Resistors (W4.1/4.2 - Chap 2.4 - 2.8)

resistivity, p (not the letter p, but the Greek lowercase letter rho)

intrinsic ability of material to RESIST the flow of charge

• depends on ONLY the material, not size or length

• units: Ohm-meter (Ω·m)

• built-in characteristic of a material

resistance, R

the opposition to the flow of current in an elem

• depends on resistivity, length, cross sectional area // thickness (see notes for diagram)

• units: Ohms (Ω)

formula for R using p (rho), L, and A

resistance (R) in relation to Ohm’s Law

Ohm’s Law: v = R*i

therefore, R = v/i where 1Ω = 1V/1A

(rmbr: elem must be LINEAR for this to apply, you can assume a resistor is linear unless stated otherwise)

conductance, G

physical property reciprocal to the resistance, describes an elem’s ability to encourage the flow of current.

• G = 1/R

• units: 1S = 1Ω_⁻¹ = 1A/1V

• S : siemens

general symbol of a resistor in circuit diagram

power of a resistor, p

power DELIVERED TO (received by) a resistor when PC applies.

• p = v*i = v*(v/R) = v²/R → helpful when you are not given the value of i (current through the resistor)

• p = v*i = (i*R)*i = i²R → helpful when you are not given the value of v (voltage across the resistor)

source 

voltage OR current generator which supplies en to a circuit

independent source

a voltage or current generator not dependent on other circuit variables

IDEAL voltage source

a perfect voltage source that maintains a fixed voltage with zero internal resistance

• only exists in theory

• always holds v exactly const even if i or any other circuit var changes

• can absorb/supply infinite current

• behaves like a short circuit if v = 0 (specified voltage = 0 V)

• ex: a 5V source w 0Ω internal resistance.

INDEPENDENT voltage source

a voltage source that provides a fixed voltage val not dep on any circuit vars

• voltage is const and does not dep on other vars

• could have internal resistance in complex circuits (this is why most ind. voltage sources are not considered IDEAL) → irl, ind. sources basically always have some amount of internal resistance

• can be realistic or ideal (depending on if it’s modelled after a real circuit or a theoretical circuit) 

• ex: a 5 V battery w unknown/non-zero internal resistance

all _____ sources are ________ but NOT all ______ sources are ________

ideal; ind; ind; ideal

IDEAL current source

current source that provides a const current no matter the voltage across it, and contains INFINITE INTERNAL RESISTANCE

• inf internal R → no current flows thru source itself other than the specified current.

• only exists in theory

• voltage across it can go up or down as required to keep current const.

• deps on no other vars (like voltage or current from elsewhere)

• behaves like an OPEN CIRCUIT if i = 0 (the specified current = 0 A)

independent current source

current source whose output current is fixed or specified and doesn’t dep on the voltage across it or any other circuit var (voltage or current elsewhere)

• current is set externally ind. of circuit

• can be realistic or ideal (depending on if it’s modelled after a real circuit or a theoretical circuit) 

• has finite internal R if non-ideal

open circuit (mathematically)

i = 0 A (no current flow)

R = very high / ideally approaching inf.

v = varying (can be any val)

open circuit defn

a break or gap in a circuit path where NO CURRENT can flow

• voltage across the open terminals can be any value

• ex. a switch that’s turned off, a cut wire, battery removed from device.

open circuit symbol 

short circuit (mathematically)

i = VERY HIGH (can be dangerously high)

v = 0 V

R = very low / ideally 0

short circuit defn

unintended low-resistance path in a circuit that allows current to flow along a SHORTCUT instead of the correct path (NOT desired)

• happens when 2 terminals in a circuit that should be at diff voltages are connected directly w vv little/0 R

• the low R means that a VERY LARGE current can flow, which can cause overheating/damage/fires

• ex. bare wires touching where they shouldn’t, damaged insulation causing wires to connect directly

short circuit symbol

see notes

ideal voltage source symbol

see notes

ideal current source symbol

see notes

voltmeter

an instrument used to measure the diff in elec pot b/w 2 points in a circuit.

• has vv high internal R so it doesn’t draw significant current from the circuit

• an IDEAL voltmeter would have no current flowing in it at all.

if R → very large, then current → ______

very small, bc resistance OPPOSES THE FLOW OF CURRENT

illustration of a voltmeter

• connect red probe to positive terminal (where it has higher electric potential)

• connect black probe to negative terminal (where it has lower elec pot)

• current would flow from higher pot → lower pot

• voltage is measured ACROSS an element so it is connected in || to the elem

how to read a voltmeter

RMBR: v_rb = v_red - v_black

• rmbr that the voltmeter gives you the elec pot DIFFERENCE

• if the voltmeter’s reading of the elec pot diff is pos, then terminal connected to the red probe is at a higher elec pot

• if neg, the terminal connected to the red probe is at a lower elec pot → this means our assumption of which terminal had higher potential was wrong.

ammeter

an instrument used to measure the current flowing thru a branch of a circuit

• vv low internal R so that current can pass through easily (current always favours the path of least R)

• an IDEAL ammeter would have a voltage of 0

if R → very small, then i → _____

very large

illustration of an ammeter

• red probe: where we assume current enters

• black probe: where we assume current leaves

• unlike a voltmeter, an ammeter must be placed in SERIES after the elem we want to analyze

how to read a ammeter

• pos reading: the current flows in the direction of the probes (enters the red probe and leaves through the black probe) → adheres to passive convention

• neg reading: the current flows oppo to the probe dir (enters the black probe and leaves through red probe) → does not adhere to passive convention.

• an ammeter would not give a reading if a current does not go through it COMPLETELY.

KCL (Kirchhoff’s Current Law)

defn: the sum of currents entering a node/junction = the sum of currents leaving the node/junction

• this is why i is the same througout a purely series circuit (see image)

KVL (Kirchhoff’s Voltage Law)

defn: the algebraic sum of all voltages around any CLOSED LOOP = 0 (the gains and losses of voltage add up to 0)

Voltage Divider Rule

• voltage drop across an elem is directly proportional to the elem’s resistance.

• you can ONLY use v_source in formula

• ONLY applies to series circuits.

a __________ pushes current thru an elem

voltage drop

earth ground

the reference point in a circuit where voltage = 0 and all the potential has been “used up”

simplified universal symbols for voltmeter + ammeter

universal symbol for battery

• pos end is always the longer end

current divider rule

• current is divided in a parallel circuit

• current is separated at a node

• the formula is simpler with G (conductance) instead of R → recall that G = 1/R

formula for adding resistances that are in PARALLEL with each other

(or you could convert to G for an easier computation)