ES250: Electrical Science Chapter 10: Sinusoidal Steady-State Analysis

Introduction Linear circuits with sinusoidal inputs that are at steady state are called ac circuits, e.g, the power system that provides us with electricity can be considered 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, as 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-domain representation” we can analyze ac circuits in the frequency domain to determine their steady-state response

  Sinusoidal 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 (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:

  Sinusoidal Signals If a circuit element has voltage and current as shown: 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: The complete response of this circuit is of the form: 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 If the input to this circuit is the voltage:

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 +  jb the notation Re{a +  jb} is read as the real part of the 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

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 frequency 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 Phasor The steps involved in transforming a function in the time domain to the frequency domain are summarized below: write the function in the time domain, y(t), as a cosine waveform with a phase angle φ as Express the cosine waveform as the real part of a complex quantity by using Euler's identity so that Drop the real part notation Suppress the e jωt term, noting the value of ω for later use, obtaining the phasor note, since it is easy to move through these steps, we usually jump directly from step 1 to step 4

The Phasor For example, let us determine the phasor notation for the 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 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 w is the frequency of the forcing inputs (sources)

Exercise 10.5-2 Find the steady-state voltage v(t) represented by the phasor: MATLAB code: >> V=80+j*75 V = 80.0000 +75.0000i >> abs(V) ans = 109.6586 >> angle(V) ans = 0.7532 >> 180*angle(V)/pi ans = 43.1524

Phasor 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 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:

Phasor 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:

Impedance 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 resistance in dc circuits impedance has units of ohms Impedance is a complex number that relates the V phasor to 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 algebra in the same way we have solved resistive circuits

Impedance 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 both R and X are measured in ohms the magnitude of the impedance is the phase angle is admittance Y is defined as

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

Exercise 10.7-1 The circuit below is shown in its time form and frequency domain form, using phasors and impedances:

Exercise 10.7-2 The circuit below is shown in its time form and frequency domain 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 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 frequency domain, it is easy to conclude that all the techniques of analysis we developed for resistive circuits hold for phasor currents 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 analysis

Kirchhoff's Laws Using Phasors Thus, the equivalent impedance for a series of impedances is the sum of the individual impedances, as shown:

Kirchhoff'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

Ex. 10.8-1: Analysis Using Impedances Find the steady-state current i(t) using phasors for the RLC circuit below when R = 9 Ω, L = 10 mH, and C = 1 mF: Mesh KVL:

Ex. 10.8-2: Analysis Using Impedances Find the steady-state output voltage vo(t) using phasors when : By voltage divider:

Questions?