Electric Circuit Analysis: Sinusoidal Steady-State Analysis

ES 2210 - Electric Circuit Analysis

Chapter 9 - Sinusoidal Steady-State Analysis

Chapter 9 - Objectives

• The student should:

  • Be able to perform a phasor transform of a sinusoid and its phasor inverse.

  • Be able to phasor-transform a circuit.

  • Solve arbitrarily complex circuits with sinusoidal sources using the phasor method.

  • Note: Sections 9.10 and 9.11 (related to transformers) will not be covered in ES 2210, but later in EE 2220.

Why Should You Care?

• Response of a circuit to a sinusoidal source (i.e., an A.C. source) is important for three reasons:

  • Importance of electrical power generation and distribution.

  • Ability to predict behavior to a non-sinusoidal source if behavior to a sinusoidal source is known, utilizing Fourier series and Fourier transforms.

  • Electrical circuit design requirements and specifications are often based on response to sinusoidal input.

Sinusoidal Steady-State Analysis Overview

• The process for solving sinusoidal steady-state analysis includes:

  • Electrical Circuit Differential Equation Classical Solution Methods

  • Solution Transform using complex numbers

  • Algebraic Solution

  • Inverse Transform

  • The chapter 9 method will be used for analysis.

Complex Numbers: A Quick Review

• Definition of j:

  • Defined as $j = <br>oot{-1}$

  • Properties of j:

  • $j^2 = -1$

  • $j^3 = j^2 imes j = -j$

  • $j^4 = j^2 imes j^2 = 1$

• Complex numbers are common in almost all engineering disciplines and a more detailed review can be found in Appendix B of the textbook.

Structure of Complex Numbers

• A complex number is expressed as:

  • $C = a + jb$

  • Where:

  • a = Real Part

  • b = Imaginary Part

  • The notation does not indicate addition but signifies association between C, a, and b:

  • Real Part: ext{Re}ig"{C} = a

  • Imaginary Part: ext{Im}ig"{C} = b

Representation of Complex Numbers

• Rectangular Form:

  • Graphically represented in the complex plane:

  • Imaginary axis (b)

  • Real axis (a)

• Polar Form:

  • Represented as:

  • $C = r ext{∠} heta$

  • Where r is the magnitude and $ heta$ is the angle from the real axis.

• Conversion Forms:

  • Rectangular to Polar conversion:

  • $r = <br>oot{(a^2 + b^2)}$

  • $heta = an^{-1}\bigg( rac{b}{a}\bigg)$

  • Polar to Rectangular conversion:

  • $a = r ext{cos}( heta)$

  • $b = r ext{sin}( heta)$

Euler's Formula

• A critical relationship summarized as:

  • $r ext{∠} heta = r ext{cos}( heta) + j r ext{sin}( heta) = r e^{j heta}$

• Used frequently in Electrical Engineering and disciplines.

Special Cases of Complex Numbers

• Positive Real Number: $a = a + j0 = a ext{∠}0^ ext{◦}$

• Positive Imaginary Number: $b = 0 + jb = b ext{∠}90^ ext{◦}$

• Negative Real Number: $-a = -a + j0 = a ext{∠}180^ ext{◦}$

• Negative Imaginary Number: $-b = 0 - jb = b ext{∠}-90^ ext{◦}$

• Conjugate: If $C = a + jb$, then $C^* = a - jb$

Operations with Complex Numbers

• Addition and Subtraction in Rectangular Form:

  • Let $C = a + jb$ and $Z = x + jy$

  • Addition: $C + Z = (a + x) + j(b + y)$

  • Subtraction: $C - Z = (a - x) + j(b - y)$

• Multiplication in Rectangular:

  • $C imes Z = (a + jb)(x + jy)$

  • Distribution results in:

  • $= ax + j(ay + bx) - by$

• Division in Rectangular Form:

  • $rac{C}{Z} = rac{a + jb}{x + jy}$

  • Multiply numerator and denominator by the conjugate: $rac{(a + jb)(x - jy)}{(x + jy)(x - jy)}$

  • Result in terms of real and imaginary components gives:

  • $rac{a x + b y}{x^2 + y^2} + j rac{b x - a y}{x^2 + y^2}$

Addition and Subtraction in Polar Form

• In polar form:

  • Let $C = r<em>1∠θ</em>1$ and $Z = r<em>2∠θ</em>2$

  • Convert to rectangular form for addition and subtraction.

Multiplication and Division in Polar Form

• Multiplication: For two phasors,

  • $C imes Z = r<em>1 r</em>2 ∠(θ<em>1 + θ</em>2)$

• Division:

  • $rac{C}{Z} = rac{r<em>1}{r</em>2} ∠(θ<em>1 - θ</em>2)$

Arithmetic for Complex Numbers

• Recommended strategies:

  • Use rectangular form for addition and subtraction.

  • Use polar form for multiplication and division.

Powers of a Complex Number

• For any integer n,

  • If $C = a + jb$

  • $C^n = (re^{jθ})^n = r^n e^{jnθ}$

  • Which can be expressed as: $C^n = r^n ext{cos}(nθ) + j r^n ext{sin}(nθ)$

Sinusoidal Source Terminology

• A sinusoidal source is given by:

  • $v(t) = V_m ext{cos}( ext{ω}t + ext{ϕ})$

  • Where:

    • $V_m$: Amplitude or magnitude

    • $ext{ω}$: Angular frequency in radians per second $[ ext{ rad/s}]$

    • $t$: Time in seconds $[ ext{ s}]$

    • $ext{ϕ}$: Phase angle in degrees $[ ext{ deg}]$

    • $f = rac{ ext{ω}}{2 ext{π}}$: Frequency in Hertz $[ ext{ Hz}]$

    • $T = rac{1}{f}$: Period in seconds $[ ext{ s}]$

Observations about Sinusoids

• A sinusoidal function is periodic, repeating every $T$ seconds.

• Frequency:

  • Cycles per second: $f = rac{1}{T}$

• Angular frequency:

  • $ext{ω} = 2 ext{π}f = rac{2 ext{π}}{T}$

• The maximum amplitude is $V_m$.

• The phase angle $ext{ϕ}$ shifts the sinusoid along the time axis.

  • $v<em>s(t) = V</em>m ext{sin}( ext{ω}t + ext{ϕ})$

Sinusoid Frequency and Phase Angle Questions

• Frequency for $i(t) = 36 ext{cos}(4 ext{π}t + 45°) mA$ is $f = 2 ext{Hz}$.

• Phase angle for $v(t) = 50 ext{cos}(3000t - 60°) V$ is $ext{ϕ} = -60°$.

Evaluating Sinusoids at Specific Times

• To find a sinusoidal current at $t = 5 ms$ for $i(t) = 0.2 ext{cos}(50 ext{π}t + 45°) A$

  • Convert all arguments to the same units:

  • Example: $50 ext{π}t$ gives radians and needs to be carefully factored with respect to degrees (conversion method shown).

  • Need careful setting for calculator units!

• Result: At $t = 0.005 s$:

  • $i(0.005) = 0.2 ext{cos}(90°) = 0 A$

Sinusoid Analysis in Circuits

• When voltage $v = V<em>m ext{sin}( ext{ω}t)$ and current $i = I</em>m ext{sin}( ext{ω}t + ext{ϕ})$:

• We indicate current leads voltage by the phase angle $ext{ϕ}$ or vice-versa.

Current Response for Sinusoidal Sources in Circuits

• Current $i(t)$ responds according to defined circuit laws, with the phasor representation providing shortcuts for analysis.

• Transient Phase Component:

  • This component decays to zero as $t o ext{∞}$.

• Steady-State Phase Component:

  • Persistently represents system behavior influenced by input voltage.

  • Has the same frequency as the input voltage, though potentially differing in magnitude and phase angle.

Circuit Analysis Techniques

• Fundamental components of circuit analysis techniques:

  • Ohm’s law

  • Kirchhoff's voltage law (KVL)

  • Kirchhoff's current law (KCL)

• Using these laws in the time domain leads to effective analytical techniques that can be employed in the phasor domain as well.

  • Relation between voltage and current in resistors, inductors, and capacitors under dynamic conditions is defined succinctly by their impedance

  • Impedance: Defined as the ratio of voltage to current, represented with units of Ohms $[ ext{Ω}]$

Impedance of Circuit Elements

• Resistor:

  • $Z_R = R$

• Inductor:

  • $Z_L = j ext{ω}L$

• Capacitor:

  • $Z_C = rac{1}{j ext{ω}C} = -j rac{1}{ ext{ω}C}$

Summary of Sinusoidal Steady-State

• In the Time Domain:

  • Resistor: $v(t) = R i(t)$

  • Inductor: $v(t) = L rac{di(t)}{dt}$

  • Capacitor: $i(t) = C rac{dv(t)}{dt}$

• In the Phasor Domain:

  • $V = Z I$

• Phasor domain allows analysis methods of circuit design under sinusoidal steady-state conditions.

AC Steady-State Analysis Steps

• Steps in AC steady-state analysis include:

  1. Redraw the circuit (phasing does not change components).

  2. Phasor transform known voltages and currents.

  3. Use $V$ and $I$ for unknown voltages and currents.

  4. Replace inductance and capacitance with their respective impedances.

  5. Write equations using circuit analysis methods, solved with a calculator.

  6. Perform inverse transformation back to the time domain.

Example Problem Summaries & Techniques

• Iterative evaluation of current and voltage in sinusoidal forms, calculation of impedances, and applying fundamental circuit equations illustrate how to achieve operational results effectively.

• Conduct operations meticulously, especially when transforming to and from phasor forms to ensure consistency of results.

• Analysis using source transformations is beneficial for simplifying complex interactions and extracting values.

• Utilize any DC circuit methods learned to derive phasor responses efficiently—fundamental understanding of electrical interactions remains key to mastering circuit behavior.