A. M. Nknejad Unversty of Calforna, Berkeley EE 100 / 42 Lecture 4 p. 1/14 EE 42/100 Lecture 4: Resstve Networks and Nodal Analyss ELECTRONICS Rev B 1/25/2012 (9:49PM) Prof. Al M. Nknejad Unversty of Calforna, Berkeley Copyrght c 2012 by Al M. Nknejad
A. M. Nknejad Unversty of Calforna, Berkeley EE 100 / 42 Lecture 4 p. 2/14 Parallel Notaton v R 1 v R 1 Last lecture we learned that two resstors n parallel can be combned as follows R eq = R 1 R 1 It s very common notaton to wrte Req = R 1, n other words we defne an operator that takes two nputs outputs x y = xy x y Note that f x y, then x y = xy x y = y 1 y x y
A. M. Nknejad Unversty of Calforna, Berkeley EE 100 / 42 Lecture 4 p. 3/14 Current Source r o A current source s the dual of a voltage source. It supples a constant current regardless of the external voltage appled to ts termnals. Ths property means that t has nfnte complance, or nfnte output resstance. In contrast, a voltage source has a constant voltage regardless of the current drawn. It has zero output resstance. Unlke a voltage source, there are no everyday examples (such as batteres) that can be used to llustrate the prncpal of a current source. Transstors n the forward actve" regon do behave lke current sources, but for that you ll have to wat untl later. Any real current source has fnte output resstance, but t s typcally very large. Contrast ths wth a real voltage source.
A. M. Nknejad Unversty of Calforna, Berkeley EE 100 / 42 Lecture 4 p. 4/14 Your Taste for Opens and Shorts vs vs Recall that voltage sources detest short crcuts, because that volates KCL. Also, two voltage sources cannot be put n parallel. Voltage sources do lke open crcuts, because they don t draw any power. The current source as the dual relatonshp. It cannot drve an open crcut, because that volates KCL. Lkewse, two current sources cannot be put n seres. Current sources love short crcuts, and they happly delver all of ther current to a short crcut. Ths results n no power draw (why?).
A. M. Nknejad Unversty of Calforna, Berkeley EE 100 / 42 Lecture 4 p. 5/14 Battery Charger V b V b t A smple battery charger can be made wth a current source. A real battery has a dscharge curve, n other words as you draw current from the battery, the output voltage drops. A current source can be used to charge the battery, and t wll delver current regardless of the battery voltage. Of course, we need provsons to turn off the current source when the battery s full. In the old days, a tmer was used to control how long the current was on. In more modern unts, the charger montors the battery voltage and detects when the battery s full.
A. M. Nknejad Unversty of Calforna, Berkeley EE 100 / 42 Lecture 4 p. 6/14 Dependent Sources v c Gv c c F c Evc R c Current and voltage sources are ndependent sources because ther output s ndependent of any other currents or voltages n the crcut. In contrast, dependent sources depend drectly on the the voltage or current n other parts of the crcut. Four flavors are possble: Voltage Controlled Voltage Source (VCVS), Voltage Controlled Current Source (VCCS), Current Controlled Voltage Source (CCVS), and Current Controlled Current Source (CCCS). A VCCS s an deal voltage amplfer or attenuator (gan < 1), whle a CCCS s an deal current amplfer. Sometmes the sgnal of nterest s a current whle the desred output s a voltage. A CCVS s an deal trans-resstance amplfer, because the gan has unts of resstance. A trans-conductance amplfer does the nverse.
A. M. Nknejad Unversty of Calforna, Berkeley EE 100 / 42 Lecture 4 p. 7/14 Voltage Dvder R 1 v 1 v 2 A voltage dvder s an extremely useful crcut snce t allows us to derve any fracton of the source voltage at the output. Notce that the current n the crcut s gven by = R 1 Suppose the output voltage v2 s connected across the termnals of. Let s calculate the output voltage v 2 n terms of the source voltage v 2 = = R 1 = α Note that the factor α 1. By changng ether R1,, or both, we can vary the attenuaton of the crcut. Between R 1 and, the largest resstor wns (gets the majorty of the voltage drop).
A. M. Nknejad Unversty of Calforna, Berkeley EE 100 / 42 Lecture 4 p. 8/14 Rheostats and Potentometers v 1 x = 1 x = 0 x R 1 v 3 v 2 Source: Wkpeda A varable resstor s called a rheostat. A related element has three termnals and t s called a potentometer. It s typcally constructed so that t presents a fxed resstance R across two of ts termnals. The thrd termnal s connected to a pont between the two other termnals n such a way that the resstance vares lnearly (or perhaps logarthmcally dependng on the applcaton).
A. M. Nknejad Unversty of Calforna, Berkeley EE 100 / 42 Lecture 4 p. 9/14 Potentometer Dvder v 1 x = 1 x = 0 x R 1 v 3 For the crcut shown, snce the total resstance s fxed, we can wrte v 3 = R 1 = x(r 1 ) R 1 = x. where x 1.
A. M. Nknejad Unversty of Calforna, Berkeley EE 100 / 42 Lecture 4 p. 10/14 Current Dvders s 1 2 G 1 G 2 Suppose two conductors are placed n parallel. The current s then splts nto the two conductors. The rato of the current nto say G 1 can be calculated as follows 1 = G 1 where s the voltage across the termnals of the conductors (equal by KVL). Ths s computed from the total equvalent conductance = (G 1 G 2 ) 1 s Substtutng the above relaton, we have that the current nto 1 s gven by 1 = G 1 G 1 G 2 s = R 1 s = β s As before, β 1 and the largest conductance (smallest resstance) wns" (get the majorty of the current).
A. M. Nknejad Unversty of Calforna, Berkeley EE 100 / 42 Lecture 4 p. 11/14 Shorts and Opens Agan s 1 G 1 k G k N G N R 1 v 1 v2 R N R N-1 v N vn-1 R k-1 vk-1 R k R k1 vk1 v k The equatons for voltage dvder and current dvder are easy to generalze to N seres or parallel resstors v k = R k P Rj k = G k P Gj s
A. M. Nknejad Unversty of Calforna, Berkeley EE 100 / 42 Lecture 4 p. 12/14 Wnners and Losers In a current dvder, a short crcut always wns! s 1 G 1 s N G N In a voltage dvder, an open crcut always wns. R 1 v 1 v2 R N R N-1 v N vn-1 R k-1 vk-1 k1 vk1
Whch means the rheostat should have a resstance of Rmax = 1440Ω 14.4Ω. Why s ths a bad dea? A. M. Nknejad Unversty of Calforna, Berkeley EE 100 / 42 Lecture 4 p. 13/14 (Bad) Applcaton: Lght Dmmer Suppose we wsh to buld a lght dmmer usng a rheostat. By addng a resstor n seres, we can control the voltage drop across the lamp, whch s modeld as an equvalent resstor. For R = 0Ω, the bulb radates at full ntensty. For R = Ω, the lght shuts off. Example: Say a 10W lght bulb works off a 12V battery. It has an equvalent DC resstance of V 2 R eq = 10W R eq = V 2 10W = 14.4Ω The lamp draws I = 10 W/12 V = 0.83A at full ntensty. To go to 10% lght level, we should reduce the current by a factor of 100. In other words, the equvalent seres combnaton of the rheostat and lamp should present R T = 12V = 12V I 10 LL 1200 = 1440Ω
A. M. Nknejad Unversty of Calforna, Berkeley EE 100 / 42 Lecture 4 p. 14/14 Dmmer wth Potentometer? R 1 You may have wondered why we ddn t use a potentometer to control the lght level. The reason s that the equaton we derved for the potentometer neglected the loadng effect of anythng connected to the thrd termnal. You can now see that the correct voltage dvder equaton s gven by V bulb = R bulb R 1 R bulb Ths s not as straghtforward as we thought. If R2 R bulb, then the desred equaton would follow but then a lot of unnecessary power would be wasted n R 1 and. Why?