Electrical Circuits
Electrical Circuits

Electrical Circuits

Lead Author(s): Linda DeBrunner, Victor DeBrunner

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Electrical Circuits provides the non-major with a knowledge of DC circuits, AC circuits, and filtering–both passive filtering and operational amplifier circuits.

Chapter 1. What is Electrical and Computer Engineering?

Electrical and Computer Engineering (ECE) is arguably the broadest of Engineering disciplines:

  • Materials to Complex Systems
  • Physics to Mathematics
  • Devices to Massive Interconnections
  • Hardware to Software

ECE practitioners design devices using novel materials. We design very complex systems such as power generation and distribution systems and communication networks. Devices can be simple, or they can be full-on Systems on a Chip (SoC). A computer chip may have 106 or 109 individual devices, and networks can have at least that many connected devices. These are complex systems that defy easy characterization. ECE practitioners must also use software as an integrated part of most engineering solutions. Real-time software running on embedded microprocessors or microprocessor systems are used in nearly every electronic device you can think of. Most instruments can be considered to be special purpose computers. Physical understanding is important, but mathematical abstraction is also required for ECE professionals. Software running in real-time can mimic differential equations so precisely that they have nearly replaced op-amp filters (usually called "active" RC filters). This is a level of abstraction that many of you do not see at the engineering level (the implementation level) in your disciplines. However, this class will focus nearly exclusively on the core basics that define all of these areas—circuit theory.

Your authors are in the systems area of Computer Engineering. This area encompasses both the theory of the systems

  • Digital Signal and Image Processing
  • Controls
  • Communications

and also their implementation—digital circuits and computing. Many students think of the former as mathematics and the latter as computers, but these are just broad strokes that hide many details. We have already mentioned the notion that software can nearly perfectly implement ordinary differential equations (more precisely, they implement ordinary integral equations for practical reasons that we will discover in this class at a later time). What we have not seen is that the "computers" can come in several different forms. 

​A microprocessor is a small, scaled down version of a computer that you are all intimately familiar with. A microprocessor is at the heart of your laptop, for instance.

Core i7 920 quad front and back [1]

Field programmable logic devices, such as FPGAs (Field Programmable Gate Arrays), have become extremely popular in current engineering practice. These devices are "programmed" using a hardware description language, such as Verilog or VHDL (VHSIC Hardware Description Language). The programming is really a field definition of the circuit connections, i.e. building a circuit in the field with billions of circuit devices/elements. For higher speed needs, industrial production uses ASICs (Application Specific Integrated Circuits) that are pre-designed for a particular application, or that can be programmed for a specific customer. 

An IC (Integrated Circuit) is a device with ever larger numbers of circuit elements built on a single "chip" or device with a common input/output interface. The IC was invented by an Electrical Engineer, Dr. Jack Kilby, in the early 1960s for the Texas Instruments, Inc. This invention won Dr. Kilby the Nobel Prize for Physics in 2000. The invention of the IC completely changed the way ECE is practiced.

  Jack Kilby's Integrated Circuit [2]

​While we will not delve deeply into this invention, we will develop some skills relating to the Operational Amplifier (or Op-Amp) in a later section of this book. Op-Amps can be purchased as ICs.

​ Operational Amplifier IC [3]

Each Op-Amp contains many transistors. Before the integrated circuit was invented, circuits were built with discrete transistors.

                    Transistor-Level Schematic for a 741 op-amp [4]

All of the circuit work has application in:

  • Sensors – e.g. Accelerometers
  • Communications – e.g. JPEG and MPEG
  • Control (feedback systems) – e.g. Vibration control
  •  Power systems

These areas, and many others in ECE, are supported by the professional society for ECE, the IEEE, the Institute for Electrical and Electronic Engineers. To help with some jargon, you might find it helpful to note that Electronic Engineering is another name for the more common (at least in the US) term, “Computer Engineering.” Let's examine each of these applications briefly.

Many of you will be involved in engineering systems or products where sensors must be connected for use by the system or product. Sensors are usually analog (this just means that they will provide a range of output values that are not discrete events). Circuits will match these analog outputs to the computational hardware such as that seen previously. An accelerometer is a sensor based on a material that transduces the energy of physical motion into an electric signal, a current or voltage. The word transduce just means to convert one energy form to another, and a transducer is a device that performs this energy conversion.

​Communication systems connect various sensors and engineering resources together. For example, a camera may provide the vision in a robotic system, and the images will be coded by the camera as JPEG files. These formats preserve memory and bandwidth, both commodities that have real value in communications systems.

CCD (charge-coupled device) sensor from a Sony video camera [5]    

Control systems are used to automate systems and processes. Vibration control, for instance, might use an accelerometer in a control loop to reduce the vibrations in a structural element.

GY-521 MPU-6050 Module 3 Axis Gyroscope + Accelerometer [6]

​However, another element, an actuator, is needed to affect the structural element. Actuators are often electric motors, but there is a wide range of these devices.   

This is a universal motor that runs on alternating current. The commutator is a rotary switch that supplies power to the rotor windings, reversing the direction of current with each half turn so the magnetic field of the winding exerts a steady torque to turn the rotor in one direction. Labeled parts: (A) commutator, (B) brush, (C) rotor (armature) winding, (D), stator (field) winding, (E) brush guide. [7]

Some actuators are connected via circuits. An interesting device is a piezo-electric sensor/actuator which acts as both sensor and actuator. Baseball bats have been designed using these for better performance, for instance.

Finally, most of you will at sometime or another interface with the electric power grid. This grid is an extremely complex circuit as built. It too, has very many devices, just like the IC. But, the power grid has devices that are spread out over an immense geographical area (for instance, North America) instead of so many tiny devices on a single chip. Most of you will not need to understand that full grid; your professional interest will lie mostly with motors and batteries.

Finally, there are other ways to categorize and sub-divide the ECE discipline. Some important ones are:

  • Open loop vs. closed loop
  • DC (direct current)/static vs. AC (alternating current)/dynamic vs. filtering
  • Physical models vs. abstract models
  • Circuits vs. software
  • Legal code vs. engineering practice and economics

Open loop systems operate with no feedback. 


Feedback is the modification of the system performance based on the effects of previous inputs that are observed via sensor. 


We will develop analysis tools for feedback systems in this course. Feedback is an integral part of nearly all ECE practice. For this course, we will see feedback when we discuss Op-Amps.

Question 1

Which of the following systems typically use feedback?


A robotic arm


An elevator


Cruise control


All of the above


None of the above

Circuits operate in several real conditions. Power systems require a steady-state, single frequency, excitation. These circuits are called AC circuits. Most information processing systems do not operate in steady-state, but rather in a combination of steady-state and transient excitations. These circuits are called filters. Analysis of these circuits requires an understanding of dynamic response. Very few practical systems involve only the use of direct current circuits, as this is the static excitation case. No information can be transmitted, and no dynamic system can be controlled with such a circuit. A few power systems, such as automobile lighting systems, are found in practice. However, the concepts that we find for these systems will directly extend to the other two cases, and so we will start with DC circuits in this text, and then proceed to the more mathematically rigorous, and more practical, filtering and AC circuits cases.

Physical models derive from the geometry and materials of the actual system or device. Most engineering problems center around physical items, like a beam or some vibrating element. In ECE, these vibrations are in the energy states of the electrons. These models are typically written using FE (Finite Element) models. Lumped parameter physical models result in ODE (Ordinary Differential Equations). The homogeneous second order differential equation is given by:

When the middle constant, b, is identically zero, the system described is called a (pure) resonator, and the solution is

where the amplitude A, the frequency ω, and the phase φ all depend on the input (non-zero term on the right hand side of the ODE) and the constants m and b.

Abstract models derive from mathematics or philosophical descriptions of systems. There exist many abstract models for a resonator. A clock pendulum can be used to describe the sinusoidal motion. Another common abstraction is to just discretize the differential equation. For that, we need a sampling time, T, so that

and then using the Euler replacement of the first finite difference for the time derivative, i.e. using

we have the approximation

With some normalization, we find

The beam is now modeled very easily using any computational device. Of course, many details are ignored here; for example, the sampling time, T, regulates the quality of the approximation.

The notion of circuits can also be abstract. In the 1960s and earlier, for instance, engineers in the signal processing area mostly worked in radar and sonar, or radio and television. These systems required circuit designers, and many circuits were built and sold for these products. By the 1990s, most of these circuits had been eliminated, as they were replaced by digital computers and hardware doing the work of the circuits using abstract models. Nowadays, a system may be designed in MatlabTM; its performance analyzed and thoroughly studied. Then, MatlabTM may be used to produce VHDL or C, which is then used for either an ASIC or an FPGA, or  it is used in an embedded microprocessor system. This work is not really hardware, nor is it really software. It is computer engineering.

Most work in ECE does not follow legal codes whose primary focus is on safety. For instance, there is a radical difference in the way circuits are treated by the legal system when compared to the manner in which buildings or other structures are treated. Some power systems engineers need to be licensed professionally, but very few others need this licensure. However, our designs are are much more susceptible to economic pressures. The cost of even a few pennies on a device or appliance that sells  just millions of units can add up to many hundreds of thousands of dollars. Bigger selling items scale up in their costs. For that reason, circuits are designed primarily to be inexpensive. Safety is a secondary concern since the risks are often minimal, or even non-existent. Usually, safety problems are handled through lawsuits.

Since ABET accreditation is the driver of this class and not ECE practice, we will be focusing on DC circuits, AC circuits, and filtering. Understanding these concepts will prepare you to interface sensors and actuators in your applications, as well as provide you an understanding of the underlying principles behind current ECE practice.

[1] Image courtesy of Atomicbre under CC BY 3.0 via Wikimedia Commons​

[2] Image courtesy of Summeyye oz under CC BY-SA 4.0  via Wikimedia Commons

[3] Image courtesy of AndyDK at en.wikiversity under CC BY-SA 2.5 via Wikimedia Commons

[4] Image courtesy of Omegatron under CC BY-SA 3.0 or GFDL via Wikimedia Commons

 [5] Image courtesy of wdwd under GFDL or CC BY 3.0  via Wikimedia Commons

[6] Image courtesy of ©Nevit Dilmen  CC BY-SA 3.0 or GFDL via Wikimedia Commons​​

[7] Image courtesy Ulfbastel (original work) and Chetvorno (derived work) under Public Domain, via Wikimedia Commons

[8] Image created by authors

[9] Image created by authors