Circuitry Articles

Article / Updated 07-29-2022

To find the total response of an RL parallel circuit such as the one shown here, you need to find the zero-input response and the zero-state response and then add them together. After fiddling with the math, you determine that the zero-input response of the sample circuit is this: Now you are ready to calculate the zero-state response for the circuit. Zero-state response means zero initial conditions. For the zero-state circuit shown earlier, zero initial conditions means looking at the circuit with zero inductor current at t < 0. You need to find the homogeneous and particular solutions to get the zero-state response. Next, you have zero initial conditions and an input current of iN(t) = u(t), where u(t) is a unit step input. When the step input u(t) = 0, the solution to the differential equation is the solution ih(t): The inductor current ih(t) is the solution to the homogeneous first-order differential equation: This solution is the general solution for the zero input. You find the constant c1 after finding the particular solution and applying the initial condition of no inductor current. After time t = 0, a unit step input describes the transient inductor current. The inductor current for this step input is called the step response. You find the particular solution ip(t) by setting the step input u(t) equal to 1. For a unit step input iN(t) = u(t), substitute u(t) = 1 into the differential equation: The particular solution ip(t) is the solution for the differential equation when the input is a unit step u(t) = 1 after t = 0. Because u(t) = 1 (a constant) after time t = 0, assume a particular solution ip(t) is a constant IA. Because the derivative of a constant is 0, the following is true: Substitute ip(t) = IA into the first-order differential equation: The particular solution eventually follows the form of the input because the zero-input (or free response) diminishes to 0 over time. You can generalize the result when the input step has strength IA or IAu(t). You need to add the homogeneous solution ih(t) and the particular solution ip(t) to get the zero-state response: At t = 0, the initial condition is 0 because this is a zero-state calculation. To find c1, apply iZS(0) = 0: Solving for c1 gives you C1 = -IA Substituting c1 into the zero-state response iZS(t), you wind up with

Article / Updated 07-29-2022

A first-order RL parallel circuit has one resistor (or network of resistors) and a single inductor. First-order circuits can be analyzed using first-order differential equations. By analyzing a first-order circuit, you can understand its timing and delays. Analyzing such a parallel RL circuit, like the one shown here, follows the same process as analyzing an RC series circuit. So if you are familiar with that procedure, this should be a breeze. If your RL parallel circuit has an inductor connected with a network of resistors rather than a single resistor, you can use the same approach to analyze the circuit. But you have to find the Norton equivalent first, reducing the resistor network to a single resistor in parallel with a single current source. Start with the simple RL parallel circuit Because the resistor and inductor are connected in parallel in the example, they must have the same voltage v(t). The resistor current iR(t) is based on Ohm’s law: The element constraint for an inductor is given as where i(t) is the inductor current and L is the inductance. You need a changing current to generate voltage across an inductor. If the inductor current doesn’t change, there’s no inductor voltage, which implies a short circuit. Now substitute v(t) = Ldi(t)/dt into Ohm’s law because you have the same voltage across the resistor and inductor: Kirchhoff’s current law (KCL) says the incoming currents are equal to the outgoing currents at a node. Use KCL at Node A of the sample circuit to get iN(t) = iR(t) =i(t). Substitute iR(t) into the KCL equation to give you The RL parallel circuit is a first-order circuit because it’s described by a first-order differential equation, where the unknown variable is the inductor current i(t). A circuit containing a single equivalent inductor and an equivalent resistor is a first-order circuit. Knowing the inductor current gives you the magnetic energy stored in an inductor. In general, the inductor current is referred to as a state variable because the inductor current describes the behavior of the circuit. Calculate the zero-input response for an RL parallel circuit Here is how the RL parallel circuit is split up into two problems: the zero-input response and the zero-state response. Here, you’ll start by analyzing the zero-input response. To simplify matters, you set the input source (or forcing function) equal to 0: iN(t) = 0 amps. This means no input current for all time — a big, fat zero. The first-order differential equation reduces to For an input source of no current, the inductor current iZI is called a zero-input response. No external forces are acting on the circuit except for its initial state (or inductor current, in this case). The output is due to some initial inductor current I0 at time t = 0. You make a reasonable guess at the solution (the natural exponential function!) and substitute your guess into the RL first-order differential equation. Assume the inductor current and solution to be iZI(t) = Bekt This is a reasonable guess because the time derivative of an exponential is also an exponential. Like a good friend, the exponential function won’t let you down when solving these differential equations. You determine the constants B and k next. Substitute your guess iZI(t) = Bekt into the differential equation: Replacing iZI(t) with Bekt and doing some math gives you the following: You have the characteristic equation after factoring out Bekt: The characteristic equation gives you an algebraic problem to solve for the constant k: Use k = –R/L and the initial inductor current I0 at t = 0. This implies that B = I0, so the zero-input response iZI(t) gives you the following: The constant L/R is called the time constant. The time constant provides a measure of how long an inductor current takes to go to 0 or change from one state to another. To analyze the RL parallel circuit further, you must calculate the circuit’s zero-state response, and then add that result to the zero-input response to find the total response for the circuit.

Cheat Sheet / Updated 01-26-2022

When doing circuit analysis, you need to know some essential laws, electrical quantities, relationships, and theorems. Ohm’s law is a key device equation that relates current, voltage, and resistance. Using Kirchhoff’s laws, you can simplify a network of resistors using a single equivalent resistor. You can also do the same type of calculation to obtain the equivalent capacitance and inductance for a network of capacitors or inductors. For more complicated circuits, the node-voltage analysis and mesh current techniques come in handy. And when you want to try different loads for a particular source circuit, you can use the Thévenin or Norton equivalent.

Article / Updated 09-27-2021

Before you start working with line voltage in your electronic circuits, you need to understand a few details about how most residential and commercial buildings are wired. The following description applies only to the United States; if you’re in a different country, you’ll need to determine the standards for your country’s wiring. Standard line voltage wiring in the United States is done with plastic-sheathed cables, which usually have three conductors. This type of cable is technically called NMB cable, but most electricians refer to it using its most popular brand name, Romex. Three conductors inside electric cables Two of the conductors in NMB cable are covered with plastic insulation (one white, the other black). The third conductor is bare copper. These conductors are designated as follows: Hot: The black wire is the hot wire, which provides a 120 VAC current source. Neutral: The white wire is called the neutral wire. It provides the return path for the current provided by the hot wire. The neutral wire is connected to an earth ground. Ground: The bare wire is called the ground wire. Like the neutral wire, the ground wire is also connected to an earth ground. However, the neutral and ground wires serve two distinct purposes. The neutral wire forms a part of the live circuit along with the hot wire. In contrast, the ground wire is connected to any metal parts in an appliance, such as a microwave oven or coffee pot. This is a safety feature, in case the hot or neutral wires somehow come in contact with metal parts. Connecting the metal parts to earth ground eliminates the shock hazard in the event of a short circuit. Note that some circuits require a fourth conductor. When a fourth conductor is used, it's covered with red insulation and is also a hot wire. How they're connected to a standard outlet The three wires in a standard NMB cable are connected to the three prongs of a standard electrical outlet (properly called a receptacle). As you can see, the neutral and hot wires are connected to the two vertical prongs at the top of the receptacle (neutral on the left, hot on the right) and the ground wire is connected to the round prong at the bottom of the receptacle. You can plug a two-prong or three-prong plug into a standard three-prong receptacle. Two-prong plugs are designed for appliances that don't require grounding. Most nongrounded appliances are double-insulated, which means that there are two layers of insulation between any live wires and any metal parts within the appliance. The first layer is the insulation on the wire itself; the second is usually in the form of a plastic case that isolates the live wiring from other metal parts. Three-prong plugs Three-prong plugs are for appliances that require the ground connection for safety. Most appliances that use a metal chassis require a separate ground connection. There is only one way to insert a three-prong plug into a three-prong receptacle. But regular two-prong plugs, which lack the ground prong, can be connected with either prong on the hot side. To prevent that from happening, the receptacles are polarized, which means that the neutral prong is wider than the hot prong. Thus, there's only one way to plug a polarized plug into a polarized receptacle. That way, you can always keep track of which wire is hot and which is neutral. You should always place switches or fuses on the hot wire rather than on the neutral wire. That way, if the switch is open or the fuse blows, the current in the hot wire will be prevented from proceeding beyond the switch or fuse into your circuit. This minimizes any risk of shock that might occur if a wire comes loose within your project.

Article / Updated 04-21-2017

There are many applications for an RLC circuit, including band-pass filters, band-reject filters, and low-/high-pass filters. You can use series and parallel RLC circuits to create band-pass and band-reject filters. An RLC circuit has a resistor, inductor, and capacitor connected in series or in parallel. You can get a transfer function for a band-pass filter with a parallel RLC circuit, like the one shown here. You can use current division to find the current transfer function of the parallel RLC circuit. By measuring the current through the resistor IR(s), you form a band-pass filter. Start with the current divider equation: A little algebraic manipulation gives you a current transfer function, T(s) = IR(s)/IS(s), for the band-pass filter: Plug in s = jω to get the frequency response T(jω): This equation has the same form as the RLC series equations. For the rest of this problem, you follow the same process as for the RLC series circuit. The transfer function is at a maximum when the denominator is minimized, which occurs when the real part of the denominator is set to 0. The cutoff frequencies are found when their gains |T(jωC)| = 0.707|T(jω)| or the –3 dB point. Therefore, ω0 is The center frequency, the cutoff frequencies, and the bandwidth have equations identical to the ones for the RLC series band-pass filter. Your cutoff frequencies are ωC1 and ωC2: The bandwidth BW and quality factor Q are

Article / Updated 04-11-2017

The easiest way to provide a voltage source for an electronic circuit is to include a battery. There are plenty of other ways to provide voltage, including AC adapters (which you can plug into the wall) and solar cells (which convert sunlight to voltage). However, batteries remain the most practical source of juice for most electronic circuits. A battery is a device that converts chemical energy into electrical energy in the form of voltage, which in turn can cause current to flow. A battery works by immersing two plates made of different metals into a special chemical solution called an electrolyte. The metals react with the electrolyte to produce a flow of charges that accumulate on the negative plate, called the anode. The positive plate, called the cathode, is sucked dry of charges. As a result, a voltage is formed between the two plates. These plates are connected to external terminals to which you can connect a circuit to cause current to flow. Batteries come in many different shapes and sizes, but for electronics projects, you need concern yourself only with a few standard types of batteries, all of which are available at any grocery, drug, or department store. Cylindrical batteries come in four standard sizes: AAA, AA, C, and D. Regardless of the size, these batteries provide 1.5 V each; the only difference between the smaller and larger sizes is that the larger batteries can provide more current. The cathode, or positive terminal, in a cylindrical battery is the end with the metal bump. The flat metal end is the anode, or negative terminal. The rectangular battery is a 9 V battery. That little rectangular box actually contains six small cells, each about half the size of a AAA cell. The 1.5 volts produced by each of these small cells combine to create a total of 9 volts. Here are a few other things you should know about batteries: Besides AAA, AA, C, D, and 9 V batteries, many other battery sizes are available. Most of those batteries are designed for special applications, such as digital cameras, hearing aids, laptop computers, and so on. All batteries contain chemicals that are toxic to you and to the environment. Treat them with care, and dispose of them properly according to your local laws. Don't just throw them in the trash. You can (and should) use your multimeter to measure the voltage produced by your batteries. Set the multimeter to an appropriate DC voltage range (such as 20 V). Then, touch the red test lead to the positive terminal of the battery and the black test lead to the negative terminal. The multimeter will tell you the voltage difference between the negative and positive terminals. For cylindrical batteries (AAA, AA, C, or D) it should be about 1.5 V. For 9 V batteries, it should be about 9 V. Rechargeable batteries cost more than non-rechargeable batteries but last longer because you can recharge them when they go dead. The easiest way to use batteries in an electronic circuit is to use a battery holder, which is a little plastic gadget designed to hold one or more batteries. Wonder why they sell AAA, AA, C, and D cells but not A or B? Actually, A cell and B cell batteries exist. However, those sizes never really caught on. Check out which tools you'll need for electronics projects.

Article / Updated 02-09-2017

In circuits, inductors resist instantaneous changes in current and store magnetic energy. Inductors are electromagnetic devices that find heavy use in radiofrequency (RF) circuits. They serve as RF “chokes,” blocking high-frequency signals. This application of inductor circuits is called filtering. Electronic filters select or block whichever frequencies the user chooses. Describe an inductor Unlike capacitors, which are electrostatic devices, inductors are electromagnetic devices. Whereas capacitors avoid an instantaneous change in voltage, inductors prevent an abrupt change in current. Inductors are wires wound into several loops to form coils. In fact, the inductor’s symbol looks like a coil of wire, as shown here. Current flowing through a wire creates a magnetic field, and the magnetic field lines encircle the wire along its axis. The concentration, or density, of the magnetic field lines is called magnetic flux. The coiled shape of inductors increases the magnetic flux that naturally occurs when current flows through a straight wire. The greater the flux, the greater the inductance. If you needed a circuit that stored more magnetic energy, you could get even larger inductance values by inserting iron into the wire coil. Here’s the defining equation for the inductor: where the inductance L is a constant measured in henries (H). Here is the same equation in graphical form. The figure shows the i-v characteristic of an inductor, where the slope of the line is the value of the inductance. The preceding equation says that the voltage across the inductor depends on the time rate of change of the current. In other words, no change in inductor current means no voltage across the inductor. To create voltage across the inductor, current must change smoothly. Otherwise, an instantaneous change in current would create one humongous voltage across the inductor. Think of inductance L as a proportionality constant, like a resistor acts as a constant in Ohm’s law. This notion of Ohm’s law for inductors (and capacitors) becomes useful when you start working with phasors. To express the current through the inductor in terms of the voltage, you integrate the preceding equation as follows: The second term in this equation is the initial current through the inductor at time t = 0. Find the energy storage of an attractive inductor To find the energy stored in the inductor, you need the following power definition, which applies to any device: The subscript L denotes an inductor device. Substituting the voltage for an inductor into the power equation gives you the following: The energy wL(t) stored per unit time is the power. Integrating the preceding equation gives you the energy stored in an inductor: The energy equation implies that the energy in the inductor is always positive. The inductor absorbs power from a circuit when storing energy, and the inductor releases the stored energy when delivering energy to the circuit. To visualize the current and energy relationship shown here, which shows the current as a function of time and the energy stored in an inductor. This also shows how you can get the current from the inductor relationship between current and voltage. Calculate total inductance for series and parallel inductors Inductors connected in series or connected in parallel can be reduced to one single inductor. Take a look at the circuit with three series inductors shown in the top diagram. Because the inductors are connected in series, they have the same currents: i1(t) = i2(t) =i3(t) = i(t) Add up the voltages from the series inductors to get the net voltage v(t), as follows: For a series inductors, you have an equivalent inductance of LEQ = L1 + L2 + L3 For a parallel connection of inductors, apply Kirchhoff’s current law (KCL) in the bottom diagram of the figure. KCL says the sum of the incoming currents and outgoing current at a node is equal to 0, giving you Because you have the same voltage v(t) across each of the parallel inductors, you can rewrite the equation as This equation shows how you can reduce the parallel inductors to one single inductor:

Article / Updated 03-26-2016

You've scrounged around your growing electronics bin and come up with wires to connect a circuit together and batteries to power the circuit. So how do you turn the power on and off? You use switches and relays. Turning current on and off with switches When you move the switch to shut off your flashlight, you disconnect the wires that run from the battery to the light bulb. All switches do the same thing: Connect wires to allow electric current to flow or disconnect wires to stop electric current from flowing. When you turn off your flashlight, you put the switch in what is called the open position. With the switch in the open position, you have a disconnected wire, and no current can flow. When you turn on the flashlight, you put the switch in the closed position. With the switch in the closed position, you've connected the wire (and completed the circuit), and current can flow. Starting with simple switches Your flashlight usually comes with something called a slide switch. With a slide switch, you slide the switch forward or backward to turn the light on or off. But toggle, rocker, and slide switches all do the same job, so grab whatever switch you have handy that you can easily use on the project that you're building. For example, a slide switch works well on a round, handheld flashlight because of the position of your thumb, but a toggle switch may work best to flip on a gadget sitting on your workbench. Push-button switches come in three versions: Normally closed (NC): This push-button switch disconnects the wire only when you push the button. Normally open (NO): This push-button switch connects the wire only when you push the button. Push on/Push off buttons: This switch connects the wire with one push and disconnects the wire with the next. You typically find push-button switches in electronics to start or stop a circuit. For example, you press a normally open push-button switch to ring a doorbell. What's inside a switch? You call the basic switches that we talk about in the previous section single-pole single-throw, or SPST types. Don't worry about all the different names: In essence, these switch types have one wire coming into the switch and one wire leaving it. Just to keep your electronics life interesting, you may come across other types of switches that are wired a bit differently, called double pole. Where single pole switches have one input wire, double pole switches have two input wires. With single throw switches you can connect or disconnect each input wire to one output wire, while double throw switches allow you to choose which of two output wires you connect each input wire to. There are a few single- and double-pole variations, including Single-pole double-throw (SPDT): In this switch, one wire comes into the switch and two wires leave the switch. When you want to choose what device a circuit turns on (for example, a green light to let people know that they can enter a room or a red light to tell them to stay out), use an SPDT switch. Double-pole single-throw (DPST): This switch has two wires coming into it and two wires leaving. You can use a DPST switch to control two separate circuits. For example, you can have one circuit with components running on 5 volts and another circuit with components running on 12 volts. With one switch, you can turn both circuits on or off. Double-pole double-throw (DPDT): This switch has two wires coming into it and four wires leaving. A DPDT switch has three positions. In the first position, the first pair of output wires connect. In the second position, all four output wires disconnect (some DPDT switches do not have this position). In the third position, the second pair of output wires connect. You can use this type of switch to reverse the polarity of DC voltage going into a motor so that the motor turns in the opposite direction. (One position makes the motor turn clockwise, one position turns off power to the motor, and one position turns the motor counterclockwise.) Let a relay flip the switch You've made a gadget to let you know when your no-good brother-in-law, Herman, is raiding the refrigerator. But there's one problem: The gadget runs on a 5-volt battery pack, and you want the gadget to turn on enough sound and light to scare the guy into the next county. No problem, just use a relay. How relays work A relay is simply an electrically powered switch. When your gadget sends 5 volts to the relay, an electromagnet turns on and then closes a switch inside the relay. If you wire that switch to 117 volts, you can power enough lights and sirens to send Herman scurrying. Exploring electromagnets So how does the electromagnet part of a relay setup work? An electromagnet can be something as simple as coiled wire around an iron bar or even a nail. When you run some current through the wire, the nail becomes magnetized. When you shut off the current, the nail loses that magnetic quality. Two magnets attract or repel each other, depending on which ends (or poles) of the magnets you put together. Part of the switch contained in a relay consists of a lever attached to a magnet. When voltage runs through the wire coil, the electromagnet pulls the lever toward it, and the switch closes, connecting the 115 volts to the lights and sirens (goodbye, Herman!). When you shut off current to the wire coil the electromagnet shuts off and a spring pulls the lever away, opening the switch. You can find relays that use 5, 12, or 24 VDC to power an electromagnet with a SPST, SPDT, or DPDT switch. Here are a few relay lingo options. Often instead of saying that a switch in the relay opens or closes, people talk about contacts opening or closing. Also, people sometimes call a lever in a relay an armature. But a relay by any other name, would work the same...

Article / Updated 03-26-2016

Operational amplifiers are one of the most common types of integrated circuits. The LM741 is a popular single op-amp IC, so you should understand the purpose of each of the pins in this integrated circuit to make your electronics project run smoothly. Pin Function 1 Not used 2 V- Inverting Input 3 V+ Non-inverting Input 4 -V Power 5 Not used 6 Vout Output 7 +V Power 8 Not used

Article / Updated 03-26-2016

If you are interested in understanding electronic circuits, one of the best ways to learn about electronics is to build a simple circuit. This simple circuit consists of just three components: a 9 V battery, a light-emitting diode (LED), and a resistor. Not only will you learn something about building circuits, but you can also you this completed circuit to practice using your multimeter. Here is the schematic for this circuit: You can build this circuit on a solderless breadboard. You'll need the following parts: Small solderless breadboard 470 Ω, 1/4 W resistor Red LED, 5 mm 9 V battery snap connector 9 V battery Short length of jumper wire (1″ or less) Here are the steps for building this circuit: Connect the battery snap connector. Insert the red lead in the top bus strip and the black lead in the bottom bus strip. Any hole will do, but it makes sense to connect the battery at the very end of the breadboard. Connect the resistor. Insert one end of the resistor into any hole in the bottom bus strip. Then, pick a row in the nearby terminal strip and insert the other end into a hole in that terminal strip. Connect the LED. Notice that the leads of the LED aren't the same length; one lead is shorter than the other. Insert the short lead into a hole in the top bus strip, and then insert the longer lead into a hole in a nearby terminal strip. Insert the LED into the same row as the resistor. Both the LED and the resistor are in row 26. Use the short jumper wire to connect the terminal strips into which you inserted the LED and the resistor. The jumper wire will hop over the gap that runs down the middle of the breadboard. Connect the battery to the snap connector. The LED will light up. If it doesn't, double-check your connections to make sure the circuit is assembled correctly. If it still doesn't light up, try reversing the leads of the LED (you may have inserted it backwards). If that doesn’t work, try a different battery. Do not connect the LED directly to the battery without a resistor. If you do, the LED will flash brightly, and then it will be dead forever.