Alexander-Sadiku
Fundamentals of Electric Circuits

	Chapter 9
	Sinusoidal Steady-State Analysis

Sinusoids and Phasor
Chapter 9

	9.1 Motivation
	9.2 Sinusoids’ features
	9.3 Phasors
	9.4 Phasor relationships for circuit elements
	9.5 Impedance and admittance
	9.6 Kirchhoff’s laws in the frequency domain
	9.7 Impedance combinations

How to determine v(t) and i(t)?

9.2 Sinusoids (1)

	A sinusoid is a signal that has the form of the sine or cosine function.
	A general expression for the sinusoid,
	where
	Vm  =  the amplitude of the sinusoid
	ω  = the angular frequency in radians/s
	Ф =  the phase

9.2 Sinusoids (2)

9.2 Sinusoids (3)

9.2 Sinusoids (4)

9.3 Phasor (1)

	A phasor is a complex number that represents the amplitude and phase of a sinusoid.
	It can be represented in one of the following three forms:

9.3 Phasor (2)

	Example 3
	Evaluate the following complex numbers:
		a.
		b.

9.3 Phasor (3)

	Mathematic operation of complex number:
	Addition
	Subtraction
	Multiplication
	Division
	Reciprocal
	Square root
	Complex conjugate
	Euler’s identity

9.3 Phasor (4)

	Transform a sinusoid to and from the time  domain to the phasor domain:
	(time domain) (phasor domain)

9.3 Phasor (5)

	Example 4
	Transform the following sinusoids to phasors:
		i = 6cos(50t – 40o) A
		v = –4sin(30t + 50o) V

9.3 Phasor (6)

	Example 5:
	Transform the sinusoids corresponding to phasors:

9.3 Phasor (7)

	The differences between v(t) and V:
	v(t) is instantaneous or time-domain representation V is the frequency or phasor-domain representation.
	v(t) is time dependent, V is not.
	v(t) is always real with no complex term, V is generally complex.
	Note: Phasor analysis applies only when frequency is  constant; when it is applied to two or more sinusoid signals only if they have the same frequency.

9.3 Phasor (8)

9.3 Phasor (9)

	Example 6
	Use phasor approach, determine the current i(t) in a circuit described by the integro-differential equation.

9.3 Phasor (10)

	In-class exercise for Unit 6a, we can derive the differential equations for the following circuit in order to solve for vo(t) in phase domain Vo.

9.3 Phasor (11)

	The answer is YES!

9.4 Phasor Relationships
for Circuit Elements (1)

9.4 Phasor Relationships
for Circuit Elements (2)

9.4 Phasor Relationships
for Circuit Elements (3)

9.5 Impedance and Admittance (1)

9.5 Impedance and Admittance (2)

9.5 Impedance and Admittance (3)

9.5 Impedance and Admittance (4)

9.5 Impedance and Admittance (5)

9.6 Kirchhoff’s Laws
in the Frequency Domain (1)

9.7 Impedance Combinations (1)

9.7 Impedance Combinations (2)