This Set o’ Slides - Day 16, Wednesday, Feb 11
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Circuit Symbols
Kirchhoff’s Rules
Resistors in Series and Parallel
Analyzing Simple Circuits
Oh yeah, (slightly) more complex ones also.
Simple Circuits
• Series. If several electrical devices (wires, batteries,
resistors, light bulbs, capacitors, etc…) are connected in
such a way that there is only one path for the current to
flow, that is, the same current flows through each device
sequentially, then the devices are said to be wired in series.
• Parallel. If several devices are connected in such a way that
the same voltage is applied across each device, then the
devices are said to be wired in parallel.
Kirchhoff’s Rules
• Junction Rule: The total current coming into a junction in a
circuit must equal the total current leaving that junction.
Σ Iin = Σ Iout
• Loop Rule: For any close-circuit loop (path), the sum of all
the potential (voltage) drops must equal the sum of all the
potential (voltage) gains.
Δ Vloop= Σ Δ Vi = 0
Simple Battery, Wire, Resistor Circuit
 The figure shows a
resistor connected to a
battery with currentcarrying wires.
 Current must be
conserved; hence the
current I through the
resistor is the same as
the current in each wire.
 Let’s look at how the
potential changes
throughout the circuit.
Simple Battery, Wire, Resistor Circuit –
Reality vs. Ideal
Ideal – Our “Reality”!
Reality
Kirchhoff’s Loop Law
Resistors in Series
• For resistors in series, the total or equivalent resistance (of
the combo) is simply the sum of the individual resistances:
RS = R1 + R2 + R3 + ….
• The equivalent resistance is the same as the one resistance
that could replace all the other resistors and still have the
same current through that part of the circuit.
• Note that the equivalent resistance will be greater than the
largest of the individual resistors.
• Each resistor must have the same current, I.
Resistors in Series (continued)
• The voltage drop across each resistor is ΔV = IR.
• The voltage drop across the combo of resistors is the sum of
the individual voltage drops and is equal to current times
the equivalent resistance, RS.
ΔVtotal = ΔV1 + ΔV2 + ΔV3 +… = I RS
• The power dissipated through each resistor is P = I2 R.
• The total power dissipated through the combo of resistors is
equal to the sum of the power dissipated through the
individual resistors and is equal to the power dissipated
through an equivalent resistor, RS.
Ptotal = P1 + P2 + P3 +… = I2 RS
Resistors in Parallel
• For resistors in parallel, the total or equivalent resistance
(of the combo) is the reciprocal of the sum of the reciprocals
of the individual resistances:
1 / RP = 1 / R1 + 1 / R2 + 1 / R3 + …
RP = [1 / R1 + 1 / R2 + 1 / R3 + …]-1
or
• The equivalent resistance is the same as the one resistance
that could replace all the other resistors and still have the
same current through that part of the circuit.
• Note that the equivalent resistance will be less than the least
of the individual resistors.
• Each resistor must have the same potential drop or voltage.
Resistors in Parallel (continued)
• The current through each resistor is I = ΔV / R.
• The current through all of the resistors is the sum of the
individual currents and is equal to voltage divided by the
equivalent resistance, RP.
Itotal = I1 + I2 + I3 +… = ΔV / RP
• The power dissipated through each resistor is P = I2 R.
• The total power dissipated through the combo of resistors is
equal to the sum of the power dissipated through the
individual resistors and is equal to the power dissipated
through an equivalent resistor, RP.
Ptotal = P1 + P2 + P3 +… = I2 RP
Household Circuits
• The outlets and most lighting in your house are wired in
parallel. How do you know?
- Everything has the same voltage.
- One thing turned off doesn’t affect other things.
- Note that switches must be wired in series with the
outlets or lights they control.
- Circuit breakers/fuses must also be in series with the
circuit they are designed to protect.
Resistors in Both Series and Parallel
• Woohoo!
• If too complex, need Kirchhoff’s Rules and a “system”.
• For simpler circuits (mostly resistors in series and parallel
with one battery), simplify and “piece together” the
solution.
• Do lots of examples. Here’s one.
Solving Circuits with Kirchhoff’s Rules
• Draw out the circuit.
• Draw the current in each branch of the circuit. Generally, in
the direction you believe the current goes, but, the direction
is arbitrary. If you choose incorrectly, your calculations will
result in a negative value, meaning the correct direction is
opposite your chosen direction.
• Mark each resistor end with a plus and minus sign, based on
your chosen current direction. Outside of a battery, current
always flows from higher (+) potential to lower (-) potential
• Apply the junction rule at all junctions.
Solving Circuits with Kirchhoff’s Rules (continued)
• Assign loop directions for each loop. Direction is arbitrary;
you’ll get the same (correct) answer either way. Generally,
loop from – to + through batteries when possible. This is for
intuition’s sake and isn’t necessary.
• Apply the loop rule going in the direction chosen.
Σ Vdrops = Σ Vrises
Potential difference in resistors is given by Ohm’s Law:
V=IR.
Potential difference in batteries is given by size of battery.
Solving Circuits with Kirchhoff’s Rules (continued)
• Looping rule continued:
Looping through any device from + to – is a Vdrop.
Looping through any device from – to + is a Vrise.
A loop that goes against the current in a resistor signifies a
potential rise, Vrise.
• Obtain as many equations as unknowns.
• Solve equations simultaneously.
• Check.
Double-check potential sum in loop.
Check if/that Pinput = Poutput.
Internal Resistance and Terminal Voltage
• Devices that provide emf, batteries and generators for example, also
have their own internal resistance.
• This resistance can be thought of as a separate resistor in series with
the device and the circuit.
• This lowers the output voltage from its nominal value.
• The output voltage is called the terminal voltage (since it’s measured
at the terminals.) It’s dependent on the total current being drawn from
the source (battery).
Vterminal = E – I Rinternal
• A voltmeter might register adequate voltage (emf), but, a voltmeter
draws little/no current. The battery looks good, but, when current is
drawn from battery, the terminal voltage drops to low levels.
• This is why the best battery test is a load test.