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Sinusoidal waveforms as explained in Chap. 3 have an amplitude r, a frequency ω, and a phase shift or phase angle ϕ and are expressed as

Electric circuits consist of several components that form a certain topology. The drivers of the circuit can be voltage sources and/or current source. They force the current pass through the circuit by generating voltage drops across elements. A circuit performs certain tasks and has a certain output as well. The sources which are considered as input to the circuit can generate various waveforms. These waveforms excite the circuit and cause different effects.

Most of the circuits introduced so far have been analyzed in time domain. This means that the input to the circuit, the circuit variables, and the responses have been presented as a function of time. All the input functions such as unit step, ramp, impulse, exponential, sinusoidal, etc. have been introduced as a time-dependent variable, and their effects on circuits have been identified directly as a function of time. This required utilization of differential equations and solutions in time domain. However, high-order circuits result in high-order differential equations, which, considering the initial conditions, sometimes are hard to solve. In addition, for circuits which are exposed to a spectrum of frequencies such as filters, the time domain analysis is a limiting factor.

Electric circuits are a method to demonstrate how an electric device receives power and operate. For instance, a flashlight has batteries to provide power to a light bulb, wires to bring the power from batteries to the bulb, and a switch to control the on-off action. The operation of this flashlight can be modeled by its electric circuit. Of course, it has mechanical components to hold the batteries, wires, the switch, and the bulb in place and make the entire unit waterproof. The wires and the switch have electric insulators to protect the flow of electric current in the wires. Nevertheless, these components have no electrical importance when it comes to the distribution of the power and modeling of the circuit. Figure 1.1 shows a flashlight with its internal components and its equivalent circuit.

Electric circuit analysis is the collection of methods and tools to determine the voltages and currents as well as the power consumption and generation in electric circuits and components. The relations among the electric elements depend on the elements of the circuit and their configuration or topology. That is how the circuit elements are connected together.

The flow of current in the circuit branches and drop of voltage across circuit elements depend on their behavior and their ability to store energy. For instance, the voltage drop across a resistor is in phase with its current passing through. But that is not the same in a capacitor or an inductor. This makes the circuit KVL and KCL equations integrodifferential equations. The order of these equations depends on the number of energy-storing elements. In this chapter, the circuit elements are introduced and their equations are discussed. The order of a circuit is discussed, and responses of first- and second-order circuits to their initial condition and to external sources are analyzed.

A network is a combination of one or several electric circuits that together perform a specific action. The network might have several inputs that receive excitations and outputs to show the results. A port is a set of two terminals that allows for a source to connect and excite the network or connect a measurement device and record the response. For instance, a voltage source connected to a port of a network may cause currents to flow through the network and voltage drops to appear across the elements. To measure any of these currents or voltages, terminals may be extended to demonstrate some measurment locations, forming an output port. This forms a two-port network. Similarly, two-port networks can be easily expanded to multiple-port networks each showing a parameter in the circuit. Figure 12.1 shows a circuit and the process to consider it as a two-port network.