We will now study dynamic systems which are subjected to sinusoidal forcing functions. Previously, in our analysis of dynamic systems, we determined both the unforced response (or homogeneous solution) and the forced response (or particular solution) to the given forcing function. In the next several chapters, however, we will restrict our attention to only the system’s forced response to a sinusoidal input; this response is commonly called the sinusoidal steady-state system response. This analysis approach is useful if we are concerned primarily with the system’s response after any initial conditions have died out, since we are ignoring any transient effects due to the system’s natural response.

Restricting our attention to the steady-state sinusoidal response allows a considerable simplification in the system analysis: we can solve algebraic equations rather than differential equations. This advantage often more than compensates for the loss of information relative to the systems natural response. For example it is often the case that a sinusoidal input is applied for a very long time relative to the time required for the natural response to die out, so that the overall effects of the initial conditions are negligible.

Steady-state sinusoidal analysis methods are important for several reasons:

• Sinusoidal inputs are an extremely important category of forcing functions. In electrical engineering, for example, sinusoids are the dominant signal in the electrical power industry. The alternating current (or AC) signals used in power transmission are, in fact, so pervasive that many electrical engineers commonly refer to any sinusoidal signal as “AC”. Carrier signals used in communications systems are also sinusoidal in nature.
• The simplification associated with the analysis of steady state sinusoidal analysis is often so desirable that system responses to non-sinusoidal inputs are interpreted in terms of their sinusoidal steady-state response. This approach will be developed when we study Fourier series.
• System design requirements are often specified in terms of the desired steady-state sinusoidal response of the system.

In section 10.1 of this chapter, we qualitatively introduce the basic concepts relative to sinusoidal steady state analyses so that readers can get the “general idea” behind the analysis approach before addressing the mathematical details in later sections. Since we will be dealing exclusively with sinusoidal signals for the next few chapters, section 10.2 provides review material relative to sinusoidal signals and complex exponentials. Recall from chapter 8 that complex exponentials are a mathematically convenient way to represent sinusoidal signals. Most of the material in section 10.2 should be review, but the reader is strongly encouraged to study section 10.2 carefully – we will be using sinusoids and complex exponentials extensively throughout the remainder of this text, and a complete understanding of the concepts and terminology is crucial. In section 10.3, we examine the forced response of electrical circuits to sinusoidal inputs; in this section, we analyze our circuits using differential equations and come to the important conclusion that steady-state response of a circuit to sinusoidal inputs is governed by algebraic equations. Section 10.4 takes advantage of this conclusion to perform steady-state sinusoidal analyses of electrical circuits without writing the governing differential equation for the circuit! Finally, in section 10.5, we characterize a system’s response purely by its effect on a sinusoidal input. This concept will be used extensively throughout the remainder of this textbook.

• State the relationship between the sinusoidal steady state system response and the forced response of a system
• For sinusoidal steady-state conditions, state the relationship between the frequencies of the input and output signals for a linear, time-invariant system
• State the two parameters used to characterize the sinusoidal steady-state response of a linear, time-invariant system
• Define periodic signals
• Define the amplitude, frequency, radian frequency, and phase of a sinusoidal signal
• Express sinusoidal signals in phasor form
• Perform frequency-domain analyses of electrical circuits
• Sketch phasor diagrams of a circuit’s input and output
• State the definition of impedance and admittance
• State, from memory, the impedance relations for resistors, capacitors, and inductors
• Calculate impedances for resistors, capacitors, and inductors
• State how to use the following analysis approaches in the frequency domain:
  • KVL and KCL
  • Voltage and current dividers
  • Circuit reduction techniques
  • Nodal and mesh analysis
  • Superposition, especially when multiple frequencies are present
  • Thévenin’s and Norton’s theorems
• Determine the load impedance necessary to deliver maximum power to a load
• Define the frequency response of a system
• Define the magnitude response and phase response of a system
• Determine the magnitude and phase responses of a circuit

In this chapter, we will be almost exclusively concerned with sinusoidal signals, which can be written in the form:

$$f(t) = A \cos (\omega t+ \theta) (Eq. 10.1)$$

where $A$ is the amplitude of the sinusoid, $\omega$ is the angular frequency (in radians/second) of the signal, and $\theta$ is the phase angle (expressed in radians or degrees) of the signal. $A$ provides the peak value of the sinusoid, $\omega$ governs the rate of oscillation of the signal, and $\theta$ affects the translation of the sinusoid in time. A typical sinusoidal signal is shown in Figure 10.1.

If the sinusoidal signal of Figure 10.1 is applied to a linear time invariant system, the response of the system will consist of the system’s natural response (due to the initial conditions on the system) superimposed on the system’s forced response (the response due to the forcing function). As we have seen in previous chapters, the forced response has the same form as the forcing function. Thus, if the input is a constant value the forced response is constant, as we have seen in the case of the step response of a system. In the case of a sinusoidal input to a system, the forced response will consist of a sinusoid of the same frequency as the input sinusoid. Since the natural response of the system decays with time, the steady state response of a linear time invariant system to a sinusoidal input is a sinusoid, as shown in Figure 10.2. The amplitude and phase of the output may be different than the input amplitude and phase, but both the input and output signals have the same frequency.

It is common to characterize a system by the ratio of the magnitudes of the input and output signals ($\frac{B}{A}$ in Figure 10.2) and the difference in phases between the input and output signals ($\phi - \theta$) in Figure 10.2) at a particular frequency. It is important to note that the ratio of magnitudes and difference in phases is dependent upon the frequency of the applied sinusoidal signal.

Consider the series RLC circuit shown in Figure 10.3 below. The input voltage to the circuit is given by:

$$v_s(t) = \begin{cases} 0, t < 0 \\ \cos(5t), t \geq 0 \\ \end{cases}$$

Thus, the input is zero prior to $t = 0$, and the sinusoidal input is suddenly “switched on” at time $t = 0$. The input forcing function is shown in Figure 10.4(a). The circuit is “relaxed” before the sinusoidal input is applied, so the circuit initial conditions are:

$$y(0^-) = \frac{dy}{dt} |_{t=0^-} = 0$$

This circuit has been analyzed previously in Chapter 8, and the derivation of the governing differential equation will not be repeated here. The full output response of the circuit is shown in Figure 10.4(b). The natural response of the circuit is readily apparent in the initial portion of the response but these transients die out quickly, leaving only the sinusoidal steady-state response of the circuit. It is only this steady state response in which we will be interested for the next several modules. With knowledge of the frequency of the signals, we can define both the input and (steady-state) output by their amplitude and phase, and characterize the circuit by the ratio of the output-to-input amplitude and the difference in the phases of the output and input.

• Sinusoidal signals can be expressed mathematically in the form:

$$f(t) = A \cos(\omega t + \theta)$$

• In the above, $A$ is the amplitude of the sinusoid, it describes the maximum and minimum values of the signal.
• In the above, $\theta$ is the phase angle of the sinusoid, it describes the time shift of the sinusoid relative to a pure cosine.
• In the above, $\omega$ is the radian frequency of the sinusoid. The sinusoid repeats itself at time intervals of $\frac{2\pi}{\omega}$ seconds.
• A sinusoidal signal is completely described by its frequency, its amplitude, and its phase angle.
• The steady-state response of a linear, time-invariant system to a sinusoidal input is a sinusoid with the same frequency.
• Since the frequencies of the input and output are the same, the relationship between the input and output sinusoids is completely characterized by the relationships between:
  • The input and output amplitudes.
  • The input and output phase angles.

1. In the circuit below, all circuit elements are linear and time invariant. The input voltage $V_{in}(t)=10 \cos(2t+40^{\circ})$. What is the radian frequency of the output voltage $V_{out}(t)$?

2. In the circuit below, all circuit elements are linear and time invariant. The input voltage is $V_{in}(t)=10 \cos(2t + 40^{\circ})$. The output voltage is of the form $V_{out}(t) = A \cos(\omega t + \phi^{\circ})$. If the ratio between the input and output, $| \frac{V_{out}}{V_{in}} | = 0.5$ and the phase difference between the input and output is $20^{\circ}$, what are:

• a. The radian frequency of the output, $\omega$?
• The amplitude of the output, $A$?
• The phase angle of the output, $\phi$?

In this section, we will review properties of sinusoidal functions and complex exponentials. We will also introduce phasor notation, which will significantly simplify the sinusoidal steady-state analysis of systems, and provide terminology which will be used in subsequent sinusoidal steady-state related modules.

Much of the material presented here has been provided previously in Chapter 8; this material is, however, important enough to bear repetition. Likewise, a brief overview of complex arithmetic, which will be essential in using complex exponentials effectively, is provided at the end of this section. Readers who need to review complex arithmetic may find it useful to peruse this overview before reading the material in this section relating to complex exponentials and phasors.

The sinusoidal signal shown in Figure 10.5 is represented mathematically by:

$$f(t)=V_P \cos(\omega t) (Eq. 10.2)$$

The amplitude or peak value of the function is $V_P$. $V_P$ is the maximum value achieved by the function; the function itself is bounded by $+ V_P$ and $- V_P$, so that $-V_P \leq f(t) \leq V_P$. The radian frequency or angular frequency of the function is $\omega$; the units of $\omega$ are radians/second. The function is said to be periodic; periodic functions repeat themselves at regular intervals, so that:

$$f(t+nT) = f(t) (Eq. 10.3)$$

Where $n$ is any integer and $T$ is the period of the signal. The sinusoidal waveform shown in Figure 10.5 goes through one complete cycle or period in $T$ seconds. Since the sinusoid of equation (10.2) repeats itself every $2\pi$ radians, the period is related to the radian frequency of the sinusoid by:

$$\omega = \frac{2\pi}{T} (Eq. 10.4)$$

It is common to define the frequency of the sinusoid in terms of the number of cycles of the waveform which occur in one second. In these terms, the frequency $f$ of the function is:

$$f= \frac{1}{T} (Eq. 10.5)$$

The units of $f$ are cycles/second or Hertz (abbreviated Hz). The frequency and radian frequency are related by:

$$f=\frac{\omega}{2\pi} (Eq. 10.6)$$

Or equivalently:

$$\omega = 2 \pi f (Eq. 10.7)$$

Regardless of whether the sinusoid’s rate of oscillation is expressed as frequency or radian frequency, it is important to realize that the argument of the sinusoid in equation (10.2) must be expressed in radians. Thus, equation (10.2) can be expressed in terms of frequency in Hz as:

$$f(t) = \cos(2 \pi ft) (Eq. 10.8)$$

To avoid confusion in our mathematics, we will almost invariably write sinusoidal functions in terms of radian frequency as shown in equation (10.2), although Hz is generally taken as the standard unit for frequency (experimental apparatus, for example, commonly express frequency in Hz).

A more general expression of a sinusoidal signal is:

$$v(t)=V_P \cos(\omega t + \theta) (Eq. 10.9)$$

Where $\theta$ is the phase angle or phase of the sinusoid. The phase angle simply translates the sinusoid along the time axis, as shown in Figure 10.6. A positive phase angle shifts the signal left in time, while a negative phase angle shifts the signal right – this is consistent with our discussion of step functions in section 6.1, where it was noted that subtracting a value from the unit step argument resulted a time delay of the function. Thus, as shown in Figure 10.6, a positive phase angle causes the sinusoid to be shifted left by $\frac{\theta}{\omega}$ seconds.

The units of phase angle should be radians, to be consistent with the units of $\omega t$ in the argument of the cosine. It is typical, however, to express phase angle in degrees, with $180^{\circ}$ corresponding to $\pi$ radians. Thus, the conversion between radians and degrees can be expressed as:

Number of degrees $= \frac{180}{\pi} x$ Number of radians

For example, we will consider the two expressions below to be equivalent, though the expression on the right- hand side of the equal sign contains a mathematical inconsistency:

$$V_P \cos(\omega t + \frac{\pi}{2}) = V_P \cos(\omega t + 90^{\circ})$$

For convenience, we introduce the terms leading and lagging when referring to the sign on the phase angle, $\theta$. A sinusoidal signal $v_1(t)$ is said to lead another sinusoid $v_2(t)$ of the same frequency if the phase difference between the two is such that $v_1(t)$ is shifted left in time relative to $v_2(t)$. Likewise, $v_1(t)$ is said to lag another sinusoid $v_2(t)$ of the same frequency if the phase difference between the two is such that $v_1(t)$ is shifted right in time relative to $v_2(t)$. This terminology is described graphically in Figure 10.7.

Finally, we note that the representation of sinusoidal signals as a phase shifted cosine function, as provided by equation (10.9), is completely general. If we are given a sinusoidal function in terms of a sine function, it can be readily converted to the form of equation (10.9) by subtracting a phase of $\frac{\pi}{2}$ (or $90^{\circ}$) from the argument, since:

$$\sin(\omega t) = \cos(\omega t - \frac{\pi}{2})$$

Likewise, sign changes can be accounted for by a $\pm \pi$ radian phase shift, since:

$$-\cos(\omega t) = \cos(\omega t \pm \pi)$$

Obviously, we could have chosen either a cosine or sine representation of a sinusoidal signal. We prefer the cosine representation, since a cosine is the real part of a complex exponential. In the next module, we will see that sinusoidal steady-state circuit analysis is simplified significantly by using complex exponentials to represent the sinusoidal functions. The cosine is the real part of a complex exponential (as we saw previously in chapter 8). Since all measurable signals are real valued, we take the real part of our complex exponential-based result as our physical response; this results in a solution of the form of equation (10.9).

Since representation of sinusoidal waveforms as complex exponentials will become important to us in circuit analysis, we devote the following subsection to a review of complex exponentials and their interpretation as sinusoidal signals.

Euler’s identity can be used to represent complex numbers as complex exponentials:

$$e^{\pm j \theta} = \cos \theta \pm j \sin \theta (Eq. 10.10)$$

If we generalize equation (9) to time-varying signals of arbitrary magnitude we can write:

$$V_Pe^{\pm j (\omega t + \theta)} = V_P \cos(\omega t + \theta) \pm jV_P \sin(\omega t + \theta) (Eq. 10.11)$$

So that:

$$V_P \cos(\omega t + \theta) = Re \left\{ V_Pe^{\pm j(\omega t + \theta)} \right\} (Eq. 10.12)$$

And:

$$V_P \sin(\omega t + \theta) = Im \left\{ V_Pe^{\pm j(\omega t + \theta)} \right\} (Eq. 10.13)$$

Where $Re \left\{ V_Pe^{\pm j(\omega + \theta)} \right\}$ and $Im \left\{ V_Pe^{\pm j(\omega t + \theta)} \right\}$ denote the real part of $V_Pe^{\pm j(\omega t + \theta)}$ and the imaginary part of $V_Pe^{\pm j(\omega t + \theta)}$, respectively.

The complex exponential of equation (10.11) can also be written as:

$$V_Pe^{\pm j(\omega t + \theta)} = V_Pe^{j \theta} e^{j \omega t} (10.14)$$

The term $V_Pe^{j \theta}$ on the right-hand side of equation (10.14) is simply a complex number which provides the magnitude and phase information of the complex exponential of equation (10.11). From equation (10.12), this magnitude and phase can be used to express the magnitude and phase angle of a sinusoidal signal of the form given in equation (10.9).

The complex number in polar coordinates which provides the magnitude and phase angle of a time-varying complex exponential, as given in equation (10.14) is called a phasor. The phasor representing $V_P \cos(\omega t + \theta)$ is defined as:

$$\underline{V} = V_Pe^{j \theta} = V_P \angle \theta (Eq. 10.15)$$

We will use a capital letter with an underscore to denote a phasor. Using bold typeface to represent phasors is more common; our notation is simply for consistency between lecture material and written material – boldface type is difficult to create on a whiteboard during lecture!

The phasor representing a sinusoid does not provide information about the frequency of the sinusoid – frequency information must be kept track of separately.

Much the material in this section has been provided previously in section 8.3. It is repeated here to emphasize its importance and to expand slightly upon some crucial topics.

In our presentation of complex exponentials, we first provide a brief review of complex numbers. A complex number contains both real and imaginary parts. Thus, we may write a complex number $\underline{A}$ as:

$$\underline{A} a + jb (Eq. 10.16)$$

Where:

$$j= \sqrt{-1} (Eq. 10.17)$$

And the underscore denotes a complex number. The complex number $\underline{A}$ can be represented on orthogonal axes representing the real and imaginary part of the number, as shown in Figure 10.8. (In Figure 10.8, we have taken the liberty of representing $\underline{A}$ as a vector, although it is really just a number.) We can also represent the complex number in polar coordinates, also shown in Figure 10.8. The polar coordinates consist of a magnitude $|A|$ and phase angle $\theta A$, defined as:

$$|A| = \sqrt{a^2+b^2} (Eq. 10.18)$$

$$\theta_A = \tan^{-1} \left( \frac{b}{a} \right) (Eq. 10.19)$$

Notice that the phase angle is defined counterclockwise from the positive real axis. Conversely, we can determine the rectangular coordinates from the polar coordinates from:

$$a=Re \left\{ \underline{A} \right\} = |A| \cos(\theta_A) (Eq. 10.20)$$

$$b=Im \left\{ \underline{A} \right\} = |A| \sin(\theta_A) (Eq. 10.21)$$

Where the notation $Re \left\{ \underline{A} \right\}$ and $Im \left\{ \underline{A} \right\}$ denote the real [art of $\underline{A}$ and the imaginary part of $\underline{A}$, respectively.

The polar coordinates of a complex number of $\underline{A}$ are often represented in the form:

$$\underline{A} |A| \angle \theta_A (Eq. 10.22)$$

An alternate method of representing complex numbers in polar coordinates employs complex exponential notation. Without proof, we claim that:

$$e^{j \theta} = 1 \angle \theta (Eq. 10.23)$$

Thus, $e^{j \theta}$ is a complex number with magnitude 1 and phase angle $\theta$. From Figure 10.8, it is easy to see that this definition of the complex exponential agrees with Euler’s equation:

$$e^{\pm j \theta} = \cos \theta \pm j \sin \theta (Eq. 10.24)$$

With the definition of equation (10.23), we can define any arbitrary complex number in terms of complex numbers. For example, our previous complex number $\underline{A}$ can be represented as:

$$\underline{A} = |A| e^{j \theta_A} (Eq. 10.25)$$

We can generalize our definition of the complex exponential to time-varying signals. If we define a time varying signal $e^{j \omega t}$, we can use equation (10.24) to write:

$$e^{j \omega t} = \cos \omega t \pm j \sin \omega t (Eq. 10.26)$$

The signal $e^{j \omega t}$ can be visualized as a unit vector rotating around the origin in the complex plane; the tip of the vector scribes a unit circle with its center at the origin of the complex plane. This is illustrated in Figure 10.9. The vector rotates at a rate defined by the quantity $\omega$ – the vector makes one complete revolution every $\frac{2 \pi}{\omega}$ seconds.

The projection of this rotating vector on the real axis traces out the signal $\cos \omega t$, as shown in Figure 10.7, while the projection of the rotating vector on the imaginary axis traces out the signal $\sin \omega t$, also shown in Figure 10.9.

Thus, we interpret the complex exponential function $e^{j \omega t}$ as an alternate “type” of sinusoidal signal. The real part of this function is $\cos \omega t$ while the imaginary part of this function is $\sin \omega t$.

Addition and subtraction of complex numbers is most easily performed in rectangular coordinates. Given two complex numbers $\underline{A}$ and $\underline{B}$, defined as:

$$\underline{A} = a + jb$$

$$\underline{B} = c + jd$$

The sum and difference of the complex number can be determined by:

$$\underline{A} + \underline{B} = (a+c) + j(b+d)$$

And:

$$\underline{A} - \underline{B} = (a-c) + j(b-d)$$

Multiplication and division, on the other hand, are probably most easily performed using polar coordinates. If we define two complex numbers as:

$$\underline{A} = |A| e^{j \theta_A} = |A| \angle \theta_A$$

$$\underline{B} = |B| e^{j \theta_B} = |B| \angle \theta_B$$

The product and quotient can be determined by:

$$\underline{A} \cdot \underline{B} = |A| e^{j \theta_A} \cdot |B| e ^{j \theta_B} = |A| \cdot |B| e^{j(\theta_A + \theta_B)} = |A| \cdot |B| \angle (\theta_A + \theta_B)$$

And:

$$\frac{\underline{A}}{\underline{B}} = \frac{|A|e^{j \theta_A}}{|B|e^{j \theta_B}} = \frac{\underline{A}}{\underline{B}} \angle (\theta_A - \theta_B)$$

The conjugate of a complex number, denoted by a *, is obtained by changing the sign on the imaginary part of the number. For example, if $\underline{A} = a + jb = |A| e^{j \theta}$, then:

$$\underline{A^*} = a-jb= |A|e^{-j \theta}$$

Conjugation does not affect the magnitude of the complex number, but it changes the sign on the phase angle. It is easy to show that:

$$\underline{A} \cdot \underline{A^*} = |A|^2$$

Several useful relationships between polar and rectangular coordinate representations of complex numbers are provided below. The reader is encouraged to prove any that are not self-evident.

$$j=1 \angle 90^{\circ} \\ -j = 1 \angle - 90^{\circ} \\ \frac{1}{j} = -j = 1 \angle - 90^{\circ} \\ 1=1 \angle 0^{\circ} \\ -1=1 \angle 180^{\circ}$$

• Periodic signals repeat themselves at a specific time interval. Sinusoidal signals are a special case of periodic signals.
• A sinusoidal signal can always be written in the form $v(t) = V_P \cos(\omega t + \theta)$.
• It is often convenient, when analyzing a system’s steady-state response to sinusoidal inputs, to express sinusoidal signals in terms of complex exponentials. This is possible because of Euler’s formula:

$$e^{\pm j \omega t} = \cos \omega t \pm j \sin \omega t$$

• From Euler’s formula, a sinusoidal signal can be expressed as the real part of a complex exponential:

$$v(t) = V_P \cos(\omega t + \theta) = Re \left\{ V_Pe^{\pm j(\omega t + \theta)} \right\}$$

• The magnitude and phase angle of a complex exponential signal are conveniently expressed as a phasor:

$$\underline{V} = V_Pe^{j \theta}$$

• Using phasor notation, the above complex exponential signal can be written as:

$$V_Pe^{\pm j(\omega t + \theta)} = \underline{V}e^{j \omega t}$$

• Phasors can then be operated on arithmetically in the same way as any other complex number. However, when operating on phasors, keep in mind that you are dealing with the amplitude and phase angle of a sinusoidal signal.

1. Express the following complex numbers in rectangular form:
   1. $3e^{j45^{\circ}}$
   2. $5\sqrt{2}e^{j135^{\circ}}$
   3. $2.5e^{j90^{\circ}}$
   4. $6e^{j \pi}$
2. Express the following complex numbers in complex exponential form:
   1. $2-j2$
   2. $-j3$
   3. $6$
   4. $3+j$
3. Evaluate the following expressions. Express your results in complex exponential form.
   1. $-\frac{j}{2(j+1)}$
   2. $\frac{2-j2}{4+j4}$
   3. $2e^{j45^{\circ}} \cdot \frac{2}{j+1}$
   4. $j+ \frac{2}{j}$
4. Represent the following sinusoids in phasor form:
   1. $3 \cos(5t-60^{\circ})$
   2. $-2 \cos(300t+45^{\circ}$
   3. $\sin(6t)$
   4. $7\cos(3t)$
5. Write the signal representing the real part of the following complex exponentials:
   1. $5\sqrt{2}e^{j(100t-45^{\circ})}$
   2. $3e^{j \pi}e^{j3t}$
   3. $2e^{j(\pi t-30^{\circ})+4e^{j(4t+20^{\circ})}$