Basic Direct-Current Circuits

Edited 7/31/13 by Stephen Albright, WL, JCH & DGH

Introduction

So far this course has covered basic electrostatics, electrostatic motors, electric fields, the force

of an electrostatic field, charge distribution, capacitance, dielectrics, the force between current

carrying conductors and the charge to mass ratio of the electron.

This lab exercise will explore the practical electrical components that have been developed from

these basic principles and how they can be used in simple direct current (DC) circuits. This lab

will also explore the basic DC motor. Standard measuring instruments will be used.

The components explored in this lab include resistors

( )

, capacitors

( )

, inductors

( )

and batteries/power supplies

( )

.

These will be explored with ammeters

( )

,

voltmeters

( )

and ohmmeters

( )

Fixed (as opposed to variable) resistors, capacitors and inductors in various configurations will

be used to establish the relations between resistance (R), voltage (V) and current (I) in simple

DC circuits. Resistance (R) is measured in ohms (Ω), capacitance (C) is measured in farads (F),

inductance (L) is measured in henries (H), voltage (V) is measured in volts (V) and current (I) is

measured in amps (A). Your results will be compared to that predicted by theory.

Note: Refer to “Introduction to Electrodynamics", by David J. Grifths, 3rd edition or "Electricity

and Magnetism", by Edward Purcell for equations relating current and voltage for resistors,

capacitors and inductors connected in series and in parallel.

Resistance Circuits

In the following schematics:

represents a voltmeter; it measures the potential difference (V) across its terminals

represents the voltmeter mounted in the power supply – values are displayed on

the front panel of the power supply

represents an ammeter; it measures the current (I) passing through a circuit element

represent an ohmmeter; it measures inherent resistance (Ω) – Note: ohmmeters can only

be used when no power is applied to the circuit element being measured

R

1

, R

2

and R

3

are resistors

represents the power source, in this case a power supply

represents an on off switch

Simple Series Circuit:

1) Use an ohmmeter to measure the resistance of R

1

, R

2

and R

3

on the module shown in Fig 1

and Schematic 1. Record these values.

2) Construct schematic 1. Set the power supply to approximately three volts DC as read on

the power supply voltmeter (V

p

– displayed on the front panel). Record this voltage.

3) Record the current (I) and the voltage across R

1

(V

R1

), R

2

(V

R2

), and R

3

(V

R3

).

4) Change the power supply voltage to about 7 volts and repeat step 3).

Figure 1 Schematic 1

Data Analysis:

Remember to specify estimated uncertainties in your experimental measurements when

comparing measured and calculated values.

a) For steps 3) & 4), compare V

p

, V

R1

, V

R2

& V

R3

. Describe and explain their relative values.

b) For steps 3) & 4), use Ohms law to calculate the current through R

1

, R

2

and R

3

. Compare I

(measured), I

R1

, I

R2

and I

R3

.

Describe and explain their relative values.

c) Use I and V

p

to determine the effective resistance of the three resistors in series. Compare

this with the combined resistance of the three resistors as measured individually.

Simple Parallel Circuit:

1) Use an ohmmeter to measure the resistance of R

1

, R

2

and R

3

on the module shown in Fig 2

and Schematic 2. Record these values.

2) Construct schematic 2. Set the power supply to approximately three volts DC as read on

the power supply voltmeter (V

p

). Record this voltage.

3) Record currents (I

1

, I

2

and I

3

) and the voltage across R

1

(V

R1

), R

2

(V

R2

), and R

3

(V

R3

).

4) Change the power supply voltage to about 7 volts and repeat step 3).

Figure 2 Schematic 2

Data Analysis:

Remember to specify estimated uncertainties in your experimental measurements when

comparing measured and calculated values.

a) For steps 3) & 4), compare V

p

, V

R1

, V

R2

& V

R3

. Describe and explain their relative values.

b) R

2

& R

3

are in parallel and are in series with R

1

. For steps 3) & 4), use Ohms law to

calculate the current through R

1

. Compare I

R1

, I

1

, I

2

and I

3

. Describe and explain their

relative values. Compare your measured values with those predicted by theory.

c) Use I

1

and V

p

to determine the effective resistance of these three resistors. Compare this

with the combined resistance of the three resistors as measured individually. Compare this

effective resistance with that predicted by theory.

d) Use the measured currents to compute the voltage drops across R

1

, R

2

and R

3

. Compare

these values with the theoretically predicted relationship among them, and with the

measured value of V

p

.

Resistance and Capacitance (RC) Circuits

Caution: The capacitors used in this lab have fairly high capacitance and it may be possible to

get a shock from one even after power to it has been removed. Always turn off the power source

and discharge all capacitors before touching the circuit connections. An insulated shorting

conductor is supplied for this purpose - be sure to hold the shorting conductor by its insulated

handle.

This lab exercise will explore the discharge rate of capacitors. Fully charged capacitors will be

discharged through a known resistor. The rate of discharge will be examined.

The discharge of capacitor C through resistor R is analyzed in the textbook. The charge q

remaining in the capacitor, at time t after discharge begins, decreases exponentially with a time

constant equal to RC. Since the time-dependent current is I = dq/dt , it is easy to show that the

current also decays exponentially. Instead of the constant current measured in Schematics 1&2

the current in an RC circuit is a function of time t:

I(t) = I

0

e

-t /RC

; log

e

(I/I

0

) = -t/RC (1)

Where t = 0 represents the start of discharge. During discharge, the capacitor is the power source.

Simple Series RC Circuit:

1) Use an ohmmeter to measure the resistance of resistor R on the module shown in Fig 3 and

Schematic 3. Record this value. Note: C

1

& C

2

are 330 microfarads each.

2) Construct schematic 3. The ammeter used in this circuit (as well as the parallel RC circuit

below) is the Agilent multimeter, set on μA. Be sure to connect the multimeter using the

Com and μA ports.

With switch S turned off, set the power supply to approximately 18 volts DC as read on the

power supply voltmeter (V

p

). Record this voltage.

3) Turn switch S on and wait for the current to become constant. Record this current as I

0

4) Simultaneously turn switch S off and start a stopwatch. Record the time required for the

current to fall to I

0

/2, I

0

/4, I

0

/8 and I

0

/16. Note: As the current decreases, it will pass through

50 μA. At approximately that value, the meter will change scales. The readings before and

after the scale change are accurate, it is just helpful to be aware the change will occur.

5) Set the power supply to approximately 36 volts. Repeat steps 3) &4)

6) Using your data from the 18 volt and 36 volt exercises, calculate the RC time constant for

these capacitors in series.

7) Compare your values of RC to that predicted by theory. Identify sources of error.

Figure 3 Schematic 3

Data Analysis:

Remember to specify estimated uncertainties in your experimental measurements when

comparing measured and calculated values. Do the following with your data from the 18 volt

and 36 volt exercises:

a) Investigate the validity of Eq. (1) by plotting the log

e

(I/I

o

) of each measurement against t.

The graph should be a straight line, reflecting the exponential form of Eq. (1).

b) From the slope of the line, determine the value of RC (ohms × farads = seconds).

c) Using the measured value of R, calculate the effective capacitance of the two capacitors in

series.

d) Compare this effective capacitance with that predicted by theory; assume the capacitors are

exactly their rated value.

Simple Parallel RC Circuit:

1) Use an ohmmeter to measure the resistance of resistor R, on the module shown in Fig 4 and

Schematic 4. Record this value. Note: C

1

& C

2

are 330 microfarad each.

2) Construct schematic 4. With switch S turned off, set the power supply to approximately 18

volts DC as read on the power supply voltmeter (V

p

). Record this voltage.

3) Turn switch S on and wait for the current to become constant. Record this current as I

0

4) Simultaneously turn switch S off and start a stopwatch. Record the time required for the

current to fall to I

0

/2, I

0

/4, I

0

/8 and I

0

/16.

5) Set the power supply to approximately 36 volts. Repeat steps 3) &4)

6) Using your data from the 18 volt and 36 volt exercises, calculate the RC time constant for

these capacitors in parallel.

7) Compare your values of RC to that predicted by theory. Identify sources of error.

Figure 4 Schematic 4

Data Analysis:

Remember to specify estimated uncertainties in your experimental measurements when

comparing measured and calculated values. Do the following with your data from the 18 volt

and 36 volt exercises:

a) Investigate the validity of Eq. (1) by plotting the log

e

(I/I

o

) of each measurement against t.

The graph should be a straight line, reflecting the exponential form of Eq. (1).

a) From the slope of the line, determine the value of RC (ohms × farads = seconds).

b) Using the measured value of R, calculate the effective capacitance of the two capacitors in

parallel.

c) Compare this effective capacitance with that predicted by theory; assume the capacitors are

exactly their rated value.

Questions:

a) Compare the effective capacitance of capacitors in series with capacitors in parallel.

b) Compare the effective value of capacitance in series and parallel to the effective value of

resistors in series and parallel.

Resistance and Inductance (RL) Circuits

This lab exercise will explore the charge and discharge rate of inductors. Inductors will be

charged and discharged through a resistor with a varying applied voltage. Charge and discharge

rates are examined in the textbook. Due to the magnetic field generated by the inductor, the

current through an inductor resists change. Therefore, when the voltage across an inductor

suddenly changes, the current will exponentially approach its new value with time constant equal

to L/R. Explicitly, the current as a function of time for charging and discharging from an applied

voltage V with resistor R in series is

I

charge

(t) = V/R(1- e

-t R/L

) (2)

I

discharge

(t) = I

0

e

-t R/L

(3)

Where t = 0 represents the start of the charge/discharge. During discharge, the inductor acts as a

current source.

Due to the limitations of the properties of real inductors, an RL circuit cannot be built that allows

measurement of a change in current with a stopwatch. To allow measurement, an oscilloscope

(or scope) will be used. Oscilloscopes are electronic devices that graphically display electrical

signals. The scope measures voltage very rapidly and plots the values against time. Imagine

plotting the measurements in the RC experiment against time – this is essentially what a scope

does, just much more rapidly.

The voltage will be supplied by a function generator, an electronic device that generates signals.

In this case, it will simply be a square wave – the signal switches between 0 and 10V. The

frequency of this switching is 600 Hz, so the voltage turns on/off every 1.67 ms.

While use of function generators and scopes will be required in subsequent lab courses, both will

be setup already; they only need to be turned on.

Schematic 5

Simple Series RL Circuit:

1) Use an ohmmeter to measure the resistance of resistor R in Schematic 5

2) Construct the circuit in Schematic 5, ensuring the black output of the function generator

is on the same side of the resistor as the black input to the scope (scope ground). Use the

inductor manufactured by Miller (not the homemade coil).

3) Turn on both the function generator and the scope. The waveform on the scope should

resemble Figure 5.

Figure 5

4) There should be yellow lines running vertically across the screen. These are cursors.

Their positions are controlled by the Multipurpose A and B knobs on the front panel of

the scope. The box in the upper right hand side of the screen displays their positions in

time, the voltage of the waveform when they intersect, and the differences between the

cursors. Put cursor A at the maximum of one discharge (right before the discharge

begins). Record this voltage as V

0

. Note: I

0

= V

0

/ R.

5) On the same discharge curve, put cursor B where the voltage is 1/2, 1/4, 1/8, and 1/16 the

maximum voltage and record the time from cursor A – the time from the beginning of the

discharge.

6) Repeat step 5) for a charging cycle, with cursor A at a charging minimum (right as

charging begins) and cursor B at some fraction of V

0

.

7) Using data collected from the charge and discharge cycles, calculate the time constant

L/R.

8) Using the time constant and the measured value of resistance, calculate the inductance of

the inductor. How does the measured value compare to the value written on the inductor?

9) Measure the resistance of the inductor with an ohmmeter. Recalculate the inductance of

by assuming the inductor resistance is in series with the resistor. How does this value

compare with the one published on the inductor?

Data Analysis:

Remember to specify estimated uncertainties in your experimental measurements when

comparing measured and calculated values. Do the following with your data from the charging

and discharging cycles.

a) Investigate the validity of Eq. (2) by plotting the log

e

(I/I

o

) of each measurement against t.

The graph should be a straight line, reflecting the exponential form of Eq. (2).

a) From the slope of the line, determine the value of L/R (henries ∕ ohms = seconds).

b) Using the measured value of R, calculate the inductance of the inductor in the circuit.

Compare with the value printed on the inductor.

c) Now including the measured resistance of the inductor, calculate the inductance. Compare

this value with the value printed on the inductor.

Self-Inductance of a Coil

This lab exercise will explore the effects of rapidly changing current in an inductor. Faraday

discovered that a time varying magnetic flux induces an electromotive force (emf) in a conductor

that opposes the change in flux. We can demonstrate this using an inductor, which in our case

will be a small handmade coil. When a current passes through the coil, a magnetic field is

produced that generates a magnetic flux through the coil. If the current through the coil is

changing, so is the flux Φ.

From Faraday’s Law, there will be a voltage drop across the coil given by:

𝑉 = −

1

𝑐

𝑑𝛷

𝑑𝑡

(4)

The flux through the coil depends only on the current I in the coil and some geometrical factors.

Thus, we can write the induced emf V as dI/dt times a constant (defined as inductance L) specific

to the coil:

𝑉 = −𝐿

𝑑𝐼

𝑑𝑡

(5)

For a long solenoid of length , radius r, and N turns of wire, with a uniform field B = 4πNI/c

inside (the flux through each loop is then πr

2

B), the self-inductance is:

𝐿 =

4𝜋

2

𝑁

2

𝑟

2

𝑙𝑐

2

sec

2

cm

−1

(6)

The above expression for inductance L is in CGS units. The SI unit of inductance is the “henry”,

where 1 henry = 1.11 × 10

-12

sec

2

cm

-1

. In SI units, formula 3 becomes

𝐿 =

𝑁

2

𝜋𝑟

2

𝜖

0

𝑐

2

𝑙

henry

(7)

Simple Series RL Circuit Part II:

a) Measure and record the number of turns, radius and length of the homemade coil.

Calculate the inductance of your coil using Eq. (7).

b) Replace the inductor in Schematic 5 with the homemade inductor.

c) Set the function generator to a triangle wave by pressing the down arrow, in the “Func”

section of the panel, once. This should produce a wave that looks like Figure 6. Set the

function generator to 10 kHz by pressing the following buttons in the “Data Entry”

section: 1, 0, Vpp/kHz. The Vpp/kHz button is located on the extreme right, with an up

arrow on it.

d) On the scope screen, there should be a time on the bottom center part of the screen.

Rotate the Horizontal Scale knob clockwise until this time reads “40.0 µs.”

e) In this configuration, the oscilloscope is measuring the voltage across the resistor. Since

V = IR, this voltage is proportional to the current in the RL circuit. Use the cursors to

find dI/dt in amps/sec. Record your value.

Figure 6: Voltage Across R

f) Now connect the oscilloscope across the homemade coil as shown in Schematic 6.

Schematic 6

g) On the bottom, left-hand side of the scope screen, there should be a voltage next to a

yellow “1.” Rotate the Vertical Scale knob under “1” clockwise until the voltage on the

screen reads “10.0 mV.” The waveform should look similar to Figure 7.

Figure 7

The sudden change in voltage across the indicator, ∆V

L

, comes from the abrupt change in

dI/dt:

Δ𝑉

𝐿

= 2𝐿

𝑑𝐼

𝑑𝑡

(8)

h) Press the Cursors button once to get horizontal lines across the screen. Use the cursors to

measure the drop in voltage, ∆V

L

.

Data Analysis:

Remember to specify estimated uncertainties in your experimental measurements when

comparing measured and calculated values.

a) Using equation 7, calculate the theoretical inductance of the homemade coil from the

measured dimensions and number of turns.

b) Calculate dI/dt from measurements made in part e).

c) Using the measured ∆V

L

and dI/dt, calculate the inductance of the homemade coil using

Equation 8. Compare this value to the one calculated from the coil’s dimensions.

Questions:

a) Why isn’t the waveform a perfect square wave?

b) How does the coil’s resistance affect the waveform?

The sudden voltage drop measured above is common when working with inductors and can

become quite dangerous with large enough currents. In a real circuit with a large inductor, one

must avoid shutting off the current supply suddenly; if the current shuts off suddenly, a huge

voltage will develop across the inductor. This voltage can be large enough to destroy other

electronics in the circuit or even discharge through the air, seriously injuring or killing someone

nearby. A good example that requires attention to this is the large electromagnet used in the

upper level labs for pulsed-NMR experiments.

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