EE3300/EE5300 Electronics Applications
Week 3 Self-Study Notes

College of Science and Engineering, James Cook University

Last updated 4 March 2025

Motivation

Suppose that you’re building a battery powered device that will perform some signal processing with op-amps. Your op-amp designs need dual power supplies (e.g. and ) but you only want to use a single battery. Therefore, you propose to split the battery voltage in half and define the midpoint to be your circuit’s ground.

You construct the circuit shown in Figure 1. The voltage divider (R1 and R2) sets the voltage midpoint, which is stabilised by C3. To ensure a low-impedance virtual ground, you set up the op-amp buffer U1. The power rails are decoupled by capacitors C1 and C2.

Figure 1:
A “supply splitter” circuit in which an op-amp is used to create a virtual ground at the midpoint of a single power supply. This is an LTSpice schematic set up with a time-varying load that switches from open circuit (10⁹ Ω) to 900 Ω after 20 μs.
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However, the circuit performance is terrible! The simulated output voltage is shown in Figure 2.

Figure 2:
Simulation result from the circuit in Figure 1, generated by LTSpice and plotted in Matlab. The output voltage is AC coupled (i.e. the DC value of 4.5 V has been subtracted for clarity).
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Changing the load current by just 5 mA causes large oscillations in the ground voltage. The >100 mV swings in ground level will mess up the accuracy of your circuit. What happened? You did everything “right”: decoupling the power supplies, ensuring a stable reference voltage, and using an op-amp buffer so that the load current’s return path is isolated from the voltage divider R1-R2.

This is an example of a circuit with marginal stability. Our goal this week is to learn what stability means and what factors affect it. Ultimately the goal is to learn how to design circuits that avoid these kinds of problems.

When feedback goes wrong (an intuitive explanation)

Any circuit with feedback has the potential to oscillate. Let’s analyse the general block diagram of feedback and see how it can go wrong.

Figure 3:
**(a)** The basic negative feedback loop. **(b)** The feedback network redrawn with the subtraction operation shown explicitly. Signal examples show the normal case. **(c)** Negative feedback can become positive feedback if the feedback signal adds constructively. The signal examples show how the feedback signal can add to the input, rather than subtracting from it.
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The standard negative feedback loop in Figure 3 (a) can be redrawn to make the negative sign more explicit, as shown in Figure 3 (b).

Starting at the top-left, the signal is added to the feedback signal to produce the error signal . This is then multiplied by the powerful gain to produce the output . Notice that the feedback network as a whole operates in such a way that is a negative replica of .

However, consider what happens if the loop gain includes a phase shift. Figure 3 (c) shows the critical situation where the feedback signal is perfectly in phase with the input signal . In this case, the error signal is large. The amplifier boosts the error signal, which in turn is fed back around and causes to grow. Again the amplifier will boost the signal, and again a larger signal will appear at the input. We see that there is a “run-away” situation where the signal will grow larger and larger until the amplifier saturates.

Therefore, we must learn how to design electronics to avoid this kind of unstable feedback. In general, whenever you design a feedback loop, you must consider whether it can exhibit oscillatory behaviour. You must check for stability to make sure that your circuits won’t enter this unstable, run-away state.

The Barkhausen stability criterion

Recall that the transfer function of a closed-loop feedback system is

where is transfer function of the open-loop system and is the transfer function of the feedback network. Recall that is called the loop gain.

What happens if ? In this case, the denominator of the transfer function vanishes and we have a division by zero. The output of the system would be predicted to grow without bound.

We are therefore interested to know if there is a frequency at which the loop gain . Expressing the negative sign as a phase shift would corresponding to an angle of -180°. The Barkhausen Stability Criterion states that a feedback system will oscillate if there exists a frequency at which both conditions are true:

The Barkhausen criteria is a sufficient but not necessary condition for instability, i.e. if the system meets these criteria, then it is definitely unstable, but there are unstable systems that do not meet these criteria.

An important case arises when the loop gain is greater than 1 at the critical phase shift of -180°. If the system satisfies the minimum phase criteria (which electronic circuits generally do), then such a situation is unstable. The minimum phase criteria is that all poles and zeros of the system are in the left half of the complex plane. For non-minimum phase systems, you need more sophisticated methods such as the Nyquist criterion to analyse stability.

The Barkhausen criteria for minimum phase systems is as follows. The circuit is unstable if there exists a frequency where the loop gain satisfies both equations:

Show non-minimum phase system examples

Equations and only apply to minimum phase systems (those with all poles and zeros in the left half of the complex plane). In general, typical electronic circuits are minimum phase systems. However, it’s important to keep in mind that there are limits to the validity of these equations. Some examples where they fail are shown below.

Example 1

Suppose you have a system with an open-loop gain of 2 and phase shift of -180° at all frequencies (i.e. ). Connect this system in unity feedback (). Then the loop gain is , and the closed loop transfer function is

This system is stable, despite the fact that there is a frequency at which the loop gain is greater than 1 at -180° phase shift.

Example 2

The system with open-loop gain

is unity-gain () stable, despite the fact that there is a frequency at which the loop gain is greater than 1 at -180° phase shift.

Try it! Here’s some Matlab code to experiment with.

s = tf('s'); % Define the Laplace variable
A = 100*(s+10)^2/s^3;
B = 1;
figure();
bodeplot(A*B);
grid on;
figure();
step(A/(1+A*B));

This is not a minimum phase system because of the poles at the origin.

How this relates to electronics

Recall that last week we learned how to analyse the loop gain of a feedback circuit. It turns out that stability will be controlled by the frequencies of the poles and zeros of the loop gain. These in turn are determined by the values of the resistors, capacitors, and inductors in the circuit. Hence, the general principle will be to analyse the circuit to find the loop gain, produce the Bode plot, and assess stability using the Barkhausen criterion.

Gain and phase margins

The gain margin and phase margin are criteria that measure how close a system comes to the edge of stability. These measures can be thought of as “safety margins.”

The gain and phase margins can be evaluated from Bode plots of the loop gain, as shown in Figure 4.

To find the gain margin, first locate the frequency where the phase shift is -180°. The system is stable if the gain at this frequency is less than 1. The gain margin is the amount by which the gain could rise before there is instability.

To find the phase margin, first locate the frequency where the gain is 0 dB. The system is stable if the phase shift at this frequency is smaller than -180° (i.e. closer to zero). The phase margin is the amount by which the phase could fall before there is instability.

Figure 4:
Bode plots of loop gains for **(a)** a stable and **(b)** unstable system. The stable system can be recognised because the gain has fallen below 1 (= 0 dB) before the phase has reached -180°.
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We also define the gain crossover frequency as the frequency at which the gain is 0 dB, and the phase crossover frequency as the frequency at which the phase is -180°. A stable amplifier is one where , i.e. the gain falls below unity before the phase reaches -180°.

It is important to emphasise that these plots show the loop gain (), not the closed loop transfer functions. We use the loop gain to assess stability.

Figure 4 (a) shows a stable amplifier. The critical phase of -180° occurs when the gain has already fallen below 0 dB.

In contrast, Figure 4 (b) shows the loop gain of an unstable amplifier. The phase crosses -180° when the gain is still very high. Consequently, the system will oscillate.

Why unity-gain is the most problematic

Feedback is more of a problem when the loop gain is increased. Consider that increasing the loop gain will shift the magnitude plot upwards. If it shifted far enough, then the the 0 dB point will coincide with -180° and the system will be unstable.

In many practical situations, the open-loop transfer function is that of a high-gain circuit such as an op-amp. The transfer function of the op-amp is fixed. As a circuit designer, you can only change the feedback network . Stability is determined by the loop gain , i.e. some transfer function that you cannot change () multiplied by the feedback network that you design ().

Let’s revisit the closed-loop transfer function:

In words, the closed-loop gain is approximately . However, stability becomes a problem when the loop gain becomes too large. This has an important implication for the design of op-amp circuits.

Evaluating stability

The example below shows how stability can be analysed from first principles.

Example 1

Consider the three-stage circuit shown in Figure 5. Assume each transistor is biased identically. Component values are , and . Determine the loop gain and sketch the Bode plot.

Is this amplifier stable? What happens if the gain is reduced by a factor of 50 or a factor of 150?

Figure 5:
Simplified three-stage amplifier circuit. The capacitor is presumably an unwanted parasitic capacitance.
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Solution

Recognise that the amplifier consists of three repeated common-source amplifiers. Each stage has a voltage gain of . Neglecting channel-length modulation and neglecting other capacitances, each stage has a pole at the output node of , which is . Therefore we can write down the open loop transfer function

This system uses voltage-controlled voltage feedback (series-shunt feedback). Therefore, we find the loop gain by injecting a test voltage as shown in green. We have

and hence the loop gain is

To produce a sketch, we need to know the loop gain at DC.

Converting to dB, .

The loop gain has three coincident poles at . This means the phase starts at 0 and falls to at high frequencies.

A sketch of the Bode plot is shown in Figure 6. From this sketch we can see that the amplifier will be unstable (because the 180° phase occurs when the gain is well above 0 dB).

Figure 6:
Rough sketch of the Bode plot of the loop gain for the three-stage system.
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In the next section, we explore what happens to this circuit when the gain is reduced.

The implications of the phase margin

The phase margin shows how much the amplifier tends to overshoot when facing a step change in input. In the language of control systems, a system with a low phase margin is underdamped.

Considering the three-stage amplifier circuit from Example 1, suppose that we adjust the feedback factor . This could be done by introducing a voltage divider so that the signal fed back is reduced.

In each case, we’ll achieve a different phase margin and consequently different settling behaviour. The results are shown in Figure 7.

Figure 7:
Drag the slider below to adjust the feedback factor . The figure shows the Bode plot of the loop gain (top) and the time-domain step response (bottom). The step response illustrates an inverting amplifier response to a unit step input (e.g. 1 mV input, with the amplitude being relative to this input).
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= 0.005

The analysis plotted in Figure 7 is based on the transfer function that was derived in Example 1. It’s plotted using the Matlab bodeplot and step functions. Of course, the transfer function does involve some simplification, e.g. the bias network and coupling capacitors are not included and the summing network is idealised. Hence it is useful to carry out a detailed simulation of the circuit.

A Micro-cap circuit model is shown in Figure 8. The transistors are generic_N devices with chosen to give the correct for the applied .

Figure 8:
Circuit model for the three-stage amplifier.
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The step response (where pulses high and then low) is shown in Figure 9.

Figure 9:
Simulated step response of the three-stage amplifier with .
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Notice the similarity to the step response in Figure 7 when you select . The differences arise because the real circuit is more complicated than the simplified structure that was used to derive the analytic transfer function. We see that the simple approach gives insight into the design trade-offs before turning to “black box” simulation methods.

Typical phase behaviour of op-amps

So far, we have learned that the loop gain must drop below 0 dB before the phase reaches -180°. However, this creates a tension for op-amp designers. Op-amps should have a very high open-loop gain but also they need to be able to handle many different circuit configurations, which may or may not introduce additional phase shift. How can we ensure that the op-amp is stable regardless of the circuit that it is placed in?

To address this challenge, general purpose op-amps use a technique called “compensation.” The op-amp designer deliberately introduces a pole at low frequencies so that the gain begins to roll off early. The intent is to sacrifice gain in order to ensure that the overall loop gain will reach 0 dB early enough to prevent oscillations.

Compensation is usually implemented by placing a capacitor on the signal path inside the op-amp. (Some op-amps have pins to allow external compensation capacitors too.) In integrated circuits, a large capacitance would take up an excessive amount of space, so the capacitor is usually placed across one of the gain stages so that it can be Miller multiplied. This is known as “Miller compensation.”

An indicative schematic of a Miller-compensated op-amp is shown in AoE Figure 4.100 (p. 283). Notice the compensation capacitor, internal to the op-amp. You might find it helpful to read the surrounding text to understand the situations where you might choose a “decompensated” op-amp.

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The open-loop gain of a typical compensated op-amp is shown in Figure 10. There is one pole at low frequencies (e.g. perhaps 10 - 100 Hz). The op-amp designer ensures that higher frequency poles and zeros are pushed above the gain-bandwidth product so that they don’t affect stability.

A consequence of this low-frequency pole is that the op-amp has an open-loop phase shift across most of its operating bandwidth. In normal situations, the phase shift is corrected by the feedback mechanism, so you don’t see this phase shift in normal closed-loop operation. However, remember that stability is determined by the loop gain . You only need to get to -180° for oscillations to occur. The op-amp’s open-loop gain has already taken you halfway there!

Figure 10:
Typical open-loop gain () for a general-purpose op-amp. The phase margin (PM) due to the op-amp itself is approximately 90°.
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Self-check quiz 2

AoE Figure 4.47 (p. 247) shows the Bode plot of the open-loop gain of the LF411 op-amp (which is one that we use in our practicals).

Assume that the LF411 is used as an inverting amplifier, i.e. the feedback network is resistive. Which of the following statements are true?

(a)

The loop gain has a pole at low frequencies (between 10 Hz and 100 Hz).

(b)

The loop gain has a zero at low frequencies (between 10 Hz and 100 Hz).

(c)

Across the typical operating bandwidth of many circuits (e.g. 100 Hz - 1 MHz), the loop gain has approximately zero phase shift.

(d)

Across the typical operating bandwidth of many circuits (e.g. 100 Hz - 1 MHz), the loop gain has approximately -90° phase shift.

(e)

Across the typical operating bandwidth of many circuits (e.g. 100 Hz - 1 MHz), the loop gain has approximately -180° phase shift.

✔️ Correct!

The 90° lagging phase shift across the operating bandwidth has very important consequences for stability. We only need 180° for oscillations. Just the op-amp by itself has taken us halfway there, and we haven’t even considered capacitances in the load yet, let alone any kind of reactance in the feedback network.

The rate of closure should be -20 dB/decade for stability

The rate of closure is the slope of the magnitude plot when it crosses the 0 dB axis. It provides a shortcut method to investigate the phase margin.

The rate of closure tells you how many poles occur at lower frequencies. The number of poles tells you how much phase shift has occurred. Therefore, the rate of closure is an indicator of stability.

Show why this is an approximation.

This is an approximation because the phase shift doesn’t happen all-at-once at the frequency of the pole or zero. The phase shift is spread out over approximately 2 decades on either side of the pole or zero. Therefore the rate of closure is most informative when the poles and zeros are well separated from each other and from the crossover frequency.

The various cases are illustrated in Table 1.

**Table 1:** The rate of closure is a shorthand method to determine whether a system will have a suitable phase margin.
Number of poles before the crossover frequency	Rate of closure (slope at 0 dB crossover)	Phase shift at 0 dB crossover	Phase margin
1	-20 dB/decade	-90°	Good
2	-40 dB/decade	-180°	Marginal (will show ringing)
3	-60 dB/decade	-270°	Unstable (will oscillate uncontrollably)

The effect of the rate of closure is illustrated in Figure 11. The figure shows the loop gain and step response for a system with one low-frequency pole (as is typical of an op-amp) plus a adjustable pole and zero introduced by the feedback network.

The controls allow you to adjust the frequency of a pole and zero that would be introduced by the design of the feedback network. By adjusting the controls, you should be able to notice:

If you push the feedback network’s poles and zeros far to the right, they have negligible impact and the response is dominated by the op-amp’s own pole.
If the rate of closure is too high, then the step response is oscillatory.
The step response is very slow when there is a low-frequency zero.
When the pole and zero are close together and not too far from the crossover frequency, the system has a fast response and minimal overshoot or ringing.

Figure 11:
Bode plot and step response for systems with various pole and zero frequencies. The first pole represents a the dominant pole of a typical compensated op-amp and is fixed at 10 Hz. The second pole and zero are adjustable.
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= 1 kHz

= 1 kHz

Design strategies to improve the phase margin

Reading task

The supply splitter circuit that we introduced at the top of this page comes from AoE Figure 4.73 (p. 263). The book includes measurements of the oscillations (complementing the simulation results that we plotted in Figure 2).

Read the discussion about this circuit in the book.
Read section 4.6.2 Capacitive loads (p. 264) and examine the suggested design remedies.

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Adding a series resistor

The reading above introduced the idea of adding a series resistor to the output of an op-amp (e.g. AoE Figure 4.78 A). The impact of this resistor may seem slightly mysterious. It’s worth examining it in more detail.

Consider the simple case of a voltage follower as shown in Figure 12 (a). Notice the position of the feedback path.

Figure 12:
**(a)** A voltage follower driving a capacitance load with included series resistance to improve phase margin. The inside of the op-amp is represented by its Thévenin equivalent circuit. **(b)** Test circuit for calculating the loop gain.
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To determine the loop gain, we cut the circuit at a convenient location (such as the feedback connection). The circuit uses voltage-controlled voltage feedback, so we introduce a test voltage as shown in Figure 12 (b).

Analysing the circuit,

where is the inherent open-loop gain of the op-amp.

Using voltage division,

Finally, the loop gain is

Notice that this transfer function has a pole at and a zero at , in addition to any poles and zeros from the op-amp itself. The impact is sketched in Figure 13.

Figure 13:
Bode plot of the loop gain showing the effect of the series resistor. Thanks to the zero, the rate-of-closure is reduced to -20 dB/decade, and the phase margin is improved. If the zero were missing, then the rate-of-closure would be -40 dB/decade (as shown by the red dashed line).
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The Bode plot allows you to determine the required size of the series resistor. Using specific numbers for the op-amp’s DC gain and dominant pole location, you can place the zero at a suitable frequency so that the rate of closure is improved to -20 dB/decade before the loop gain drops to 1.

Split feedback

The problem with the simple series resistor is that the resistor will affect the output voltage. The feedback network isn’t monitoring the DC output voltage.

A solution is to introduce a split feedback network in which the DC feedback comes from the actual output voltage while high frequency feedback comes from the op-amp’s output. This can be achieved using a capacitor as shown in AoE Figure 4.78 B (p. 264).

We will investigate this design in the tutorial.

Oscillators

An oscillator is a circuit designed to produce a periodic waveform, for example, a sine wave or a square wave. Often the design goal is to produce a circuit that oscillates at a single specific frequency. Other times, the requirement may be to have an adjustable frequency. The output should be free from distortions and track the desired shape, e.g. a sine wave or a square wave.

Relaxation oscillators

A relaxation oscillator uses the timing of a capacitor charging and discharging through a resistor .

A simple example of a relaxation oscillator using a comparator is shown in AoE Figure 7.1 (p. 426). Spend some time studying this circuit. Make sure you understand:

Which resistor and capacitor determine the frequency of oscillation?
How the reference voltage adjusts depending on whether the output voltage is high or low.
Why you would prefer to use a comparator instead of an op-amp for this circuit? (Hint: comparators don’t typically have internal compensation capacitors; so think about what this means in terms of circuit behaviour).
What is the mechanism behind the “speed up” capacitors mentioned in the book? How do they help?

Hint: if any of these questions are not clear to you, then you should ask the questions in the workshop classes.

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The 555 timer

The 555 timer is a famous integrated circuit that serves as a versatile oscillator.

Some typical 555 circuits are described in AoE section 7.1.3 (p. 428). Skim through this section. It’s helpful to know about these circuits so that they come to your mind if you need to a simple oscillator.

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Phase shift oscillators

A phase shift oscillator uses an network to produce 180° of phase shift.

A simple example is shown in AoE Figure 7.23 (p. 439). Notice the repeated RC stages that create such a phase shift as to deliberately violate the Barkhausen criterion.

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LC resonant circuits

For high frequency () oscillators, the most common designs use (inductor-capacitor) resonant circuits, also called LC tank circuits. Recall from circuit theory that the resonant frequency of an circuit is

Therefore, if we choose the inductor and capacitor values appropriately, we can select the desired frequency of oscillation.

Colpitts oscillator

A Colpitts oscillator uses an resonant circuit with the capacitor split into two to act as a voltage divider. There are many possible topologies, but an example is shown in Figure 14. This particular realisation uses a common base amplifier. The series capacitor network acts as a voltage divider to ensure that the transistor sees only a small input signal.

The bias network is not shown. For normal discrete implementations the current source would be replaced by a resistor, and would be generated appropriately (e.g. a voltage divider or current mirror).

The oscillation frequency is close to the resonant frequency. Let be the total capacitance, i.e. the series combination of the two capacitors:

Then the resonant frequency is

The approximation arises because of factors such as the input impedance seen at the emitter of and the input impedance of the bias network .

Treating this circuit as a common-base amplifier, the output can be taken from the collector of . However, the network acts as a filter, so a cleaner sinusoid arises if the output is actually the midpoint of the two capacitors. To produce a high quality sine wave, include a buffer (e.g. source follower) and further filtering to remove frequencies above .

Summary

Here are some key ideas from this week:

The downside of feedback is that it can cause oscillations. We can use the gain and phase margin as criteria to evaluate how close a circuit is to the edge of stability.
Stability is controlled by the loop gain where is the open-loop transfer function and is the feedback network’s transfer function. This means unity gain feedback () is the most problematic for stability.
Most op-amps have built-in compensation capacitors that introduce a low-frequency pole, implying that the open-loop gain has a phase shift of -90° across most of the operating bandwidth.
If the feedback network introduces additional lagging phase shift within the bandwidth of the op-amp, then the result may oscillate.
An oscillator is a circuit deliberately designed to produce periodic waveforms.

Motivation

When feedback goes wrong (an intuitive explanation)

The Barkhausen stability criterion

Example 1

Example 2

How this relates to electronics

Gain and phase margins

Why unity-gain is the most problematic

Self-check quiz 1

Evaluating stability

Example 1

Solution

The implications of the phase margin

Typical phase behaviour of op-amps

Self-check quiz 2

The rate of closure should be -20 dB/decade for stability

Design strategies to improve the phase margin

Reading task

Adding a series resistor

Split feedback

Oscillators

Relaxation oscillators

The 555 timer

Phase shift oscillators

LC resonant circuits

Colpitts oscillator

Summary