ES250:Electrical Science

Chapter 10: Sinusoidal Steady‐State Analysis

• Linear circuits with sinusoidal inputs that are at steady stateIntroduction

Linear circuits with sinusoidal inputs that are at steady state are called ac circuits, e.g, the power system that provides us with electricity is a very large ac circuit• In particular, we will see that for AC circuits:

– it's useful to associate a complex number with a sinusoid, d i ll t d fi h d i das doing so allows us to define phasors and impedances

– using phasors and impedances, we obtain a new representation of the linear circuit called the “frequency‐representation of the linear circuit, called the frequencydomain representation”

– we can analyze ac circuits in the frequency domain to determine their steady‐state response

• Assume a sinusoidal voltage source vs = Vm sin (ωt + φ) asSinusoidal Signals

Assume a sinusoidal voltage source vs Vm sin (ωt φ) as shown:

– the amplitude of the sinusoid is Vm (volts), and the radian frequency is ω (rad/s), and the phase angle φ (measured in degrees or radians)

– e.g.,

– using the trigonometric formulas of Appendix C, it can be shown that:

• If a circuit element has voltage and current as shown:Sinusoidal Signals

If a circuit element has voltage and current as shown:

h h l d h l b φ d– we say that the current leads the voltage by φ radians, or the voltage lags the current by φ radians

Steady‐State Response of Circuits with Sinusoidal Forcing Functions

• Consider the consider the RL circuit shown below:Sinusoidal Forcing Functions

• If the input to this circuit is the voltage:

• The complete response of this circuit is of the form:The complete response of this circuit is of the form:

the values of the real constants K τ I and φ are to be– the values of the real constants K, τ, Im, and φ are to be determined with K dependent on the initial condition i(0)

•– Note: the steady‐state output is a scaled and phased

shifted version of the input at the same frequency

• Using Euler's identity we can relate a complex exponential

Complex Exponential Forcing Functions• Using Euler s identity, we can relate a complex exponential

signal to a sinusoidal signal:

– where Euler's identity is: = a + jbwhere Euler s identity is: a + jb

– the notation Re{a + jb} is read as the real part of the complex number (a + jb), e.g.:complex number (a jb), e.g.:

• A sinusoidal current or voltage at a given frequency is characterized by its amplitude and phase angle, e.g.:

∼– where I is called a phasor– a phasor is a complex number that represents the

magnitude and phase of a sinusoid and may be written in exponential form, polar form, or rectangular form

• Phasors may be used to represent a linear circuit when its

The Phasor• Phasors may be used to represent a linear circuit when its

steady‐state response is sought and all independent sources are sinusoidal and have the same frequencyq y

• Although the phasor notation drops (or suppressed) the complex frequency e jωt, we continue to note that we are using a complex frequency representation of the circuit and thus are performing calculations in the frequency domain

– we have transformed the problem from the time domain to the frequency domain by the use of phasor notation

– a transform is a change in the mathematical description of a physical variable to facilitate computation

• The steps involved in transforming a function in the time

The Phasor• The steps involved in transforming a function in the time

domain to the frequency domain are summarized below:

1 write the function in the time domain y(t) as a cosine1. write the function in the time domain, y(t), as a cosine waveform with a phase angle φ as

2. Express the cosine waveform as the real part of a2. Express the cosine waveform as the real part of a complex quantity by using Euler's identity so that

3. Drop the real part notation

4. Suppress the e jωt term, noting the value of ω for later4. Suppress the e term, noting the value of ω for later use, obtaining the phasor

– note, since it is easy to move through these steps, we , y g p ,usually jump directly from step 1 to step 4

• For example let us determine the phasor notation for

The Phasor• For example, let us determine the phasor notation for

– the associated phasor is given bythe associated phasor is given by

• The process of going from phasor notation to time notation is exactly the reverse of the steps required to go from theis exactly the reverse of the steps required to go from the time to the phasor notation, e.g., if we have a voltage in phasor notation given by

– the associated time‐domain waveform is given by

where ω is the frequency of the forcing inputs (sources)

• Find the steady‐state voltage v(t) represented by the

Exercise 10.5‐2• Find the steady‐state voltage v(t) represented by the

phasor:

80 75V jMATLAB code:

80 j*

2 2

80 75

7580 75 arctan

V j

= +

= + ∠

>> V=80+j*75

V = 80.0000 +75.0000i80 75 arctan 80

109.7 0.753rad

= + ∠

= ∠

>> abs(V)

ans = 109.6586109.7 0.753rad109.7 43.2

( ) 109 7 ( 43 2 )t t

∠= ∠ °

+ °

>> angle(V)

ans = 0.7532( ) 109.7cos( 43.2 )v t tω= + °∼ >> 180*angle(V)/pi

ans = 43.1524

• We now show the relationship between the phasor voltagePhasor Relationships for R, L, and C Elements

• We now show the relationship between the phasor voltage and the phasor current of the R, L, and C elements

– we use the transformation from time to the frequencywe use the transformation from time to the frequency domain and then solve the phasor relationship for each element

– using this approach, we move from solving differential equations (harder) to solving algebraic equations (easier)

• For example, the voltage−current relationship for a resistor in the time domain and frequency domains is given by:

• The voltage−current relationship for an inductor in the timePhasor Relationships for R, L, and C Elements

The voltage current relationship for an inductor in the time domain and frequency domains is given by:

• The same relationship for a capacitor is given by:

• Ohm's law expressed in phasor notation is called theImpedance

• Ohm s law expressed in phasor notation is called the impedance of an element , defined as:

– impedance in ac circuits has a role similar to the role of– impedance in ac circuits has a role similar to the role of resistance in dc circuits

– impedance has units of ohmsimpedance has units of ohms

– Impedance is a complex number that relates the V phasorto the I phasor, but it has no meaning in the time domainto the I phasor, but it has no meaning in the time domain

• Using the impedance concept, we can solve for the response of sinusoidally excited circuits using complex p y g palgebra in the same way we have solved resistive circuits

• Since the impedance is a complex number it may be writtenImpedance

• Since the impedance is a complex number, it may be written in several forms, as follows:

– R = Re Z is called the resistive part of the impedance

X = Im Z is called the reactive part of the impedance– X = Im Z is called the reactive part of the impedance

– both R and X are measured in ohms

th it d f th i d i– the magnitude of the impedance is

– the phase angle is

– admittance Y is defined as

• These relationships can be visualized graphically in theImpedance

• These relationships can be visualized graphically in the complex plane; e.g., for:

2 22 2 8 2 828 andZ = + = =2 2 8 2.828, and

2arctan 0.707rad 452

Z

Z

= + = =

∠ =∠ = = °⇒

2

2=

2=

• The circuit below is shown in its time form and frequencyExercise 10.7‐1

• The circuit below is shown in its time form and frequency domain form, using phasors and impedances:

• The circuit below is shown in its time form and frequency d i f i h d i d

Exercise 10.7‐2domain form, using phasors and impedances:

Note, the impedance of a capacitor is purely reactive < 0, while the impedance of an inductor is purely reactive > 0

• Kirchhoff's voltage and current laws hold in the frequencyKirchhoff's Laws Using Phasors

• Kirchhoff s voltage and current laws hold in the frequency domain with phasor voltages and currents, respectively

• Since both the KVL and the KCL hold in the frequencySince both the KVL and the KCL hold in the frequency domain, it is easy to conclude that all the techniques of analysis we developed for resistive circuits hold for phasorcurrents and voltages as long as the circuit is linear, e.g.:

– principle of superposition

– source transformations

– series and parallel combinations

– Thévenin and Norton equivalent circuits

– node voltage and mesh current analysisg y

• Thus the equivalent impedance for a series of impedancesKirchhoff's Laws Using Phasors

• Thus, the equivalent impedance for a series of impedances is the sum of the individual impedances, as shown:

• Thus the equivalent admittance for parallel admittances isKirchhoff's Laws Using Phasors

• Thus, the equivalent admittance for parallel admittances is the sum of the individual admittances, as shown:

• In the case of two parallel admittances, we have:

– the corresponding equivalent impedance is:

Voltage Division in the Frequency Domain

Current Division in the Frequency Domain

• Find the steady‐state current i(t) using phasors for the RLCEx. 10.8‐1: Analysis Using Impedances

Find the steady state current i(t) using phasors for the RLCcircuit below when R = 9 Ω, L = 10 mH, and C = 1 mF:

( )sv t

( ) 100cos(100 ) 100 0 , 100rad/secsv t t ω= = ∠ ° =sV( ) 100cos(100 ) 100 0 , 100rad/secsv t t ω∠sV

• Mesh KVL:

⇒⇒

⇒

• Find the steady‐state output voltage vo(t) using phasors forEx. 10.8‐2: Analysis Using Impedances

Find the steady state output voltage vo(t) using phasors for the when :

( )sv t

B l di id• By voltage divider:

⇒⇒

⇒

Questions?Questions?