# ATAT 105 Basic Electricity > # [[T105 Week 4| ◀️ ]] &nbsp;[[T105 Home| Home ]] &nbsp;[[T105 Week 6| ▶️ ]] &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; [[QR T105 Week 5| 🌐 ]] ># [[T105 Week 5|Week 5]] >- [[T105 Week 5#Power|Power]] >- [[T105 Week 5#Parallel Circuits|Parallel Circuits]] ># [[T105 Week 5#Lab|Lab]] >- [[T105 Week 5#1 Calculate power in a series circuit|Calculate power in a series circuit]] >- [[T105 Week 5#2 Another circuit|Another circuit]] >- [[T105 Week 5#3 Parallel circuit 1 matches Circuit 1|Parallel circuit 1]] >- [[T105 Week 5#4 Parallel circuit 2 matches Circuit 2|Parallel circuit 2]] >- [[T105 Week 5#5 Calculations in a Parallel Circuit 1|Calculations in a Parallel Circuit 1]] >- [[T105 Week 5#6 Calculations in a Parallel Circuit 2|Calculations in a Parallel Circuit 2]] >- [[T105 Week 5#7 Variable Resistance in Parallel|Variable Resistance in Parallel]] >[!jbplus|c-blue]- Lesson Intro >### What >In this lesson you will learn about power formulas and parallel circuits. Some of the things you observed in the lab will now become clear. > >### Why > >The relationships between voltage, current and resistance and the mathematical calculations that allow us to derive values in a circuit will be used in the electrical lab, and when troubleshooting avionics or electrical snags. We have learned series formulas and laws, and now we learn parallel to complete this level of learning. > >### Approach and Objectives > >By understanding the following topics, you will have achieved the learning outcome for this lesson. Consult your course outline for the learning outcomes and other details of this course. > >#### Course Learning Objectives >- CLO 3. Express numerical quantities in scientific and metric notation. >- CLO 7. Validate using Kirchoff's Laws the values of voltage and current in a DC circuit. >- CLO 8. Calculate the equivalent resistance for Series, Parallel and Series-Parallel circuits. >- CLO 9. Calculate the power dissipated across an electrical load given other basic values. >- CLO 10. Assemble a functional electrical circuit using components according to a given circuit diagram. >- CLO 11. Show using a Digital Multimeter (DMM) the measurement of voltage, current and resistance in a circuit. > > >### Testing > >You will be tested on this material on the midterm test and the final test. Details [[T105 Intro#Testing and Grades|here]]. >[!jbplus|c-blue]- Prof >### Objectives > >This week we introduce the power formula, and all its derivatives, and then a jump into parallel circuits with both feet. All the laws and formulas all at once. This is made easier by making comparisons to series circuits. > > ### Theory >For the theory course, calculators out! Students should be doing all calculations in class as you go. You can warn them that this is direct preparation for the MP next week. >### Lab >For the lab, calculate and measure, with an emphasis on pointing out the differences now seen in parallel circuits. Notice how some of the last exercises are really just practice, i.e. no new concepts. If time is tight, moving directly to the [[T105 Week 5#7 Variable Resistance in Parallel|last exercise]] may be more valuable. ## Power When a circuit is functioning, which would mean that voltage and current are present, power is produced. Note that an open circuit is not functioning, and has no current. Power is the product of both voltage and current, and its relationship is proportional as per this diagram.  The format at least should remind you of the similar [[T105 Week 4#Ohm's Law as a Formula|Ohm's Law.]] The mathematical implications should also be readily understandable. By reading the graphic above, and applying our knowledge we can come up with the following formulas easily: P = ? I = ? E = ? [[A Power Law|🅰]] Some useful things become apparent. We now have a different formula to use to solve for current. We already know that $I = \frac{V}{R}$ and now we know it is also $I = \frac{P}{E}$. This is extremely useful as you have seen in the [[T110T SSGW05#Solving for missing information|math course]]. We also have a new way to solve for voltage: $E = \frac{P}{I}$ #### Power Examples We have a 12V battery driving a lamp at 3A. Calculate the power produced in this circuit. P = IV = 3A x 12V = 36W ![[Pasted image 20210808131355.png|350]] We have a 12V battery driving a 3Ω lamp. Calculate the power produced in this circuit. Here we must perform an intermediate step. We are given voltage and resistance, but the power formula does not account for resistance. What to do? Go back to Ohm's law first. With voltage and resistance we can solve for current: I = V/R = 12V/3Ω = 4A And then we have both voltage and current, allowing us to solve for power: P = IV = 4A x 12V = 48W ![[Pasted image 20210808131413.png|350]] We have a battery driving a 3Ω lamp at 5A. Calculate the Power. ![[Pasted image 20210808131435.png|350]] [[A Power Law 2|🅰]] ### Ohm's Law and the Power Wheel This wheel shows you all of the formulas that result from the combination of the Ohm's Law and Power Law formulas. You may see a reference like this in a lab or near a workbench where these calculations may be necessary. For our purposes however, we will be very comfortable deriving each formula using our math and electricity skills. This falls under the category of shortcuts. Why learn it when it is all right there on the wheel? Because experts know how this wheel was derived in every detail, they can use it rapidly but correctly. You also should understand every single aspect of this graphic. For this reason, this graphic will not be provided to you during testing. ![[Pasted image 20211005061210.png|350]] Use the practice questions this week, and check out the following math to fully get a handle on the values that are produced by an electrical circuit. ### Ohm's Law and Power Law Formulas #### Calculating Voltage First, let's place the [[T105 Week 4#Ohm's Law as a Formula|Ohm's Law]] and Power Law in ready view:   $V = I \times R$ and $P = I \times V$ Let's start with all the ways to calculate voltage: The laws give us the first two easily: $V = I \times R$ $V = \frac {P}{I}$ But since $I = \frac {P}{V}$, we can use this to replace I in the first equation, giving us: $V = \frac {P}{V} \times R$ To remove the V on the right side, we multiply both sides by V to get: $V^2 = P \times R$ To solve for V, we need to find the square root of both sides, giving us our final: $V = \sqrt {P \times R}$ ##### Voltage Formula Recap: $V = I \times R$ $V = \frac {P}{I}$ $V = \sqrt {P \times R}$ #### Calculating Current We are given: $I = \frac {V}{R}$ and $I = \frac {P}{V}$ Since $V = I \times R$ we can replace V like this: $I = \frac {P}{I \times R}$ To remove the I on the right side, we multiply both sides by I giving us: $I^2 = \frac {P}{R}$ We find the square root of both sides to give us: $I = \sqrt \frac {P}{R}$ ##### Current Formula Recap: $I = \frac {V}{R}$ $I = \frac {P}{V}$ $I = \sqrt \frac {P}{R}$ #### Calculating Resistance Ohm's Law gives us: $R = \frac {V}{I}$ Since $V = \frac {P}{I}$, we replace V in the equation: $R = \frac {\frac {P}{I}} {I}$ which reduces to: $R = \frac {P}{I^2}$ Since $I = \frac {P}{V}$ we replace I in the equation: $R = \frac {V}{\frac {P}{V}}$ which reduces to: $R = \frac {V^2}{P}$ ##### Resistance Formula Recap: $R = \frac {V}{I}$ $R = \frac {P}{I^2}$ $R = \frac {V^2}{P}$ #### Calculating Power %% [[T105T SSGW05CHANGE#Calculating Current]] %% The Power Law gives us: $P = I \times V$ Since $V = I \times R$ we replace V: $P = I \times (I \times R)$ which reduces to: $P = I^2 \times R$ And finally, since $I = \frac {V}{R}$ we replace I in the equation to get: $P = \frac {V}{R} \times V$ reduced to: $P = \frac {V^2}{R}$ ##### Power Formula Recap: $P = I \times V$ $P = I^2 \times R$ $P = \frac {V^2}{R}$ You see how we have derived each and every formula from the Ohm's Law and Power Wheel: ![[Pasted image 20211005061210.png|350]] $P = I \times V$ $P = I^2 \times R$ $P = \frac {V^2}{R}$ $I = \frac {V}{R}$ $I = \frac {P}{V}$ $I = \sqrt \frac {P}{R}$ $R = \frac {V}{I}$ $R = \frac {P}{I^2}$ $R = \frac {V^2}{P}$ $V = I \times R$ $V = \frac {P}{I}$ $V = \sqrt {P \times R}$ At this time I would like to point out something important in terms of your studying strategy. If you build your understanding step by step as we have just done, you will easily be able to derive and use these formulas. Memorizing these 12 formulas without understanding them will be much harder, take much longer, and you are much more likely to make a mistake. ## Parallel Circuits Parallel circuits are constructed such that each load is connected directly to the power source. This means there is more than one path the current can follow. This then is a major departure from the series circuit which had only one path. A parallel circuit has more than one current path (branch) connected to a common voltage source. ![[Pasted image 20210808131712.png|350]] For example, the four resistors (below) are connected in parallel. Each resistor will see the same source at A & B. Put another way, each resistor is directly wired to the source. ![[Pasted image 20210808131744.png|350]] In this circuit, you can see quite clearly that every single component is connected directly to the battery. ![[Pasted image 20210808131759.png|350]] ### Parallel Circuit Rule for Voltage Because all components are connected across the same voltage source, the voltage across each is the same. ![[Pasted image 20210808131835.png|350]] For example, the source voltage is 5.0 V. What will a voltmeter read if it is placed across each of the resistors? Especially if you remember that a wire represents no resistance and thus brings two points together electrically as if they were at the same spot, you can follow the wires from the top and bottom of the resistor over to the battery and see that it has to be the case that the voltage is the entire voltage at the battery. Make sure you tie this in to what you saw in the lab [[T105 Week 2#4 Add a Battery|here]]. We observed these things in action, and did not explain too much. Now you can look back and solve what might have been mysterious behaviour to you. ![[Pasted image 20210808131853.png|350]] ### Parallel Circuit Rule for Resistance The story for current is quite different however. Electric current will follow the path of least resistance, that is true, but when the resistances are varied, the path is shared, and the electricity will go most where there is lower resistance, and least where there is higher resistance. Let's begin by examining the rules of calculations for resistance in a parallel circuit. >The total resistance of resistors in parallel is the reciprocal of the sum of the reciprocals of the individual resistors. Looking at this expressed formulaicly, note the mentioned reciprocals on both sides of the equation. You should immediately find out how this is done on your calculator if you haven't yet. $\frac {1}{R_T} = \frac {1}{R_1} + \frac {1}{R_2} + \frac {1}{R_3}... $ For example, the resistors in a parallel circuit are 680 Ω, 1.5 k Ω, and 2.2 k Ω. What is the total resistance? ![[Pasted image 20210808131952.png|350]] We place the values into the equation to solve for R_T: $\frac {1}{R_T} = \frac {1}{680\Omega} + \frac {1}{1.5k\Omega} + \frac {1}{2.2k\Omega} $ $R_T = 386\Omega $ This is only our first example, but look at the total resistance and notice that it is lower than the lowest of the resistances in the circuit. This will always be the case. This is a helpful bit of knowledge when checking the answer from a calculator, and for estimating the resistance. ### Calculating Values in a Parallel Circuit So we will proceed with more calculation and measurement exercises, to get you familiar with the peculiarities of parallel circuits. We will fill out a table of the circuit values, much as we did with the series circuits. We will introduce the new laws as they come up. #### Parallel Circuit Calculations Example 1 In this example, we are given the resistance values, and the voltage that is applied to the circuit. ![[Pasted image 20210808132144.png|350]] As we did in previous exercises, we will fill in the values we have been given. ![[Pasted image 20221003092938.png|350]] We have learned that the voltage across any resistor will be the same as the source voltage. And so, we have a few more cells filled in. Notice that this is not like series circuits. ![[Pasted image 20221003093101.png|350]] We go back to Ohm's Law to solve for the individual currents in the circuit. Again, notice that we are not seeing results like we did in series circuits. ![[Pasted image 20221003093254.png|350]] We now have all we need to solve for Power, and so our table is complete: ![[Pasted image 20221003094237.png|350]] Notice the calculations for current. If you add I₁, I₂, and I₃, what do you get? You get proof of... ### Kirchhoff's Current Law >"The algebraic sum of the currents at any junction of conductors in a circuit is equal to zero" Is this not exactly what was stated for series circuits? It is. But you must watch the language. A junction is where two or more wires meet. And so, just like before what goes in, must go out. But now we have more than just two wires meeting, we have places in the circuit where there are more. Another way to think about this or describe it is that each path's current will add together. #### Parallel Circuit Calculations Example 2 Here we have resistances of equal value. This allows for some interesting shortcuts. ![[Pasted image 20210808132727.png|350]] We use the same formula: ![[Pasted image 20210808132757.png|350]] Notice what happened to our total resistance: >The total resistance of a parallel circuit with equal resistances is equal to one resistor's value divided by the number of equal resistors 300Ω / 3 resistors = 100Ω Do you need to use the formula or do any math if you are asked what the total resistance is of two 100Ω resistors in parallel? [[A Resistance|🅰]] #### Power in Parallel Circuits The power dissipated by a resistor can be calculated with any of the standard power formulas. Most of the time, the voltage is known, so the most convenient equation is: ![[Pasted image 20210808132913.png|250]] > In both series and parallel circuits, the total power is the sum of the power dissipated by each resistor. #### Parallel Circuit Calculations Example 3 What is the total power if 10 V is applied to the parallel combination of R1 = 400 Ω and R2 = 100 Ω? 1)Calculate total resistance of the parallel circuit RT=80 Ω 2)Calculate the power P = V2 / R = 1.25 W ### Parallel Circuit Applications Assume there are 8 parallel resistive wires that form a windshield heater for an aircraft. ![[Pasted image 20210808133122.png|350]] If the defroster dissipates 80W when connected to a 28 V source, what power is dissipated by each resistive wire? [[A Power|🅰]] What is the total resistance of the defroster? [[A Power 2|🅰]] %% #JB this graphic is not fully supported. It should be. Until it is, leave out ![[Pasted image 20210808133211.png|350]] %% ## In the Lab In the lab this week, we will build a parallel circuit and prove that all you have learned this week in the theory class is actually true in real life. Remember to follow all safety precautions at all times, there is no such thing as slacking off on safety in the electrical lab. # Lab [[T105L WS05.pdf|Lab Worksheet]] | [[T105L EQ W05|Equipment List]] | [[T105L SAFETY|Safety Briefing]] ## 1. Calculate power in a series circuit a series circuit with 2 resistors, calculate and measure including power  [➡](<TA_504.png>) ## 2. Another circuit Another with different values, calculate and measure including power  [➡](<TA_503.png>) ## 3. Parallel circuit 1 (matches Circuit 1) Compare with earlier ones.  [➡](<TA_505.png>) Put circuit 1 in parallel, what's the difference? Calculate and measure Answer the questions on the worksheet. Switch the resistors. Does it make any difference?  [➡](<TA_506.png>) ## 4. Parallel circuit 2 (matches Circuit 2) ![[Pasted image 20221002203018.png|350]] Put circuit 2 in parallel, what's the difference? Calculate and measure. With resistors in parallel, the voltage is the same, how is the balance maintained? ## 5. Calculations in a Parallel Circuit 1 ## 6. Calculations in a Parallel Circuit 2 Both of these circuits are to give you more practice with both calculations and measurements. Fill in the appropriate table on the worksheet. ## 7. Variable Resistance in Parallel  [➡](<TA_507.png>) Build this circuit. Before you apply power, ask yourself what you expect to happen when you adjust the resistance. With power on, observe and measure to see what happens in this circuit when you change the resistance. Was this what you expected? Can you explain this?