26 C H A P T E R 0: From the Ground Up!

can be expressed as

i(t)=Ccos( 0 t+γ)

whereCandγare to be determined (the sinusoidal components ofi(t)must depend on a unique

frequency 0 ; if that was not the case the concept of phasors would not apply). To obtain the equiv-

alent representation, we first obtain the phasor corresponding toAcos( 0 t), which isI 1 =Aej^0 =A,

and forBsin( 0 t)the corresponding phasor isI 2 =Be−jπ/^2 , so that

i(t)=Re[(I 1 +I 2 )ej^0 t]

Thus, the problem has been transformed into the addition of two vectorsI 1 andI 2 , which gives a

vector

I=

√

A^2 +B^2 e−jtan

− (^1) (B/A)
so that
i(t)=Re[Iej^0 t]
=Re[

√

A^2 +B^2 e−jtan

− (^1) (B/A)
ej^0 t]

√

A^2 +B^2 cos( 0 t−tan−^1 (B/A))

Or, an equivalent source with amplitudeC=

√

A^2 +B^2 , phaseγ=−tan−^1 (B/A), and frequency 0 –

that is, an equivalent phasor that generatesi(t)and has the magnitudeC, the angleγ, and rotates at

frequency 0.

In Figure 0.13 we display the result of adding two phasors (frequencyf 0 =20 Hz) and the sinusoid

that is generated by the phasorI=I 1 +I 2 =27.98ej30.4

o

0.4.4 Phasor Connection

The fundamental property of a circuit made up of constant resistors, capacitors, and inductors is that

its response to a sinusoid is also a sinusoid of the same frequency in steady state. The effect of the

circuit on the input sinusoid is on its magnitude and phase and depends on the frequency of the input

sinusoid. This is due to the linear and time-invariant nature of the circuit, and can be generalized to

more complex continuous-time as well as discrete-time systems as we will see in Chapters 3, 4, 5, 9

and 10.

To illustrate the connection of phasors with dynamic systems consider a simple RC circuit (R= 1 

andC=1F). If the input to the circuit is a sinusoidal voltage sourcevi(t)=Acos( 0 t)and the voltage

across the capacitorvc(t)is the output of interest, the circuit can be easily represented by the first-order

differential equation

dvc(t)

dt

+vc(t)=vi(t)