Charge and CurrentCharge and CurrentCounting ChargesMotion of Real ChargesAside: Conservation of ChargeDC versus AC CurrentsCurrent Flow Through a ComponentCurrent into a ComponentVoltageVoltege Across a ComponentThe Concept of GroundAn Ideal SwitchBatterySki Lift AnalogyPower and EnergyExample 1: Power Through a ComponentExample 2: Power Through Another ComponentExample 3: Instantaneous PowerAn Aside on Common SI PrefixesExample 4: An Ideal Switch / A Short CircuitKCL: Kirchhoff's Current LawAside: Origin of KCLKVL: Kirchhoff's Voltage LawAside: E&M ConnectionKVL ImplicationsEE 42/100Lecture 2: Charge, Current, Voltage, and CircuitsELECTRONICSRevised 1/18/2012 (9:04PM)Prof. Ali M. NiknejadUniversity of California, BerkeleyCopyrightc 2012 by Ali M. NiknejadA. M. Niknejad University of California, Berkeley EE 100 / 42 Lecture 2 p. 1/26 – p. 1/26Charge and Current•A conductor is a material where chargers are free to move about. Even in “rest",the charge carriers are in rapid motion due to the thermal energy. Typical carriersinclude electrons, ions, and “holes" (in semiconductors).•Current is charge in motion. When positive charges move in the positive direction,we say the current is positive. If negative charges move in the same direction, wesay the current is negative. In other words, the current flowing through a surface isdefined asI =Net charge cross ing surface in time ∆t∆twhere ∆t is a small time interval. The units of current are [I] = [C/s] = [A], orampere (after André-Marie Ampère).A. M. Niknejad University of California, Berkeley EE 100 / 42 Lecture 2 p. 2/26 – p. 2/26Charge and Current•When both positive and negative charge are moving, the net charge motiondetermines the overall current.q1i1q2i2v1v2+_q1i1q2i2v1v2+_Net current is i = i1+ i2Net current is i = i1− |i2|A. M. Niknejad University of California, Berkeley EE 100 / 42 Lecture 2 p. 3/26 – p. 3/26Counting Charges+++++++Av∆t•Suppose that the chargecarriers each have acharge of q. Let’s countthe number of charges (c)crossing a surface in time∆t and multiply by theelectrical chargeI = qc∆t•To find c, let’s make the simple assumption that all the charges are moving at aspeed of v to the right.•Then the distance traversed by the charges in time ∆t is simply v∆t, or in otherwords if we move back from the surface this distance, all the charges in the volumeformed by the cross-sectional surface A and the distance v∆t will cross thesurface in time ∆t. This means thatI = qv∆tNA∆t= q(NA)vwhere N is the density of electrons per unit volume.A. M. Niknejad University of California, Berkeley EE 100 / 42 Lecture 2 p. 4/26 – p. 4/26Motion of Real Charges¯v =1NXivi=0¯v =1NXivi≈ ˆxvd•The above result emphasizes that current is associated with motion. In our simpleexample, we assumed all carriers move at a velocity v. In reality, as you may know,electrons move very rapidly in random directions due to thermal motion(mv2∼ kT ) and v is the net drift velocity.A. M. Niknejad University of California, Berkeley EE 100 / 42 Lecture 2 p. 5/26 – p. 5/26Aside: Conservation of Charge~Jρ(t)•We know from fundamental physics that charge is conserved. That means that if ina given region the charge is changing in time, it must be due the net flow of currentinto that region. This is expressed by the current continuity relation in physics(which can be derived from Maxwell’s equations)∇ · J = −∂ρ∂t•The divergence is an expression of spatial variation of current density whereas theright-hand-side is the change in charge density at a given point.A. M. Niknejad University of California, Berkeley EE 100 / 42 Lecture 2 p. 6/26 – p. 6/26DC versus AC Currentsi(t) i(t)i(t) i(t)t tt toffset currentDCAC: SineAC: Square WaveAC: Arbitrary•A constant current is called a “Direct Current" (DC). Otherwise it’s AC.•Some AC typical waveforms are shown above. Sine waves are the waveformscoming out of an electric outlet. A square wave is the clock signal in a digital circuit.•Any time-varying current is known as an AC, or alternating current. Note that thesign of the current does not necessarily have to change (the current does not haveto alter direction), as the name implies.A. M. Niknejad University of California, Berkeley EE 100 / 42 Lecture 2 p. 7/26 – p. 7/26Current Flow Through a Componentiababiba•When current flows into a component (resistor, lamp, motor) from node a to b, wecall t his current iab. Note that the current iabis the same as −iba.•When several components are connected in a circuit, we call the componentsbranches and associate a current with each branch.iAiBiCABCBranchesNodeA. M. Niknejad University of California, Berkeley EE 100 / 42 Lecture 2 p. 8/26 – p. 8/26Current into a Componenti(t)q(t)•Suppose that we now consider the current flow into a component. If we count theamount of charge ∆q flowing into the component in a time interval ∆t, then in thelimit as ∆t → 0, the ratio is exactly the current flowing into the componentI = lim∆t→0∆q∆t=dqdtA. M. Niknejad University of California, Berkeley EE 100 / 42 Lecture 2 p. 9/26 – p. 9/26VoltageVABAB•The voltage difference VABbetween A and B is the amount of energy gained orlost per unit of charge in moving between two points.•Voltage is a relative quantity. An absolute voltage is meaningless and usually isimplicitly referenced to a known point in the circuit (ground) or in some cases apoint at infinity.•If a total charge of ∆q is moved from A → B, the energy required isE = ∆qVAB•If the energy is positive, then by definition energy is gained by the charges as theymove “downhill". If the energy is negative, then energy must be supplied externallyto move the charges “uphill".•The units of voltage are Volts (after the Italian physicist Alessandro Volta), orJoules/Coulomb.A. M. Niknejad University of California, Berkeley EE 100 / 42 Lecture 2 p. 10/26 – p.Voltege Across a Component+_+__+“Uphill”“Downhill”AB•In electrical circuits, the path of motion is well defined by wires/circuit components(also known as elements). We usually label the terminals of a component aspositive and negative to denote the voltage drop across the component.•Sometimes we don’t know the actually polarity of the voltage but we just define areference direction. In our subsequent calculations, we may discover that we werewrong and the voltage will turn out to be negative. This
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