# Direct Current Circuits Any circuit with one or

Direct Current Circuits Any circuit with one or more constant-voltage sources (batteries, power supplies, etc.) is known as a direct current, or D.C., circuit. This is in contrast to alternating current, or A.C., circuits, where the voltage sources vary sinusoidally with time. We shall study such circuits later. Combining resistors in series or parallel Do

Equivalent resistance of resistors in series or parallel Series: Req R1 R2 ... Parallel: 1 1 1

... Req R1 R2 Compare to the analogous equations for series and parallel combinations of capacitors, and be sure you can recall the forms of these equations for both Rs and Cs! Light bulb filaments are resistors operating at high temperature The brightness of a light bulb increases with the emitted power,

P=VI. We can analyze networks of identical light bulbs as resistor networks, and predict their relative brightness based on power. Objects at high temperature emit heat and light as a thermal, or blackbody, spectrum. If the object is hot enough to glow in the visible spectrum, the color we see is directly related to its temperature. A light bulb the same color as the sun has a filament the same temperature as the

surface of the sun. Demo Find the equivalent resistance of each of these circuits Full analysis of basic resistor circuits Notice the similarities to the analysis of pure capacitor networks: (1) combine all resistors to find Req , (2) find the total network current from I=V/ R, (3) work back through the circuit with this I to find the voltage drops, knowing that the current splits up through parallel sections.

Examples of circuits where basic analysis wont work: These resistor connections are neither series nor parallel. This circuit has two D.C. voltage sources in different legs of the circuit. This circuit contains two types of circuit elements: an R and a C. Kirchhoffs Rules to the rescue!

Gustav Robert Kirchhoff, 1824-1887 German physicist: In 1845, when he was only twenty-one, Kirchhoff announced the laws that allow calculation of the currents, voltages, and resistances of complex electrical networks. In later work he demonstrated that signals flow through a conductor at close to the speed of light; and he established the technique of spectrum analysis, which he then applied to determine the composition of the Sun. Rule 1: The sum of currents into any junction is zero. (Conservation of current.)

I 0 Rule 2: The sum of potential differences around any loop is zero. (Path independence of V.) V 0 Each loop and junction yields an equation (not all independent), which are then solved simultaneously.

Steps in Kirchhoff analysis of circuits (1) Label all legs of the circuit with currents, choosing the direction arbitrarily. Algebra will tell you the signs at the end. For each junction, write the junction equation, with incoming currents positive and outgoing currents negative. For N junctions, the number of independent equations will be N-1. (2) Write a loop equation for each interior loop (or an identical number of independent loops). In forming each sum, choose a direction around the loop and travel in that direction, summing the voltage differences. Note: for a resistor, the sign of the

voltage difference depends on the direction for I you chose in (1); whereas, for a voltage source, it depends only on the travel direction. (3) Solve all equations simultaneously. There must be enough givens to solve for the unknown quantities! (# of unknowns) <= (# of eqns # of givens) Complicated resistor circuit First, to get a feeling what to expect, notice that the resistor circuit is almost symmetric. What would all the currents be if the 2 ohm resistor

were turned into a 1 ohm resistor. (You can do this in your head.) Now, solve the problem generally, and put in the given numbers. Circuit with more than one voltage source Like the previous problem, this one would simplify greatly with a small change: imagine that the unknown emf were in the top leg of the circuit instead of the center one. How could we solve this rapidly? Now to solve the problem weve

been given. The simplest RC example: discharging a capacitor A capacitor is connected in series with a resistor. First it is charged, then the switch is closed and the capacitor discharges through the resistor. I I

I=0 Apply Kirchhoffs loop rule to the right hand VC RI 0 diagram and separate: Q dQ R 0

C dt +Q Q dQ 1 dt Q RC

Q is decreasing with time Integrate, then Q 1 ln(Q) ln(Q0 ) ln t exponentiate: RC Q0 Differentiate Q to find I:

Q Q0 e t t RC t

Q0 RC RC I I 0e e RC Defining RC time constant: = RC Graphing I and Q vs. time: Q0

I 0 RC and I I 0 e t / We can find the time to half-height: e t / .5 t ln(.5) t .693 .693RC

The right choice of RC gives us any time constant we would like. For example: 10k 1F 10 10 3 s 10ms Q Q0 e t / The next-simplest RC example: charging a capacitor V We start with an uncharged capacitor, and this time the

circuit includes a battery, to charge the capacitor when the switch is closed. Kirchhoffs loop rule has three terms this time: V Q=0 V IR

Q 0 C Follow the same steps as before. Separate, integrate, and exponentiate: Differentiate Q to find I: Discuss short and long times. +Q Q

dQ V Q 1 (Q CV ) dt R RC RC Q Q f (1 e t / RC ) with Q f CV

I I 0e t RC with I 0 V / R

Graphing in terms of RC time constant: = RC Graphing I and Q vs. time: I I 0 e t / with I 0 V / R Q Q f (1 e t / ) with Q f CV Discuss short and long times. Demo.

Could we solve this easily with equations we have already? The voltage divider circuit I R Vref +

L x,r V Measuring currents the old-fashioned way: the galvanometer Typical full scale current ~ 100 A The galvanometer is constructed so that the deflection of the needle is proportional to the current through the coil (an electromagnet). For null measurements, the zero point is usually set to center scale.

Measuring unknown voltage: the potentiometer The essential new element in this circuit is a variable resistor with three terminals. Measuring unknown resistance: the Wheatstone bridge A Wheatstone bridge is a measuring instrument invented by Samuel Hunter Christie in 1833, and improved and popularized by Sir Charles Wheatstone in 1843.

Detecting zero current can be done to extremely high accuracy with a galvanometer. Therefore, if R1, R2 and R3 are known to high precision, then Rx can be measured to high precision. Very small changes in Rx disrupt the balance and are readily detected. Do Meters interact with the circuits they are measuring Ammeters and voltmeters can be made from galvanometers, with a coil

resistance Rc, typically in the range of 100-200 ohms. An ammeter can be made from a galvanometer in parallel with a shunt resistor, Rsh. If Rsh << Rc ,then most of the current goes through the shunt with a small voltage drop. The shunt resistance must be small compared to resistance in the circuit. A voltmeter can be made from a galvanometer in series with a resistor Rs. If Rsh >> Rc , then most of the

voltage drop occurs across the series resistor. The meter resistance must also be large compared to R of the circuit element. Checking the imperfection of ammeters and voltmeters Ohmmeters Digital multimeter In voltmeter mode, the input resistance of these devices is very

high: in the 10 megohm to 1 gigohm range. Almost perfect! Power distribution: loads in parallel (actually A.C.) House wiring: loads in parallel Two ways to avoid electric shock Novel way to get an electric shock

Simple resistor networks: Req Do A Do A Do

A Do A Do A Do

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