siemens(S = 1/W).Y = 1 / Z = I / V • As a complex quantity, we may write Y asY = G + jBwhereG= Re(Y) is called the conductanceandB = Im(Y) is called the susceptance.

31. REVIEW Complex Numbers REVIEW Complex Numbers Source: Wikipedia

32. REVIEW Complex Numbers REVIEW Complex Numbers • Addition • (a+bi) + (c+di) = (a+c) + (b+d)*i • Multiplication • (a+bi) * (c+di) = (ac–bd) + (bc+ad)*i • Conjugate • z’ = Re(z) – Im(z)*i • Note • i2 = i * i = –1 • 1/i = i/i2 = –i Source: Wikipedia

33. REVIEW Complex Numbers Complex polar conversion Remember Pythagoras: r = √(x2 + y2) tan θ = y/xθ = atan2(y,x) x = r * cosθ y = r * sin θ x + yj= r*(cosθ + j sinθ) = r ∠θ cartesian polar Source and practice:http://www.intmath.com/complex-numbers/4-polar-form.php

34. Phasor Arithmetic Better: ϕ = atan2(B,A) https://en.wikibooks.org/wiki/Circuit_Theory/Phasor_Arithmetic

35. Phasor Arithmetic https://en.wikibooks.org/wiki/Circuit_Theory/Phasor_Arithmetic

36. Phasor Arithmetic Multiplication Division . https://en.wikibooks.org/wiki/Circuit_Theory/Phasor_Arithmetic

37. Phasor Arithmetic Inversion Property Complex Conjugate. https://en.wikibooks.org/wiki/Circuit_Theory/Phasor_Arithmetic

38. Simple Example Make node 3 the datum/ground and apply KCL at node 2: Need to solve a differential equation For the special caseVm= 10V, w = 100Hz, F=30° of a sinusoidal excitation input V = v(t) = 10* cos(100*t + 30°)we can use phasorsto calculate the steady-state solution. Assume R = 1kΩ, C = 20μF Convert excitation to phasors and components to impedances: given case

39. Simple Example Let’s calculate the current. Same techniques as with resistive circuits except that you now use complex numbers

40. Simple Example We have I = 8.9mA∠57° (for case F=30°) Convert phasor solution back to time domain (w=100Hz) i(t) = 8.9mA * cos(100*t + 57°) Voltage across the capacitor VC = ZC* I = –j*0.5kΩ * 8.9mA∠57° = 0.5kΩ∠–90° * 8.9mA∠57° = 4.5V∠–33° Convert back to time domain vc(t) = 4.5V * cos(100t – 33°) Voltage across resistor VR = ZR* I = 1kΩ∠0° * 8.9mA∠57° = 8.9V∠57° Note: VC + VR = Vsupply Note: 90° phase shift!

41. Simple Example over Time Capacitor+Resistor (Vsupply=10V∠30°; I=8.9mA∠+57°; VC=4.5V∠–33°) ∠30° Supply voltage V = Vm * cos(wt) 1st Current through capacitor 2nd Voltage over capacitor Real Values∠30° I= 8.9mA*cos(57°) = 8.9*0.55mA = 4.8mA VC= 4.5V*cos(–33°) = 4.5*0.84V = 3.8V t

42. Simple Example over Time Capacitor+Resistor Check for Vsupply∠0° I = V/ZT = 10V∠0° / (1’118Ω∠–27°) = 8.945mA ∠+27° VC = ZC * I = 0.5kΩ∠–90° * 8.945mA∠27° = 4.5V∠–63° ∠0° Real Values ∠0° I= 8.945mA*cos(27°) = 8.945*0.89mA = 8.0mA VC= 4.5V*cos(–63°) = 4.5*0.45V = 2.0V t Note: 90° phase shift!

43. Simple Example over Time Capacitor+Resistor Check VC+ VR = Vsupplyfor ∠0° Vsupply= 10V * cos(0°) = 10.0V I = 8.945mA * cos(27°) = 8.0mA VC = 4.5V * cos(–63°) = 2.0V VR = 1kΩ * 8mA = 8.0V ∠0° t Sum is 10V ! Always in sync with current!

44. Impedance Combinations • Consider the N series-connected impedances shown below • If N=2, as shown below, • The equivalent impedance at the input terminals is • Zeq = V / I = Z1 + Z2 + …+ ZN Use the voltage-division relationship V1 = Z1 / (Z1+Z2) * V and V2 = Z2 / (Z1+Z2) * V

45. Impedance Combinations • Consider the N parallel-connected impedances shown below • If N=2, as shown below, • The equivalent impedance at the input terminals is • 1/Zeq = 1/V = 1/Z1 + 1/Z2 + … + 1/ZN • or:The equivalent admittance at the input terminals is • Yeq = Y1 + Y2 + … + YN • Usethecurrent-division relationship: • I1 = Z2 / (Z1 + Z2) * I • I2 = Z1 / (Z1 + Z2) * I

46. Wye-to-delta transformation Wye-to-delta transformation • The wye-to-delta transformation that we applied to resistive circuits is also valid for impedances. Y-D Conversion A delta or wye circuit is said to be balanced if it has equal impedances in all three branches.

47. Delta-to-wye transformation Wye-to-delta transformation • The delta-to-wye transformation that we applied to resistive circuits is also valid for impedances. D-Y Conversion A delta or wye circuit is said to be balanced if it has equal impedances in all three branches.