Basic Circuit Components and Analysis

1. Circuits AP Physics C

2. A Basic Circuit • All electric circuits have three main parts • A source of energy • A closed path • A device which uses the energy • If any part of the circuit is open the device will not work!

3. Potential Difference • In a battery, a series of chemical reactions occur in which electrons are transferred from one terminal to another. There is a potential difference (voltage) between these poles. • The maximum potential difference a power source can have is called the electromotive force or (EMF), ε. The term isn't actually a force, simply the amount of energy per charge (J/C or V). • The unit for potential difference is volt (V).

4. Current • If Q is the total amount of charge that has moved past a point in a wire, we define the current I in the wire to be the rate of charge flow: • Current is the rate at which charge flows • The SI unit for current is the coulomb per second, which is called the ampere. • 1 ampere = 1 A = 1 C/s. • Conventional current flow is the flow of positive charge.

5. Modeling Current • When a voltage is applied to a circuit, there is an electric field present inside the wire. • This field causes electrons to move through the wire. • The electrons frequently collide with positive charges, which causes them to drift through the wire very slowly.

6. Resistance • Resistance (R) is defined as the restriction of electron flow. • Each time charges collide with nuclei, some of their kinetic energy is converted into thermal energy. • The unit for resistance is Ohm (Ω).

7. Ohm’s Law • For most materials, increasing the amount of voltage increases the current through the material. • Our constant of proportionality between these two values is the resistance of the material. • This formula is referred to as Ohm’s Law.

8. Electrical Power • When current runs through a lightbulb or a resistor, it dissipates electrical energy into heat. • The dissipated electrical power can be calculated using one of the following equations: • The unit for power is a watt (W), just like in mechanics.

9. Basic Circuit Components • Before you begin to understand circuits you need to be able to draw what they look like using a set of standard symbols understood anywhere in the world • For the battery symbol, the long line is considered to be the positive terminal and the short line, negative. • The voltmeter and ammeter are special devices you place in or around the circuit to measure the voltage and current.

10. Using a Voltmeter and Ammeter • The voltmeter and ammeter cannot be just placed anywhere in the circuit. They must be used according to their definition. • Since a voltmeter measures voltage or potential difference it must be placed across the device you want to measure. That way you can measure the change on either side of the device. • Since the ammeter measures the current or flow it must be placed in such a way as the charges go through the device.

11. Ways to Wire Circuits • There are 2 basic ways to wire a circuit. Keep in mind that a resistor could be anything (bulb, toaster, ceramic material…etc) • Series – One after another • Parallel – between a set of junctions and parallel to each other

12. Strategy For Solving Circuits • Circuit problems will often give you a battery and various resistors. The resistors will be in series, parallel, or a combination of the two. • The goal is generally to calculate the total resistance, which essentially turns all of your resistors into one resistor. • With one battery and one resistor, you can use Ohm’s Law to calculate the total current in the circuit. By knowing this total current, the battery voltage, and the resistance of each resistor, you can calculate voltage across resistors and current through resistors.

13. Series Circuit • In a series circuit, the resistors are wired one after another. Since they are all part of the same loop they each experience the same amount of current. • The sum of the voltages across each resistor is equal to the total voltage of the battery.

14. Equivalent Resistance in Series • As the current goes through the circuit, the charges must use energy to get through the resistor. • Each individual resistor will eat up some electric potential. We call this voltage drop.

15. Example Calculate the equivalent resistance of the circuit, the current in the circuit, and the voltage drop across each resistor.

16. Parallel Circuit • In a parallel circuit, we have multiple loops. So the current splits up among the loops with the individual loop currents adding to the total current. • It is important to understand that parallel circuits will all have some position where the current splits and comes back together. We call these junctions. • The current going in to a junction will always equal the current going out of a junction.

17. Equivalent Resistance in Parallel • Notice that the junctions both touch the positive and negative terminals of the battery. That means you have the same potential difference down each individual branch of the parallel circuit. This means that the individual voltages drops are equal.

18. Example Calculate the equivalent resistance of the circuit, the current in each resistor, and the voltage drop across each resistor.

19. Complex Circuits • Combinations of resistors can often be reduced to a single equivalent resistance through a step-by-step application of the series and parallel rules. • Once you have found the total resistance of the circuit, you can determine the total current in the circuit. That current can be used to calculate the voltage drop across each resistor.

20. Example Calculate the current through each resistor and the voltage drop across each resistor.

21. Kirchhoff’s Voltage Law (KVL) • Kirchhoff’s voltage law states that if you travel along any closed loop, the sum of the potential differences will be zero. • Each element in the loop (battery, resistor, etc.) creates a potential difference; sometimes they will be a rise in potential, sometimes they will be a drop in potential.

22. Kirchhoff’s Voltage Law (KVL) • In general, when we apply KVL we choose a loop (many circuits will have multiple loops) and we will add together all of our voltage differences and set it equal to zero. • Because we often know the battery voltages, we write out the potential difference as the voltage of the battery. For resistors, we will normally represent the potential difference as the current through that resistor times its resistance (Ohm’s Law).

23. Kirchhoff’s Current Law (KCL) • Kirchhoff’s Current Law states that the current flowing into a junction is equal to the sum the currents flowing out. • You can think of the current as water and the wires as pipes. If water is in one pipe and encounters two pipes, some of the water will go into one and the rest will go into the other. The important idea is that we are not gaining or loosing any water, but simply redirecting it.

24. Example There is a current of 1.0 A in the following circuit. What is the resistance of the unknown circuit element? The diagram below shows a segment of a circuit. What is the current in the 200 Ω resistor?

25. Applying Kirchhoff’s Laws • In general, we use Kirchhoff’s laws to determine the currents that run through various resistors in larger circuits. • By writing out KVL for different loops as well as KCL for the currents, we create a system of equations. • Using your system of equations, you can solve for various currents in the circuit. • When writing out KVL, you need to be careful how you assign your signs (based on the direction of current flow). Batteries Resistors • If current goes from NEGATIVE to POSITIVE, the potential difference is POSITIVE. • If current goes from POSITIVE to NEGATIVE, the potential difference is NEGATIVE. • If the direction of your current AGREES with the direction of your loop, then the potential difference is NEGATIVE. • If the direction of your current DISAGREES with the direction of your loop, then the potential difference is POSITIVE.

26. Kirchhoff example Apply the rules to this circuit. Choose the upper loop and the lower loop for the first two equations, and use the current rule for the third equation.

27. Internal Resistance • All components in a circuit offer some type of resistance regardless of how large or small it is. • Real batteries have what is called an internal resistance. This can be modeled as an additional resistor in series with the battery.

28. Example Suppose we have a car battery with an emf = 13.8 V, under a resistive load of 20 Ω ,the voltage sags to 11.8 V . a) What is the battery's resistance? b) What is the rate at which energy is dissipated in the battery?

29. Short Circuit • The figure shows an ideal wire shorting out a battery. • If the battery were ideal, shorting it with an ideal wire (R = 0 ) would cause the current to be infinite! • In reality, the battery’s internal resistance r becomes the only resistance in the circuit.

30. Example What is the short circuit current of a 12V car battery with an internal resistance of 0.020Ω? What would be the power dissipated?

31. Resistivity and Conductivity • All wires exhibit some amount of resistance, and it is dependent primarily on the geometry and the resistivity (a material property) of the wire. • In many circuits we can consider the wire to be of negligible resistance, but we can calculate the resistance nonetheless. • The resistance of a material is the inverse of the material’s conductivity.

32. Example Calculate the resistance of a one meter length of 24 SWG Nichrome wire.

33. Superconductivity • In 1911, the Dutch physicist KamerlinghOnnes discovered that certain materials suddenly and dramatically lose all resistance to current when cooled below a certain temperature. • This complete loss of resistance at low temperatures is called superconductivity. • Superconductors have unusual magnetic properties. Here a small permanent magnet levitates above a disk of the high temperature superconductor YBa2Cu3O7 that has been cooled to liquid-nitrogen temperature.

34. RC Circuits • The figure to the right shows a charged capacitor, a switch, and a resistor. • At t = 0, the switch closes and the capacitor begins to discharge through the resistor. • A circuit such as this, with resistors and capacitors, is called an RC circuit. • We wish to determine how the current through the resistor will vary as a function of time after the switch is closed.

35. Discharging RC Circuits • The figure to the right shows an RC circuit, some time after the switch was closed. • Kirchhoff’s loop law applied to this circuit clockwise is: • Q and I in this equation are the instantaneous values of the capacitor charge and the resistor current. • The resistor current is the rate at which charge is removed from the capacitor:

36. Discharging RC Circuits • Knowing that I = dQ/dt, the loop law for a simple closed RC circuit is: • Rearranging and integrating:    where the time constant  is:

37. Discharging RC Circuits • The charge on the capacitor of an RC circuit is: • where Q0 is the charge at t = 0, and  = RC is the time constant. • The capacitor voltage is directly proportional to the charge, so: • Where V0 is the voltage at t = 0. • The current also can be found to decay exponentially:

38. Charging RC Circuits • When solving for charging functions of RC circuits, we apply the same mathematical process. • To simplify things, the charging and discharging formulas are listed below. • You may have to derive this when working a problem, they will NOT be on the formula sheet. Discharging Charging

39. Steady State vs. Transient State • A transient state RC circuit is one in which the capacitor is not yet fully charged. • If this is the case, we must use our charging and discharging equations to analyze the circuit. • A steady state RC circuit is one in which the capacitor is fully charged. • In this case, we can act as if the capacitor is a resistor with infinite resistance, because no current flows through it.