RC-RL Circuits Justin Boyd Billy Stack Austin Lee Physics 231, Savan Kharel March 24, 2009-March 31, 2009 Objectives: The objectives of this experiment are: 1) to understand how capacitors and inductors affect the flow of current when changes in potential occur, 2) to learn how to analyze the time behavior of voltages and currents in resistor-capacitor and resistor-inductor circuits (RC and RL circuits), and 3) to measure the time constants of simple RC and RL circuits. Apparatus: The apparatus for this experiment includes 1) a Pasco Model CI-6512 FLC circuit board and the Pasco Science Workshop 750 computer data acquisition system and 2) Pasco’s DataStudio software. Error Analysis: In the experiment there is a measure of error that is difficult to avoid given the circumstances and nature of obtaining measurements with faulty equipment. There is also a measure of error with any experiment involving measurements of resistors and capacitors and inductors because of their inherent tendencies to over-heat and fully charge and discharge, respectively. Conclusion: Capacitors affect the flow of current when changes in potential occur by. Inductors affect the flow of current when changes in potential occur by. It was determined that the value for the time constant was measured correctly by graphical analysis. By using our measurements obtained by the equipment we were able to successfully graph our results with comparable accuracy and precision. Our calculations for the time constant voltages are supported by the data as well. The data collected can also be proven true by the equation relationships determined in the theory above. Theory: A circuit consisting of a resistor, a capacitor, and an EMF source is called a RC circuit. A circuit consisting of a resistor, an inductor, and an EMF source is called a RL circuit. Capacitors act as reservoirs for charge and as a result RC circuits can accommodate large initial changes in current. This changes as the capacitor becomes fully charged and no current flows in the circuit. Inductors are the exact opposite. Inductors resist large current changes, but after these initial changes, the current will reach a steady state value. These types of properties can be used to change time behaviors of output voltages when an input voltage is applied to a circuit. Each RC and RL circuit has a fundamental characteristic time response, and the time constants can be expressed simply in terms of the values of the resistors, capacitors, and inductors. The current flowing through a resistor depends directly on the potential difference across it and inversely on the value of the resistor (I=V/R). Capacitors are made up of two conductors, each with a large surface area and separated from each other by a thin piece of insulating, dielectric material. With its value for capacitance relying on the surface area, the separation of the conductors, and the dielectric constant of the insulating material, the given amount of charge held by each capacitor is circumstantial on the potential difference that is applied (q=CV). Inductors are usually made up of coils of wire and sometimes a ferromagnetic material is inserted into the coil to increase its inductance. Magnetic fields are created by flowing current. The back Emf, which opposes the change in current, is directly proportional to the change in current per unit time by: V = -L(delta i’/ delta t’), where V is voltage, L is inductance, and (delta i’/ delta t’) is change in current flowing. The Simple RC Circuit: One of the principles of Kirchhoff’s Laws for electrical circuits states that in any closed circuit, the algebraic sum of the increases in potential difference and the decreases in potential difference around a closed loop are equal to zero. Applying this to the RC circuit we get E - V’r(t) – V’c(t) = 0. In this equation, E is constant, but –V’r(t) and –V’c(t) are functions of time. These two variables are related to each other by time and the defining relationship, i(t) = delta’q’/delta’t’. The potential drop across the resistor is given by Ohm’s law: ‘V’r(t) = -i(t) R. The potential drop across the capacitor is given by –V’c’(t) = -q(t)/C. Inserting these equations into our Kirchhoff’s Law equation we have delta’q(t)’/(q(t)-EC) = -delta’t’/(RC). Solving this by the methods of calculus we finally derive V’c’(t) = E e^(-t/(RC). There are two cases to this equation: t=0 and t=infinity. This means that at t=0 the capacitor is uncharged and has no potential difference across it and at t=infinity the capacitor is fully charged with the potential difference equal to that of the Emf source. The RC time constant for a simple RC circuit is defined as the time that the potential difference across the capacitor will increase to value equal to (1-(1/e)) of the maximum change, E. Knowing this, ‘T’ is the product of R and C. With this we derive V’R’(t) = E e^(-t/T) which is the equation for the potential difference across the resistor as a function of time after the switch connects the power supply to the circuit. Case 2: The Emf source is zero and the capacitor is initially charged to the full potential difference of the Emf. By setting the Emf source equal to zero, we get dq(t)/q(t) = -1/(RC) dt which can be solved using similar procedures as above to yield the relationship, q(t) = EC e^(-t/T). From this we can get our final equation for case two which is V’R’(t) = -E e^(-t/T). The Simple RL Circuit: Case 1: Switch is initially closed connecting the Emf source in the circuit. Initially the inductor has no current flowing through it. Kirchhoff’s principle as applied to this gives E-V’R’(t)-V’L’(t) = 0. In this equation, E is a constant, but –V’R’(t) and -V’L’(t) are functions of time. Using the same method as described above in “The Simple RC Circuit,” our potential drop across the resistor is given by Ohm’s Law, -V’R’(t) = -i(t) R and the potential drop across the inductor is given by –V’L’(t) = -L (delta’i’ (t)/delta’t’). Inserting this into our Kirchhoff equation and solving by calculus gives us i(t) = E/R (1-e^(-t/T)). With this equation we can deduce that V’L’(t) = E e^(-t/T). There are two cases to this equation: t=0 and t=infinity. This means that at t=0 the capacitor is uncharged and has no potential difference across it and at t=infinity the capacitor is fully charged with the potential difference equal to that of the Emf source. The RL time constant for a simple RL circuit is defined as the time that the potential difference across the capacitor will increase to value equal to (1-(1/e)) of the maximum change, E. Knowing this, ‘T’ is L/R. With this we derive V’L’(t) = E e^(-t/T) which is the equation for the potential difference across the resistor as a function of time after the switch connects the power supply to the circuit. Case 2: The Emf source is zero and the capacitor is initially charged to the full potential difference of the Emf. By setting the Emf source equal to zero, we get di(t)/i(t) = -1/(L/R) dt which can be solved using similar procedures as above to yield the relationship, i(t) = E/L e^(-t/T). From this we can get our final equation for case two which is V’L’(t) = -E e^(-t/T). Data/Results: