So far, we have considered circuits that have been governed by algebraic relations. These circuits have, in general, contained only power sources and resistive elements. All elements in these circuits, therefore, have either supplied power from external sources or dissipated power. For these resistive circuits, we can apply either time-varying or constant signals to the circuit without really affecting our analysis approach. Ohm’s law, for example, is equally applicable to time-varying or constant voltages and currents:

$$V = I \cdot R \Leftrightarrow v(t) = i(t) \cdot R$$

Since the governing equation is algebraic, it is applicable at every point in time – voltages and currents at a point in time are affected only by voltages and currents at the same point in time.

We will now begin to consider circuit elements, which are governed by differential equations. These circuit elements are called dynamic circuit elements or energy storage elements. Physically, these circuit elements store energy, which they can later release back to the circuit. The response, at a given time, of circuits that contain these elements is not only related to other circuit parameters at the same time; it may also depend upon the parameters at other times.

This chapter begins with an overview of the basic concepts associated with energy storage. This discussion focuses not on electrical systems, but instead introduces the topic qualitatively in the context of systems with which the reader is already familiar. The goal is to provide a basis for the mathematics, which will be introduced subsequently. Since we will now be concerned with time-varying signals, section 6.2 introduces the basic signals that we will be dealing with in the immediate future. This chapter concludes with presentations of the two electrical energy storage elements that we will be concerned with: capacitors and inductors. The method by which energy is stored in these elements is presented in sections 6.3 and 6.4, along with the governing equations relating voltage and current for these elements.

• Qualitatively state the effect of energy storage on the type of mathematics governing a system
• Define transient response
• Define steady-state response
• Write the mathematical expression for a unit step function
• Sketch the unit step function
• Sketch shifted and scaled versions of the unit step function
• Write the mathematical expression for a decaying exponential function
• Define the time constant of an exponential function
• Sketch a decaying exponential function, given the function’s initial value and time constant
• Use a unit step function to restrict an exponential function to times greater than zero
• Write the circuit symbol for a capacitor
• State the mechanism by which a capacitor stores energy
• State the voltage-current relationship for a capacitor in both differential and integral form
• State the response of a capacitor to constant voltages and instantaneous voltage changes
• Write the mathematical expression describing energy storage in a capacitor
• Determine the equivalent capacitance of series and parallel combinations of capacitors
• Sketch a circuit describing a non-ideal capacitor
• Write the circuit symbol for an inductor
• State the mechanism by which an inductor stores energy
• State the voltage-current relationship for an inductor in both differential and integral form
• State the response of an inductor to constant voltages and instantaneous current changes
• Write the mathematical expression describing energy storage in an inductor
• Determine the equivalent inductance of series and parallel combinations of inductors
• Sketch a circuit describing a non-ideal inductor

This section provides a brief overview of what it meant by energy storage in terms of a system-level description of some physical process. Several examples of energy storage elements are presented, for which the reader should have an intuitive understanding. These examples are intended to introduce the basic concepts in a qualitative manner; the mathematical analysis of dynamic systems will be provided in later chapters.

We have previously introduced the concept of representing a physical process as a system. In this viewpoint, the physical process has an input and an output. The input to the system is generated from sources external to the system – we will consider the input to the system to be a known function of time. The output of the system is the system’s response to the input. The input-output equation governing the system provides the relationship between the system’s input and output. A general input-output equation has the form:

$$y(t) = f \{ u(t) \} (Eq. 6.1)$$

The process is shown in block diagram form in Figure 6.1.

The system of Figure 6.1 transfers the energy in the system input to the system output. This process transforms the input signal $u(t)$ into the output signal $y(t)$. In order to perform this energy transfer, the system will, in general, contain elements that both store and dissipate energy. To date, we have analyzed systems which contain only energy dissipation elements. We review these systems briefly below in a systems context. Subsequently, we introduce systems that store energy; our discussion of energy storage elements is mainly qualitative in this chapter and presents systems for which the reader should have an intuitive understanding.

In previous chapters, we considered cases in which the input-output equation is algebraic. This implies that the processes being performed by the system involve only sources and components which dissipate energy. For example, output voltage of the inverting voltage amplifier of Figure 6.2 is:

$$V_{OUT} = - \left( \frac{R_f}{R_{in}}V_{in} (Eq. 6.2)$$

This circuit contains only resistors (in the form of $R_f$ and $R_{in}$) and sources (in the form of $V_{in}$ and the op-amp power supplies) and the equation relating the input and output is algebraic. Note that the op-amp power supplies do not appear in equation (6.2), since linear operation of the circuit of Figure 6.2 implies that the output voltage is independent of the op-amp power supplies.

One side effect of an algebraic input-output equation is that the output responds instantaneously to any changes in the input. For example, consider the circuit shown in Figure 6.3. The input voltage is based on the position of a switch; when the switch closes, the input voltage applied to the circuit increases instantaneously from 0V to 2V. Figure 6.3 indicates that the switch closes at time $t = 5$ seconds; thus, the input voltage as a function of time is as shown in Figure 6.4(a). For the values of $R_f$ and $R_{in}$ shown in Figure 6.3, the input-output equation becomes:

$$V_{OUT}(t) = -5V_{in}(t) (Eq. 6.3)$$

and the output voltage as a function of time is as shown in Figure 6.4(b). The output voltage responds immediately to the change in the input voltage.

We now consider systems, which contain energy storage elements. The inclusion of energy storage elements results in the input-output equation for the system, which is a differential equation. We present the concepts in terms of two examples for which the reader most likely has some expectations based on experience and intuition.

As an example of a system, which includes energy storage elements, consider the mass-damper system shown in Figure 6.5. The applied force $F(t)$ pushes the mass to the right. The mass’s velocity is $v(t)$. The mass slides on a surface with sliding coefficient of friction $b$, which induces a force, which opposes the mass’s motion. We will consider the applied force to be the input to our system and the mass’s velocity to be the output, as shown by the block diagram of Figure 6.6. This system models, for example, pushing a stalled automobile.

The system of Figure 6.5 contains both energy storage and energy dissipation elements. Kinetic energy is stored in the form of the velocity of the mass. The sliding coefficient of friction dissipates energy. Thus, the system has a single energy storage element (the mass) and a single energy dissipation element (the sliding friction). In section 4.1, we determined that the governing equation for the system was the first order differential equation:

$$m\frac{dv(t)}{dt} + bv(t) = F(t) (Eq. 6.4)$$

The presence of the energy storage element causes the input-output equation to be a differential equation.

We will examine the effect that the energy storage element has upon the system response in qualitative terms, rather than explicitly solving equation (6.4). If we increase the force applied to the mass, the mass will accelerate and the velocity of the mass increases. The system, therefore, is converting the energy in the input force to a kinetic energy of the mass. This energy transfer results in a change in the output variable, velocity.

The energy storage elements of the system of Figure 6.5 do not, however, allow an instantaneous change in velocity to an instantaneous change in force. For example, say that before time t = 0 no force is applied to the mass and the mass is at rest. At time t = 0 we suddenly apply a force to the mass, as shown in Figure 6.7(a) below. At time t = 0 the mass begins to accelerate but it takes time for the mass to approach its final velocity, as shown in Figure 6.7(b). This transitory stage, when the system is in transition from one constant operating condition to another is called the transient response. After a time, the energy input from the external force is balanced by the energy dissipated by the sliding friction, and the velocity of the mass remains constant. When the operating conditions are constant, the energy input is exactly balanced by the energy dissipation, and the system’s response is said to be in steady-state. We will discuss these terms in more depth in later chapters when we perform the mathematical analysis of dynamic systems.