Download Complicated Circuits

Lab 4: More Complicated Circuits
Objectives: to practice using Kirchhoff’s Current Law at circuit nodes;
to practice using Kirchhoff’s Voltage Law around circuit loops;
to distinguish between series, and parallel, and bridge topologies;
to verify formulas for adding resistance (and inverse capacitance) in series,
recognizing devices in series as voltage dividers, with a load line;
to verify formulas for adding conductance (and capacitance) in parallel,
recognizing devices in parallel as current dividers.
Apparatus:
variable resistor, dry cell or photovoltaic cell, capacitors
Equipment: DC Power Supply, Digital Voltmeter & Ohmmeter; signal generator, oscilloscope
Background: There are two fundamental ways to connect two devices together in a circuit:
series, and parallel. Devices in series must have the same current flowing
through them (since there is nowhere else for the current to go). The potential
difference across a pair of series devices (from the top of the first device to
the bottom of the last device) is Vseries = Va – Vc = (Va – Vb) + (Vb – Vc) ...
the sum of the two device voltages. Kirchhoff’s Voltage Law says that this is
independent of what path is taken from point a to point c, at any instant. This means
that the voltage across all the devices (from a to c) must be the negative of the PS
voltage (from c to a) ... or, the sum of voltages around a closed loop must be zero.
We have seen that charge does not accumulate in any circuit devices.
Kirchhoff’s Current Law says that the amount of current going into any
device or junction is the amount flowing out ... or, that the total current
flowing in thru all paths is zero. Current from a PS splits up at junction a
so different amounts go thru each (||) branch, but the currents re-join at junction b,
so the same amount of current enters the PS as left it.
Because all of the charges in a circuit are the same, the KVL is Energy conservation.
The Power Supply voltage is divided among the passive series devices, depending on
each device’s resistance; the most voltage is dropped (most electric Energy lost) within
the most resistive device in that branch.
The sum of voltages along each branch equals the PS voltage, so if the devices have
the same current (if they’re in series) then the total effective resistance of that branch is
Rseries = δVseries /I = R1 + R2 + R3 +... . A useful consequence is that a variable resistor,
in series with a load, can vary the voltage across the load (and the Power to the load).
Conducting paths “in parallel” from point A to point B have the same voltage,
δV=VA – VB across them, but the currents in each branch add up (at junctions A and B)
to the total current. So S|| = (Σ Ii )/V = Σ δV/Ri => 1/R|| = 1/R1 + 1/R2 + 1/R3 + ... .
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Activity 1. Variable Resistor – Load Line
Inspect the variable resistor to determine its (total) resistance, and its max ratings.
Connect a Power Supply to the variable resistor (center pin – to – one side) in series
with a fixed commercial resistor, R = ______ ___
Adjust the PS voltage to some safe voltage (for both devices!): ________ ___.
For 5 very different knob orientations of the variable resistor, use a Digital Voltmeter
to measure the voltage across each device; calculate I , Rvariable , Rtotal , and Pvariable .
angle Vvariable[V] Vfixed [V]
Sample calculations:
Activity 2. Internal Resistance (output impedance) of a Power Source
Use the digital Voltmeter to measure the voltage of a dry cell: δVo = _______ ____ .
use a DMM as Ohmmeter to estimate the Voltmeter’s “Resistance” RV : _______ ____
calculate the current thru the Voltmeter (while measuring cell voltage): ______ ____
connect two 50[Ω] resistors, in series ; use DMM as Ohmmeter to determine their R
then connect them to the dry cell to the dry cell, and measure the dry cell voltage.
Measure the dry cell voltage while connected to a single 50[Ω] resistor.
Connect two 50[Ω] resistors in parallel ; use DMM as ohmmeter to determine their R
then connect them to the dry cell, and measure the dry cell voltage.
Rexternal []
δVcell [V]
I= δVcell/Rext [mA]
r = (δVo – δVI )/I
Calculate external currents and estimate the resistance inside the dry cell.
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Activity 3. Resistor – Capacitor (RC) Charging & Discharging
Connect a “50 Ω” R _______ ____ in series with a “100 μF” C ______ ____ .
to “signal generator”, set to produce square wave-forms with frequency f ≈ 100 Hz.
The oscilloscope screen will show us a graph of both voltages as a function of time ...
but all of the BNC jacks’ shield wires (black) are connected to their instrument case.
First, use 3-prong adapter to float the oscilloscope case above signal generator ground.
Then connect both o’scope shield wires (black) to the connection between R and C .
Use Channel 1 Red at the other side of R, and Channel 2 red at the other side of C .
Turn the oscilloscope “on”, and (once it displays data) press chan 2 [menu] => invert
Adjust the “vertical position” of both input channels so you know which is VR and VC.
Adjust the vertical “voltage scales” so that each graph trace fills a few screen divisions
suggestion: adjust generator “Output” so “1” fills top half and “2” fills bottom half.
Adjust oscilloscope horizontal “time scale” so an entire wave ( |¯|_| ) fits on the screen.
Sketch Resistor voltage
carefully, to scale,
including δV and t values
The resistor’s peak voltage
is ______ [divisions]
× _____ [mV/division]
= _______ ___ .
The time needed for δVR to
decrease to ½ this maximum
is ______ [divisions] × _____ [ms/division] = _______ ___ ± ___% (uncertainty).
Mean decay time τ , the “1/e time”, is 1.4427× = (1/ℓn[2])× longer than this “½ time”
Compute τ : ________ ____ ; also compute the product R·C: _________ ____ .
pay careful attention to units! compute the % difference: ______ . (discuss in report)
draw a straight line, tangent to δVR curve at t=0; when does it cross δVR =0? _____ ___
Sketch Capacitor voltage
carefully, to scale,
including δV and t values
The capacitor’s peak voltage
is ______ [divisions]
× _____ [mV/division]
= _______ ___ .
The time needed for δVC to
increase to ½ this maximum
is ______ [divisions] × _____ [ms/division] = _______ ___ ± ____ (uncertainty).
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Connect another “identical” capacitor _______ ____ parallel the original C1 .
What happens to the time required to half-fill the pair of capacitors?
So, the effective capacitance of two in parallel is (formula!) C|| =
remove C2 . Connect another “identical” resistor _______ ____ parallel to original R1.
What happens to the time required to half-fill the original capacitor?
So, the effective resistance of two in parallel is (formula!)
Activity 4. Capacitors in Series, as a Voltage Divider
Connect C1 ______ ____ in series with a different capacitor C2 = _______ ____.
Connect to the signal generator, set to produce sine waves.
Draw the circuit, including labels for each device.
Set DMM to measure AC voltage. For two frequencies, and two output levels at each,
measure voltages for C1 , for C2 , and both (C1,C2) .
δVC1 [V]
δVC2 [V]
δVboth [V]
δV1+δV2 [V]
δV1/ δVboth
δV2/ δVboth
Lab 4 Summary
1. Graph Voltage across variable resistor (↑) vs. Current thru variable resistor (→) .
determine the slope, and both intercepts , of the “best-fit” line. Label the intercepts
“short-circuit I”, “open-circuit V”. What is the meaning of the slope (or its inverse)?
2. The Voltmeter’s measured impedance was not zero, so there was some current into it,
so the internal resistance had some voltage drop while Vo was being measured.
Estimate this voltage drop … and the % error that this causes in Vo .
3. (Answer on Conclusion sheet) Compute the maximum current thru the 50Ω resistor;
at this rate, how long would it take to fill the capacitor? Why did the current decrease?
Why did the current not decrease along a straight line?
4. Compare & Contrast: series and parallel Resistors, with series and parallel Capacitors.
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