In previous chapters, a number of approaches have been presented for analyzing electrical circuits. In these analysis approaches, we have been provided with a circuit consisting of a number of elements (resistors, power supplies, etc.) and determined some circuit variable of interest (a voltage or current, for example). In the process of determining this variable, we have written equations which allow us to determine any and all variables in the system. For a complex circuit, with many elements, this approach can result in a very large number of equations and a correspondingly large amount of effort expended in the solution of these equations. Unfortunately, much of the physical insight about the overall operation of the circuit may be lost in the detailed description of all of the individual circuit elements. This limitation becomes particularly serious when we attempt to design a circuit to perform some task.

In this chapter, we introduce the concept of a systems level approach to circuit analysis. In this type of approach, we represent the circuit as a system with some inputs and outputs. We then characterize the system by the mathematical relationship between the system inputs and the system outputs. This relationship is called the input-output relation for the system. This representation of a system leads to several network theorems whose use can simplify the analysis of these systems. The network theorems essentially allow us to model a portion of a complicated circuit as a much simpler (but equivalent) circuit. This simplified model can then be used to facilitate the design or analysis of the remainder of the circuit.

The above approach for representing circuits is particularly useful in circuit design; successful design approaches for large circuits typically use a top-down strategy. In this design approach, the overall system is broken down into a number of interconnected subsystems, each of which performs some specific task. This input-output relationships for these individual subsystems can be determined based on the task performed. The subsystems can then be designed to implement the desired input-output relation. An audio compact disc player, for example, will include subsystems to perform filtering, digital-to-analog conversion, and amplification processes. It is significantly easier to designate the subsystems based on their individual requirements than to attempt to design the entire system all at once. We will thus begin to think of the circuits we analyze as systems which perform some overall task, rather than as a collection of individual circuit elements.

• Define signals and systems
• Represent systems in block diagram form
• Identify system inputs and outputs
• Write input-output equations for systems
• State the defining properties of linear systems
• Determine whether a system is linear
• State conditions under which superposition can be applied to circuit analysis
• Analyze electrical circuits using the principle of superposition
• Define the i-v characteristic for a circuit
• Represent a resistive circuit in terms of its i-v characteristic
• Represent a resistive circuit as a two-terminal network
• Determine Thévenin and Norton equivalent circuits for circuits containing power sources and resistors
• Relate Thévenin and Norton equivalent circuits to i-v characteristics of two-terminal networks
• Determine a load resistance which will maximize the power transfer from a circuit

In this section, we introduce basic concepts relative to systems-level descriptions of general physical systems. Later sections will address application of these concepts specifically to electrical circuits.

A system is commonly represented as shown in the block diagram of Fig. 4.1. The system has some input, u(t), and some output, y(t). In general, both the input and output can be functions of time; the case of constant values is a special case of a time-varying function. The output will be represented as some arbitrary function of the input:

$$y \left( t \right) = f \{ u \left( t \right) \} (Eq. 4.1)$$

Equation (4.1) is said to be the input-output equation governing the system. The above relationship has only one input and one output - the system is said to be a single-input-single-output (SISO) system. Systems can have multiple inputs and multiple outputs, in these cases there will be an input-output equation for each system output and each of these equations may be a function of several inputs. We will concern ourselves only with SISO systems for now.

Once important aspect of the systems-level approach represented by equation (4.1) and Fig. (4.1) is that we are representing our system as a “black box”. We really have no idea what the system itself is, beyond a mathematical dependence of the output variable on the input variable. The physical system itself could be mechanical, thermal, electrical, or fluidic. In fact, it is fairly common to represent a mechanical system as an “equivalent” electrical system (or vice-versa), if doing so increases the physical insight into the system's operation.

The circuits we analyze can now be thought of as systems which perform some overall task, rather than as collections of individual circuit elements. We will also think of the inputs and outputs of the system as signals, rather than specific circuit parameters such as voltages or currents. This approach is somewhat more abstract than we are perhaps used to, so we will provide some additional discussion of what we mean by these terms.

Generally, most people think of a system as a group of interrelated “elements” which perform some task. This viewpoint, though intuitively correct, is not specific enough to be useful from an engineering standpoint. In these chapters, we will define a system as a collection of elements which store and dissipate energy. The system transfers the energy in the system inputs to the system outputs; the process of energy transfer is represented by the input-output equation for the system. Examples of the energy transfer can include mechanical systems (the kinetic energy resulting from using a force to accelerate a mass, or the potential energy resulting from using a force to compress a spring), thermal systems (applying heat to change a mass's temperature), and electrical systems (dissipating electrical power with the filament in a light bulb to produce light).

The task to be performed by the system of Fig. 4.1 is thus the transformation of some input signal u(t) into an output signal y(t). Signals, for us, will be any waveform which can vary as a function of time. This is an extremely broad definition - examples of signals include:

• The force applied to mass.
• The velocity of the mass as it accelerates in response to the applied force.
• The current applied to a circuit by a power supply.
• The voltage difference across a resistor which is subjected to some current flow.
• The electrical power supplied to a heating element.
• The temperature of a mass which is being heated by an electric coil.

The transformation of the input signal to the output signal is performed by the input-output relation governing the system. The input-output relation can be a combination of algebraic, differential, and integral equations.

To provide some concrete examples of the above concepts, several examples of system-level representations of common processes are provided below.

Consider the mass-damper system shown in the figure below. The applied force F(t) pushes the mass to the right. The mass's velocity resulting from the applied force is v(t). The mass slides on a surface wityh sliding coefficient of friction b, which induces a force $F_b=bv(t)$ which opposes the mass's motion. The mass is initially at rest and the applied force is zero for time before time t=0.

The governing equation for the system (obtained by drawing a free body diagram of the mass and applying $ \sum F = ma $) is:

$$m \frac{dv \left( t \right)}{dt} + bv \left( t \right) = F \left( t \right)$$

The governing equation for the system is a first order differential equation. Knowledge of the externally applied force $F \left( t \right)$ and the initial velocity of the mass allows us to determine the velocity of the mass at all subsequent times. Thus, we can model the system as having an input signal $F \left( t \right)$ - which is known - and an output velocity $v \left( t \right)$ which can be determined from the input signal and the properties of the system (the mass, m, and coefficient of fiction, b). The system can then be represented by the block diagram below:

(It is rather unusual to place the system governing equation directly in a block diagram; we do it here to illustrate a point.)

For the electrical circuit below, write the equations governing the input-output relationship for the circuit. The applied input to the circuit is the voltage source $V_{in}$ and the output is the voltage $V$ across the $2\Omega$ resistor.

We previously wrote mesh equations for this circuit (for a specific value of $V_{in}$) in Chapter 3.2. We repeat these mesh equations here, along with our definitions of the mesh currents:

$$4i_1 + 2 \left( i_1 - 3V_X \right) + V_{in} = 0$$

$$-V_{in} +3 \left(i_2 - 3V_X \right) + 5i_2 = 0$$

$$V_X = 5i_2$$

The output voltage V is related to the mesh currents by:

$$V= 2 \left( i_1 - 3V_x \right)$$

The above four equations provide an inpuot-output description of the circuit. If desired, they can be combined to eliminate all variables except $V_{in}$ and $V$ and re-written in the form $V = f \{ V_{in} \}$ per equation (1). Note that all information about the original system, except the relationship between the input and output signals, is lost once we do this.

The system-level block diagram for the circuit might then be drawn as:

Our final example is of a temperature control system. This example illustrates the representation of a complex system as a set of interacting subsystems.

A typical temperature control system for a building will have a thermostat which allows the occupants to set a desired temperature, a furnace (or air conditioner) which provides a means of adjusting the building's temperature, some way of measuring the actual building temperature, and a controller which decides whether to turn the furnace or air conditioner on or off, based on the difference between the desired and actual temperatures. The block diagram below provides one possible approach toward interconnecting these subsystems into an overall temperature control system. This block diagram can be used to identify individual subsystems, and provide specifications for the subsystems, which can allow the design to proceed efficiently. For example:

1. The temperature measurement system might be required to produce a voltage, which is a function of the temperature in the building. The thermistor-based temperature measurement systems we have designed and constructed in the lab are good examples of this type of system.
2. The controller might operate by comparing the desired temperature (generally represented by a voltage level) with the voltage indicating the actual temperature. For a heating system, if the actual temperature is lower than desired by some minimum amount, the controller will make a decision to switch the furnace on. Design decisions might be made to determine what minimum temperature difference is required to turn the furnace on, and whether to base the decision to turn on the furnace strictly upon a temperature difference or on a rate of change in temperature difference.
3. When the furnace turns on it will apply heat to the building, causing the building's temperature to increase. Once the building temperature is high enough, the controller will then typically turn the furnace back off. The furnace must be designed to provide appropriate heat input to the building, based on the building size and the anticipated heat losses to the building's surroundings. (For example, a larger building or a building in a colder climate will require a larger furnace.)
4. A model of the building's heat losses will generally be necessary in order to size the furnace correctly and choose an appropriate control scheme. Design choices for the building itself may include insulation requirements necessary to satisfy desired heating costs.

Designs for the above subsystems can now proceed somewhat independently, with proper coordination between the design activities.

• Systems are a set of components which work together to perform some task. Systems are typically considered to have one or more inputs (which are provided to the system from the external environment) and one or more outputs (which the system provides to the environment).
• Generically, the inputs and outputs of systems are signals. Signals are simply time-varying functions. They can be voltages, currents, velocities, pressures, etc.
• Systems are often characterized by their input-output equations. The input-output equation for a system simply provides a mathematical relationship between the input to the system and the output from the system. Once the input is defined as a particular number or function of time, that value or function can be substituted into the input-output equation to determine the system's response to that input.

1. The input to the circuit below is the current, U. The output is the current through the $10\Omega$ resistor, I. Determine an input-output equation for the circuit.

2. The input to the circuit is the voltage U. The output is the voltage $V_1$. Determine an input-output relation for the circuit.

We have so far introduced a number of approaches for analyzing electrical circuits, including: Kirchhoff's current law, Kirchhoff's voltage law, circuit reduction techniques, nodal analysis, and mesh analysis. When we have applied the above analysis methods, we have generally assumed that any circuit elements operate linearly. For example, we have used Ohm's law to model the voltage-current relationship for resistors. Ohm's law is applicable only for linear resistors - that is, for resistors whose voltage-current relationship is a straight line described by the equation $v=Ri$. Non-linear resistors have been mentioned briefly; in Lab Assignment 1, for example, we forced a resistor to dissipate an excessive amount of power, thereby causing the resistor to burn out and display nonlinear operating characteristics. All circuit elements will display some degree of non-linearity, at least under extreme operating conditions.

Unfortunately, the analysis of non-linear circuits is considerably more complicated than analysis of linear circuits. Additionally, in subsequent chapters we will introduce a number of analysis methods which are applicable only to linear circuits. The analysis of linear circuits is thus very pervasive - for example, designing linear circuits is much simpler than the design of non-linear circuits. For this reason, many non-linear circuits are assumed to operate linearly for design purposes; non-linear effects are accounted for subsequently during design validation and testing phases.