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Kirchhoff’s Rules, RC and RL Circuits Notes – Physics C Circuit Basics V=IR – Ohm’s Law; applies to both the entire circuit and any resistive element within it. The voltage across any two elements in parallel is the same; the current through any two elements in series is the same. The voltage across elements in series adds up to the total voltage; the current through any elements in parallel adds up to the total current. R L A ; resistance of a conductor of length L, area, A and resistivity, . Rs R1 R2 ... 1 1 1 Resistors in series and parallel; ... Rp R1 R2 Q=CV - definition of capacitance; applies to the entire circuit and any capacitor within the circuit; The charge on any capacitors in series is the same and the voltage across these elements adds up to the total voltage; the charge on any capacitors in a parallel adds up to the total charge and the voltage across these elements is the same. C o A - capacitance of a parallel plate capacitor of area A, separation d and dielectric d constant . C p C1 C2 ... Capacitors in series and parallel; 1 1 1 ... C s C1 C 2 Students should be able to find the equivalent resistance or capacitance of a simple series and parallel circuit. They should also be able to find the current through or voltage across any resistor in the circuit, and the charge on or voltage across any capacitor in the circuit, both under steady state conditions. P=IV - the power dissipated or generated by any element is equal to the current through it multiplied by the voltage across it. Kirchhoff’s Rules Used when a circuit is not a simple series and parallel combination of resistors; often used when there is more than one voltage source in a circuit. Point Rule: the sum of all the currents into any point in the circuit is equal to the sum of all of the currents out of the same point. I I in out Loop Rule: the sum of the voltage drops around any closed loop is zero. V 0 Problem Solving Strategy: 1) Label all junctions and corners in order to make solution easier. 2) Guess current directions in each branch of the circuit. 3) Apply point rule to n-1 junctions where n is the number of junctions. 4) Apply loop rule to as many loops as necessary such the total number of equations is equal to the number of unknown currents. a. When going with the current, V IR through a resistor; when going against the current, V IR b. When going from the positive terminal to the negative terminal of a battery, V=-V and when going from negative to positive terminal, V=+V. 5) Solve the system of equations for the unknown currents. Rarely are the students asked to solve an entire system on an AP exam, but they are often asked to apply these principles to parts of circuits on multiple choice problems. RC Circuits The simplest RC circuit contains a single resistor R in series with a capacitor C and a battery of voltage V. Apply Kirchhoff’s loop rule to the loop. V = VR+VC q V iR C dq q V R dt C - substituting Ohm’s Law and the definition of capacitance; - substituting the relationship between current and charge; Solve the differential equation by method of separation of variables: dq dt VC q RC dq dt VC q RC ln(VC q ) const t / RC q (t ) VC (1 e t / RC ) The constant comes from the initial conditions: q=0 at t=0. For a discharging circuit, the method of solution is similar: the only difference is that V=0. Plug that in, make the same substitutions as above, solve by separation of variables and the result is q (t ) VCe t / RC . In circuits with more than one branch, or a resistor in parallel with the capacitor, it helps to teach the students about the initial and steady state behavior of the capacitor. When the capacitor is empty, it offers no “resistance” to the flow of current; when it is full, no current will flow in the branch with the capacitor in it. Students should be able to solve more complex RC circuits for the initial and steady state currents in each part of the circuit. Rarely will they be asked to solve the differential equation for the current as a function of time in these more complex circuits. They should however, be familiar with the graphs of charging and discharging circuits. RL Circuits A The picture at right represents a resistor in series with an inductor and a battery. B The switch is initially moved to position A and the inductor is “charged.” The voltage across the inductor varies (according to Faraday’s Law) with the rate of change of current according to the equation VL L di (magnitude only, the dt sign due to Lenz’s law has been omitted). Again we start with Kirchhoff’s Loop Rule. V = VR+VL di V iR L dt Solve by separation of variables: di V iR L dt di Rdt V /R i L di Rdt V /R i L V Rt ln( i ) const - the constant comes from the initial conditions: i=0 at t=0. R L V i (t ) (1 e Rt / L ) R When the switch is flipped to position B, the inductor begins to “discharge.” Again, for the discharging equation, simply start with V=0 in the loop equation as above. Then solve by separation V of variables as above to get the equation i (t ) e Rt / L where the initial current in the inductor is R the same as it was just before the switch was opened causing the inductor to discharge. Again, for more complex circuits involving more than one resistor in a circuit with an inductor, either in series or parallel with the inductor, it is important for the students to understand the initial behavior of the circuit after a switch is moved, and the long-term behavior after the switch is moved (either opened or closed). Initially, it is useful for the students to think of the inductor as an open switch: it prevents any sudden changes in current so if there was no current just before the switch is moved, there will be no current immediately after. However, the ideal inductor behaves like a wire when the current has reached a steady state. These two situations will help the student determine the current in any part of the circuit just after a switch is moved and in steady state conditions. Students should be able to solve complex RL circuits for initial and steady state conditions, and graph the current in each branch as a function of time. However, they will only be required to solve the differential equation for a simple RL circuit with one resistor and one inductor.