Prof. Anyes Taffard. Physics 120/220. Foundations Circuit elements Resistors: series & parallel Ohm s law Kirchhoff s laws Complex numbers

Transcription

1 Prof. Anyes Taffard Physics 120/220 Foundations Circuit elements Resistors: series & parallel Ohm s law Kirchhoff s laws Complex numbers

2 Foundations Units: ü Q: charge [Coulomb] ü V: voltage = potential [Volt] ü I: current = flow of charge [Ampere] ü P: power = energy per unit time [Watts] 2 Energy: ü Electrical: QV= potential energy of charge Q ü Mechanical: mgh= potential energy of mass m Concepts: zero is arbitrary for potential energy. Mechanical: h referenced to sea level/floor Electrical: V referenced to what?

3 Foundations (cont.) 3 Ground: As a convention, zero volts = ground potential It a large body capable of accumulating or losing lots of electrons. Example: metal enclosure of a battery powered circuit. Circuit symbol: Note: Use common ground in your circuits. Voltage drop: difference in potential between two points Bias: a voltage, usually DC (direct current) Current: I is positive in the direction which positive charges move. ü Positive current moves from more positive voltage to more negative voltage. I = dq dt

4 DC vs- AC DC: Direct Current. Constant voltage versus time. 4 Electrical power AC (Alternative Current): 60Hz: North America 50Hz: Europe, Asia. Amplitude for AC (Alternative current): V peak V pp =V peak-to-peak V peak = 1 2 V pp V rms = 1 2 V peak Useful to describe quantity that are positive and negative. See Young & Freedman p1063

5 About Schematic Circuit diagrams 5 Learn how to read (and draw) them. Wire: assume electric potential is uniform along wire Node: Connected Not connected Battery: AC power supply (PS) or function generator: Resistor:

6 About Schematic Circuit diagrams (cont.) Capacitor: Polarized capacitor 6 Diode: Amplifier:

7 Convention for drawing schematic 7 Source on left Load on right Positive voltage at top Label components Lines representing wires are horizontal or vertical Negative voltage at bottom In Out Signal flows left to right 4-way nodes Preferred style Not preferred

8 Resistors in series and parallel 8 Series Parallel Equivalent Resistors in series: R eq = R 1 + R 2, always get larger R eq Same current across each resistors 1 Resistors in parallel: = 1 + 1, R R, always get smaller R eq R 1 R eq = R 1R 2 2 R eq 1 + R 2 Same voltage across each resistor Large R in series (in parallel) with a small R has the resistance of the largest (smaller) one

9 Resistors in series and parallel (cont.) Example: 5k in parallel with 10k 5k is about two 10k in parallel, therefore it s about equivalent to three 10k in parallel. 1 = 3 R eq 10 9 R = 10 3 = 3.33k Learn how to do back of the envelope calculation (accuracy within 5%): If two resistors in parallel differ by a factor of 10, then we can ignore the larger of the two.

10 Ohm s law & Power 10 Ohm s law: V=IR or I= V/R V = voltage drop across resistor I = current thru resistor Ohm s law is empirical, good for some material but not all. Power: P=VI For a resistor: P=RI 2 P=V 2 /R Note Resistor: 3 parameters Resistance R Tolerance % Power rating: Don t exceed this or you will burn it

11 Kirchhoff's rules 11 Rule 1: Sum of the voltage around a close loop is zero. ( V 2 V ) 1 + ( V 3 V ) 2 + ( V 1 V ) 3 = 0 Voltage drop across R 1 Voltage drop across R 2 Voltage drop across battery To use, choose either CCW or CW direction. When a positive test charge heads towards the negative of the battery, when it passes across R 1 (and R 2 ), there is a decrease in potential (V 2 <V 1 ), thus use negative potential. Across the battery, go from negative to positive voltage, so use positive potential.

12 Kirchhoff's rules (cont.) Rule 2: Sum of the current into a node is zero 12 I 3 I 1 + I 2 + I 3 = 0 I 1 I 2 To use, decide (arbitrarily) a positive direction for the current. If you get a negative current as answer, the current direction is opposite to the one chosen.

13 Complex Numbers 13

14 Complex Numbers Notation: N or N!!N = a + bi ü Complex conjugate!n * = a bi i = j = 1 or i 2 = j 2 = 1 a: real part b: imaginary part 14 Rules: Addition Subtraction Multiplication Division ( a + bi) + ( c + di) = ( a + c) + ( b + d)i ( a + bi) ( c + di) = ( a c) + ( b d)i ( a + bi) ( c + di) = ( ac bd) + ( bc + ad)i ( a + bi) c + di ( ) = ( a + bi )( c di) ( c + di) ( c di) ac + bd ac ad = + c d c 2 + d i 2

15 Complex Numbers (cont.) Magnitude (modulus): 15 1!N = a + bi = a + bi 2 = a 2 + b 2!N = ( N! N! * ) 1 2 Real part: Imaginary part:!n=re ( N! ) ( )( a bi)!n=im! N ( ) ( ) 1 2 = a 2 + b 2

16 Complex plane X-axis: real part Y-axis: imaginary part Can also use polar coordinate (R,θ): a + bi = ( R,θ ) R = ( a 2 + b 2 ) 1 2 =!N = (!N!N * ) 1 2 b θ = tan 1 a Since e ix = cos x + isin x!n = a + bi = R e iθ Multiply (divide) using polar coord.: multiply (divide) magnitude and add (subtract) θ à ( ) ( ae iθ ) ce iθ ( ) = ac e i b+d Im b θ a R Re 16 Euler s formula: ae iθ = acosb + isinb ( ) = acosb ( ) = asinb Re ae ib Im ae ib