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 =
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 =
oot{(a^2 + b^2)}heta = an^{-1}igg( rac{b}{a}igg)
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 = r1∠θ1 and Z = r2∠θ2
Convert to rectangular form for addition and subtraction.
Multiplication and Division in Polar Form
Multiplication: For two phasors,
C imes Z = r1 r2 ∠(θ1 + θ2)
Division:
rac{C}{Z} = rac{r1}{r2} ∠(θ1 - θ2)
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.
vs(t) = Vm 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 = Vm ext{sin}( ext{ω}t) and current i = Im 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:
Redraw the circuit (phasing does not change components).
Phasor transform known voltages and currents.
Use $V$ and $I$ for unknown voltages and currents.
Replace inductance and capacitance with their respective impedances.
Write equations using circuit analysis methods, solved with a calculator.
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.