Benjamin Griffiths (160159871)

AMPLIFIERS REPORT
[WORD COUNT = 2401]

Introduction

The purpose of this lab is to show how the performance of an operational amplifier circuit in the frequency domain
can be represented by a first order model. Different power amplifier circuits will be constructed, to compare
differences in performance and investigate the effects of feedback to correct defects. Furthermore, the effects of
measuring equipment such as oscilloscope probes and BNC cables on a circuits performance will be investigated.

Background

The first part of the lab looks into the effects of cable impedance and capacitance on the circuits performance it is
measuring from. It is important to interpret measurements carefully because the measured values shown on the
oscilloscope may not necessarily be accurate due to the impedance and capacitance of the probes interacting with
the circuit.

For example, when measuring the rise time of a pulsed signal through an RC circuit, the rise time changes when
multiple BNC cables are used as opposed to one. Hence great care must be taking when interpreting measurements,
since the values were different despite there being no physical change to the circuit. The purpose is to demonstrate
how measuring equipment effectively becomes part of the circuit, and how the effects can be alleviated by using a
balanced 10:1 oscilloscope probe.

The second part of the lab focusses on showing how an operational amplifier with feedback can be characterised by
a first order system, where mathematical relationships link time domain and frequency domain measurements
together. This will be shown by taking measurements in the time domain and frequency domain, where they can be
compared to their theoretical values using mathematical relationships.

For example, for a non-inverting amplifier, the rise time and corner frequency can be measured using an oscilloscope.
The relationship between the rise time and time constant can be used to calculate the corner frequency, which, if
equal to the measured value, proves the relationship.

The third part of the lab focusses on power amplifiers, and identifies why there are defects and how they can be
avoided. This is important in many applications such as audio amplification where you want to amplify a signal, while
maintaining its shape.

For example, when a push-pull amplifier configuration is constructed and tested, a defect in the output signal is seen.

Theory

Operational amplifiers, also known as op-amps are high gain DC and AC signal amplifiers which have a range of uses
and can be configured in ways to exhibit specific behaviours such as voltage adders and differential amplifiers.

As seen in figure 2, op-amps are 3 terminal devices, with an inverting input, non-inverting input and output terminal,
where the input terminals having infinite impedance and the output terminal having a very low impedance. Op-
amps operate by amplifying the difference in voltage at their inputs by its open-loop gain, which is ideally infinite.

The operational amplifier equation that shows this is:

𝑉𝑂𝑈𝑇 = 𝐴𝑜 (𝑉 + − 𝑉 − ) (1)

Where V+ and V- are the non-inverting and inverting voltage respectively, and Ao is open-loop gain (explained later).

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Operational amplifiers can be modelled as first order systems, where the corner frequency (fo), time constant (τ)
and phase angle (ɸ) have a defined relationships.

The corner frequency is where the signal is attenuated by -3dB.

The voltage gain decibel:

𝑉𝑜𝑢𝑡
𝑑𝐵 = 20 log10 ( ) (2)
𝑉𝑖𝑛

Hence the voltage-gain at the -3dB point will be approximately 70.8% of the maximum gain value.

The corner frequency:

1
𝑓0 = (3)
2𝜋𝜏
Where 𝜏 = 𝑅𝐶, the time constant of the circuit.

The time constant has the following relationship to the rise time:

𝑅𝑖𝑠𝑒 𝑇𝑖𝑚𝑒 = 2.2𝜏 (4)

Where the rise time (seconds) is the time it takes for a square wave pulse to go from 10% to 90% of its maximum
value.

The phase angle is related to the frequency of the input signal (figure 1).

Figure 1: Phase angle at half power frequency.

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The phase angle varies with frequency, with the corner frequency having a phase shift of +45°.

In the second part of the lab, a standard non-inverting op-amp amplifier set-up was used as shown in figure 2.

Op-Amp
VIN
VOUT

R2
R1

Figure 2: Non-inverting op-amp circuit

Where the gain of this amplifier is given by:

𝑉𝑂𝑈𝑇 𝑅2
𝐺𝑎𝑖𝑛 = =1+ (5)
𝑉𝐼𝑁 𝑅1
The gain equation (4) assumes the open loop gain is infinite. The open loop gain is defined as the gain when there is
no feedback, typically around 20,000. However, due to internal parasitic capacitances within the op-amp itself, as
the frequency increases, this capacitance exhibits a lower reactance, effectively reducing the gain. The effects of
this can be clearly seen in figure 3, where the open loop gain is only maintained at low frequencies below 10Hz, and
is then attenuated at a rate of 20db per frequency decade. [1]

The effect of the op amps parasitic capacitance explains why the output is phase shifted as the frequency increases,
because the reduced reactance of the capacitances causes the voltage to lag behind the current.

Figure 3: Open loop gain against frequency.

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In order to achieved a constant gain with a better frequency response, negative feedback is used. Negative feedback
involves taking a fraction of the output signal and feeding it into the inverting input of the op-amp. Since the op-
amp has an exceedingly high gain, to get an output voltage to be within the supply rails, the difference in the input
voltages must essentially be zero.

In order for this to happen, the op-amp will try to make its inputs the same, and will adjust its output to ensure
this happens. This produces a closed-loop gain, which is lower than the open-loop gain, but with a greater
bandwidth, enabling a stable gain across a wider frequency range. [2]

Open-Loop Response

60dB Closed-Loop Response

Low Bandwidth
20dB Closed-Loop Response

High Bandwidth

Figure 4: Closed loop gain against frequency.

Effectively, the lower the gain of the amplifier, the larger the bandwidth, visualised in figure 4.

For audio applications, the reduced gain is still large enough, and can easily be increased through further
amplification stages.

The gain-bandwidth product is calculated by:

𝐺𝑎𝑖𝑛 𝐵𝑎𝑛𝑑𝑤𝑖𝑑𝑡ℎ = 𝐿𝑜𝑤 𝐹𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝐺𝑎𝑖𝑛 × 𝐵𝑎𝑛𝑑𝑤𝑖𝑑𝑡ℎ (6)

Where bandwidth is the -3dB Corner Frequency.

The value obtained should be constant, independent of the gain it is measured at, and gives a measure of the gain
that can be achieved at a certain frequency.

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The third part of the lab investigates the use of op-amps as power amplifiers, with the first circuit shown below in
figure 5.
+15V
+15V

VIN
NPN
Op-Amp
470Ω

-15V PNP RL

R2
-15V
R1

Figure 5: Power amplifier – feedback from op-amp output.

The op-amp, R1 and R2 are configured as a non-inverting amplifier, and provide voltage gain, while the NPN/PNP
transistors configured as back-to-back emitter followers provide power gain.

The NPN/ PNP transistors in a ‘push-pull’ configuration operate as a class B amplifier, where one transistor
amplifies the positive part of the input signal while the other amplifies the negative part. These two signals are
then combined to produce the complete waveform (figure 6). [3]

Positive
Output

Input Signal Combined

Current Output Signal
Current

Negative
Output

Figure 6: Signals in Push-Pull configuration.

However, the output signal is not a direct replica of the input signal, because the base-emitter voltage (VBE) causes
a voltage drop of 0.7V, causing the output signals amplitude to always be 0.7V less than the input. When the input
voltage falls below 0.7V, the transistor will shut off, and will only conduct again when the input falls to -0.7V.

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This causes ‘cross-over distortion’, visualised in figure 7. [4]

VPEAK

Input Signal

VPEAK – 0.7
Crossover Distortion
– flat spot at 0V cross
over point

Output Signal

Figure 7: Example of Crossover Distortion.

There is a flat spot at each crossover point, caused by the transistors switching in response to the input signal.

The second circuit in figure 8, has feedback from the output of the power stage.
+15V
+15V

NPN
VIN
Op-Amp
470Ω

PNP RL

-15V
-15V

R2

R1

Figure 8: Power amplifier – feedback from power stage output.

The feedback will cause the op-amps output to increase by 0.7V, eliminating the voltage drop associated with the
previous circuit.

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Method & Results

Lab 1 – Experiment 1:

Experiment 1 involved looking into the effects of measuring equipment on a circuits performance.

Set up the circuit in figure 9, where the capacitor value is to be determined.

Use a 20kHz square wave at 1Vpk-pk as the input signal.

Connect a standard BNC cable from the circuit to the oscilloscope, and measure the voltage and rise time.

Add a second BNC-BNC from the same point to a different oscilloscope channel, so that 2 BNC cables are measuring
simultaneously. Measure and record to new voltage and rise time.

10kΩ
BNC to Oscilloscope

20kHz, 1Vpk-pk Square Wave C

0V
Figure 9: RC Circuit.

Results below (figure 10):

Pk-Pk Voltage Across Capacitor (mV) Rise Time (µS)

1x BNC-BNC Cable (One Channel) 950 5.11
2x BNC-BNC Cable (Two Channel) 950 6.70
Figure 10: How BNC Cables affect Rise time.

The addition of a second BNC cable has no influence on the output voltage, however it does affect the rise time.
This is because the BNC cable has capacitance, which becomes part of the circuit as a capacitor in parallel with the
unknown capacitor in the circuit. The equivalent circuits for both scenarios are shown below in figure 11.

10kΩ 10kΩ

C CM C 2CM

One Channel CM = CBNC+COSCLLISCOPE Two Channel

Figure 11: Equivalent Circuit.

Calculate the unknown capacitances using the relationship between the rise-time and the time-constant from (4) via
simultaneous equations.

2.2 × 10𝑘𝛺 × (𝐶 + 𝐶𝑀 ) = 5.11𝜇𝑆

2.2 × 10𝑘𝛺 × (𝐶 + 2𝐶𝑀 ) = 6.70𝜇𝑆
𝐶 = 160𝑝𝐹
𝐶𝑀 = 72.3𝑝𝐹
Considering the size of the unmarked capacitor, it is clear that the combination of the oscilloscope and BNC
capacitance is quite significant. The value of C calculated turned out to be exact value of the capacitor.

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Lab 1 – Experiment 2:

Experiment 2 looks at how the resistance of measuring equipment affects a circuits performance.

Replace the 10kΩ resistor from experiment 1 with a 1MΩ resistor, and reduce input signal frequency to 200Hz.

Repeat the measurements for one and two BNC cables.

Results (figure 12):

Pk-Pk Voltage Across Capacitor (mV) Rise Time (µS)

1x BNC-BNC Cable (One Channel) 478 258.9
2x BNC-BNC Cable (Two Channel) 324 233.6
Figure 12: How BNC Cables affect Rise time (Oscilloscope Input Resistance).

Both the voltage and rise times are different, due to the oscilloscope input having a resistance (Ri) parallel with its
capacitance.

Equivalent circuits (figure 13).

1MΩ 1MΩ
C CM C 2CM
160pF Ri 160pF Ri/2
72.3pF 144.6pF

One Channel Two Channel

Figure 13: Equivalent Circuit with Input Resistance.

The 1MΩ and Ri act as a potential divider, hence Ri was calculated from the voltage measurements.

From the two-channel data:

𝑅𝑖⁄
0.324 = 2
𝑅
1𝑀𝛺 + 𝑖⁄2
Hence:

𝑅𝑖 = 959𝑘𝛺
Which is close to the actual labelled value of approximately 1MΩ.

Next, the rise time can be calculated from the values of R i and CM calculated to be compared with the measured
values.

The Thevenin equivalent circuit for both the one channel and two channel configurations are shown below in figure
14.

C + CM 1MΩ // Ri C + 2CM 1MΩ // (Ri/2)

Figure 14: Thevenin Equivalent Circuit.

The total time constant for each circuit can now be calculated, and the rise times can be calculated using (4).

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One-channel:
−1
1 1
2.2 × ( + ) × (160𝑝𝐹 + 72.3𝑝𝐹) = 250.14𝜇𝑆
1𝑀𝛺 0.959𝑀𝛺
Two-channel:
−1
1 1
2.2 × ( + ) × (160𝑝𝐹 + 2(72.3𝑝𝐹)) = 217.14𝜇𝑆
1𝑀𝛺 (0.959𝑀𝛺⁄ )
2
Observe that these values are very close to their measured counterparts.

Lab 2 – Experiment 1:

Experiment 1 investigates modelling op-amps as first order systems, and how feedback is used to control the
bandwidth.

Using (5), calculate values of R1 and R2 to get a gain of 100, and construct the circuit as previously seen in figure 2.

Using a 0.1Vpk-pk square wave at 1kHz as the input signal, measure the voltage and rise time of the output.

From this, the actual gain can be calculated as well as the time constant using (4).

Next, change the signal to a sinusoid, and starting at a low frequency, increase the frequency until an attenuation
of -3dB is observed, or 70.8% of the output voltage. Measure and record this frequency, as well as the phase shift
of the output signal.

Results shown in figure 15:

Voltage Gain 99.08

Rise Time 13.88 µS
Time Constant 6.309 µS
-3dB Frequency 24.8 kHz
Phase Shift -45.2°
Figure 15: Non-inverting amplifier Measurements (Gain =100).

Use (3) prove that there is a relationship between the time constant and corner frequency:

1
𝑓0 = = 25.23𝑘𝐻𝑧
2𝜋 × 6.309 × 10−6

Use (6) to calculated the gain-bandwidth product:

𝐺𝑎𝑖𝑛 𝐵𝑎𝑛𝑑𝑤𝑖𝑑𝑡ℎ = 99.08 × 24.8𝑘𝐻𝑧 = 2.457𝑀𝐻𝑧

Next, using an amplifier with a gain of 500, measure the gain and -3dB frequency.

Results shown in figure 16:

Voltage Gain 296.75

-3dB Frequency 17.02 kHz
Figure 16: Non-inverting amplifier Measurements (Gain =500).

The gain-bandwidth can now be calculated again.

𝐺𝑎𝑖𝑛 𝐵𝑎𝑛𝑑𝑤𝑖𝑑𝑡ℎ = 296.75 × 17.02𝑘𝐻𝑧 = 5.05𝑀𝐻𝑧

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Lab 3 – Experiment 1:

Construct the circuit as seen in figure 5, using (5) to calculate the values or R1 and R2 to give a voltage gain of 20.

Using a sinusoidal 1kHz input signal at a suitable amplitude, view the op-amp output and push-pull output voltages
signals together on the oscilloscope.

The waveform is shown in figure 17.

VA

VOUT

24.52Vpk-pk
25.92Vpk-pk

Figure 17: Voltage amp and power amplifier output (feedback from op amp).

The crossover distortion as mentioned in the theory can be clearly seen, and the output voltage is always trailing
0.7V less than the input voltage due to the base-emitter voltage.

Lab 3 - Experiment 2:

For experiment 2, change the feedback so that it is from the output of the push-pull stage, as shown in figure 8.

Repeat the procedure from experiment 1.

The new waveform is shown in figure 18.

VA

VOUT

4Vpk-pk

Figure 18: Voltage amp and power amplifier output (feedback from push-pull).

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 Benjamin Griffiths (160159871)

Discussion

In lab 1, when using standard coaxial BNC cables, neither of the results could be trusted because the introduction
of the second BNC cable changed the performance of the circuit despite there being no physical change. The
outcome was to realise how the 10:1 oscilloscope probe overcame these problems via capacitance compensation to
take accurate measurements.

In the lab 2 experiments, the measurements led to results that were exactly as expected. For example, using the
first-order relationships gave a corner frequency very close to the measured value, and the phase shift was exactly
as predicted.

However, from the theory, the gain-bandwidth product should have been the same for both amplifier gains, but a
human error during the experiment meant the input voltage was not low enough, hence the output was saturating.
This meant the gain was not 500, and therefore the corner frequency was not the desired value.

If the gain was 500, the -3dB frequency would be approximately 4.9kHz.

In lab 3 experiment 1, the shape of the output waveforms clearly shows the cross-over distortion as discussed in the
theory, where an unwanted 0.7V drop occurs at the output of the push-pulls stage.

In experiment 2, while the amplitude of the output was corrected to be 0.7V higher, it produced a defect in the
output signal of the op-amp. This could be caused due to the slew rate, which is a measure of how quickly the op
amp can react to changes in the input, and since the output suddenly turns off due to cross-over distortion, the
op-amps output overcompensating, producing those vertical lines in figure 18.

Neither of these circuits would be ideal for audio amplification since the output it still distorted compared to the
input. s

Conclusion

Overall, the experiment met its aims and gave me an understanding of the limitations that have to be overcome in
amplifier design. For example, the measurements in lab 2 led to results that were almost identical to their
theoretical counterparts, giving a very convincing display of how op-amps can be modelled as first-order systems.

Furthermore, lab 3 very clearly demonstrated how the effects of feedback can alter a circuits performance, and
displayed the unwanted signal defects that occur with some circuit designs. Due to time constraints, I wasn’t able
to investigate a power amplifier design that overcame the previous circuits defects, but I still gained a solid
understanding of why these defects occur.

References

[1] J O. Bird. “Operational Amplifier,” Electronical and Electrical Principles and Technology, 5th ed. Abingdon, Oxford,
UK: Routledge, 2014, pp. 303-304. [Online]. Available: https://www.dawsonera.com/readonline/9781315882871

[2] I. Poole, “Non-inverting amplifier circuit using an op-amp,” in Radio-Electronics, Unknown Date. [Online].
Available: http://www.radio-electronics.com/info/circuits/opamp_non_inverting/op_amp_non-inverting.php.
Accessed on: Oct. 25th, 2017.

[3] Unknown Author, “Class AB Amplifier,” in Electronic Tutorials, Unknown Date. [Online]. Available:
http://www.electronics-tutorials.ws/amplifier/class-ab-amplifier.html. Accessed on Oct. 26th, 2017.

[4] P. Horowitz and W. Hill, “Bipolar Transistors”, in The Art of Electronics, 3rd ed. New York, USA: Cambridge
University Press, 2015, pp. 107.

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