Objectives: To analyzeresistive, direct currentcircuits withKirchhoff’s rules andOhms law.
Junction Rule‰ The sum of the currents entering anyjunction must equal the sum of the currentsleaving that junction„ -A statement of Conservation of Charge„ Loop Rule‰ The sum of the potential differences acrossall the elements around any closed circuit loopmust be zero„ -A statement of Conservation of Energy
Junction RuleThe sum of the currents entering any junction mustequal the sum of the currents leaving that junction.The algebraic sum of the changes in potential across allof the elements around any closed circuit loop must bezero.A junction is any point in a circuit where the currenthas a choice about which way to go. The first rule, alsoknown as the point rule, is a statement of conservationof charge. If current splits at a junction in a circuit, thesum of the currents leaving the junction must be thesame as the current entering the junction.
• I1 =I2 + I3• From Conservation ofCharge• Diagram (b) shows amechanical analog ‰ In general, thenumber oftimes the junction rulecan be used is one fewerthan the number ofjunction points in thecircuit
The second rule, also known as theLOOP RULE, is a statement ofconservation of energy. Recall thatalthough charge is not "used up" ascurrent flows through resistors in acircuit, potential is. As current flowsthrough each resistor of a resistivecircuit the potential drops. The sumof the potential drops must be thesame as the applied potential.
• Traveling around theloop froma to b• In (a), the resistor istraversed in thedirection of the current,the potential across theresistor is – IR• In (b), the resistor istraversed in thedirection opposite of thecurrent, the potentialacross the resistor is +IR
LoopRule,final• In (c), the source of emfis traversed in the directionof the emf (from – to +),and the change in theelectric potential is +ε• In (d), the source of emfis traversed in the directionopposite of the emf (from+ to-), and the change in theelectric potential is -ε
Loop Equations from Kirchhoff’s Rules• The loop rule can be used as often asneeded so long as a new circuitelement (resistor or battery) or a newcurrent appears in each new equation• You need as many independentequations as you have unknowns
Figure 3.Schematic of an RC circuit. Thecomponents in the dotted box are analogous to asquare-wave generator with outputs at points and. The switch continuously moves between pointsand creating a square wave as shown in Figure 4a.
Suppose we connect a battery, with voltage, , across aresistor and capacitor in series as shown by Figure 3.This is commonly known as an RC circuit and is usedoften in electronic timing circuits. When the switch (S)is moved to position 1, the battery is connected to thecircuit and a time-varying current I (t) begins flowingthrough the circuit as the capacitor charges. When theswitch is then moved to position 2, the battery is takenout of the circuit and the capacitor discharges throughthe resistor. If the switch is moved alternately betweenpositions 1 and 2 , the voltage across points A and Bcan be plotted and would resemble Figure 4.
Figure 4. A voltage pattern known as a square wave. Movingthe switch in Figure 3 alternatively between positions 1 and2 can produce this voltage pattern. When the switch is inposition 1, the input voltage is the peak voltage is Vo .When the switch is moves to position 2 , the input voltagedrops to zero. A function generator more commonlyproduces square-wave voltages.
This voltage pattern is known as a square wave, for obvious reasons, and is commonly produced by a function generator. The function generator is capable of producing voltages that behave like a sine, square or saw-tooth functions. Additionally, thefrequency of the wave may be varied with the function generator. The dotted-box in Figure 3 may be thought of as a function generator with points A and B as outputs. We will use a two-channel oscilloscope to monitor the important voltages throughout the experiment. An oscilloscope is an invaluable tool for testing electronic circuits by measuring voltages over time, and Figure 5 shows the schematic for monitoring an RC circuit with an oscilloscope. As shown in thefigure below, the input voltage from the square-wave generator is monitored by channel one (CH 1) and the voltage across the capacitor is monitored by channel two (CH 2).
Figure 5. The RC circuit diagram. Theoscilloscopes Channel 1 monitors the function generator while Channel 2monitors the voltage drop across the capacitor.
The capacitor responds to the square-wave voltage input bygoing through a process of charging and discharging. It isshown below that during the charging cycle, the voltageacross the capacitor . When the switch is inposition , the square-wave generator outputs a zero voltageand the capacitor discharges. It can also be shown thatduring the discharging cycle, the voltage across the capacitorisCircuit designers must be careful to ensure that the period ofthe square wave gives sufficient time for the capacitor tofully charge and discharge. It can be shown3 that, as ageneral rule of thumb, the time necessary for the capacitorof an RC circuit to nearly completely charge to Vo, ordischarge to zero, is 4RC .
Here it should be noted that the product RC is knownas the time constant,t, and has units of time4. The timeconstant is the characteristic time of the charging anddischarging behavior of an RC circuit and represents thetime it takes the current to decrease to of itsinitial value, whether the capacitor is charging ordischarging. Over the period of one t, the voltage acrossthe charging capacitor increases by a factorConversely the voltage across the discharging capacitordecreases by a factor of over the same period,Put another way, in 1t the voltage across a chargingcapacitor grows to 63.2% of its maximum voltage,Vo ,and in 1t the voltage across a discharging capacitorshrinks to 36.8% of Vo .
Figure 6a. The square wave that drives the RC circuit.When the switch in Figure 3 is in position , the inputvoltage is the peak voltage is . When the switch is movesto position , the input voltage drops to zero. In thisexperiment, this input voltage is read by theoscilloscopes CH 1.
Figure 6b. The voltage drop across the capacitor of Figure 3 asread by the oscilloscopes CH 2. The capacitor alternately chargestoward and discharges toward zero according to the input voltageshown in Figure 6a. Here, the frequency (and therefore period) ofthe input square wave voltage is exactly such that the capacitor isallowed to fully charge and discharge. The time constant, , isequivalent to , and is defined by Equations 11 or 14.
An RC Circuit: ChargingCircuits with resistors and batteries have time-independent solutions: the current doesnt change astime goes by. Adding one or more capacitors changesthis. The solution is then time-dependent: the currentis a function of time.Consider a series RC circuit with a battery, resistor,and capacitor in series. The capacitor is initiallyuncharged, but starts to charge when the switch isclosed. Initially the potential difference across theresistor is the battery emf, but that steadily drops (asdoes the current) as the potential difference across thecapacitor increases.
Applying Kirchoffs loop rule: e - IR - Q/C = 0As Q increases I decreases, but Q changesbecause there is a current I. As the current decreases Q changes more slowly.I = dQ/dt, so the equation can be written: e - R (dQ/dt) - Q/C = 0
This is a differential equation that can be solved for Q as a function of time. The solution (derived in the text) is: Q(t) = Qo [ 1 - e-t/t ] where Qo = C e and the time constant t = RC.Differentiating this expression to get the current as a function of time gives: I(t) = (Qo/RC) e-t/t = Io e-t/t where Io = e/R is the maximum current possible in the circuit. The time constant t = RC determines how quickly the capacitorcharges. If RC is small the capacitor charges quickly; if RC is large the capacitor charges more slowly.
TIME CURRENT 0 Io 1*t Io/e = 0.368 Io 2*t Io/e2 = 0.135 Io3*t Io/e3 = 0.050 Io
What happens if the capacitoris now fully charged and isthen discharged through theresistor? Now the potentialdifference across the resistor isthe capacitor voltage, but thatdecreases (as does the current)as time goes by.
Applying Kirchoffs loop rule: -IR - Q/C = 0 I = dQ/dt, so the equation can be written: R (dQ/dt) = -Q/CThis is a differential equation that can be solved for Q as a function of time. The solution is: Q(t) = Qo e-t/twhere Qo is the initial charge on the capacitor and the time constant t = RC.
Differentiating this expression to get the current as a function of time gives: I(t) = -(Qo/RC) e-t/t = -Io e-t/t where Io = Qo/RC Note that, except for the minus sign, this is the same expression for current we had when the capacitor wascharging. The minus sign simply indicates that the charge flows in the opposite direction.Here the time constant t = RC determines how quickly thecapacitor discharges. If RC is small the capacitor dischargesquickly; if RC is large the capacitor discharges more slowly.