EE3300/EE5300 Electronics Applications

Week 5 Self-Study Notes

Motivation (circuit theory is an approximation)

The speed of light is approximately 30 cm per nanosecond. With that in mind, consider the circuit shown in Figure 1. In this circuit, the voltage source rises from 0 V to 10 V at a time . What is the current in the moment immediately after the 10 V rising edge?

Circuit theory says that you analyse the current by looking at the resistor at the far end of the circuit, then apply Ohm’s law to obtain However, this is only true if you wait long enough for the system to settle into a steady state. In the moment immediately after the power supply is turned on, the initial “surge current” flowing into the cable cannot depend on the load resistor because the size of that resistor is information that must be communicated through the circuit.

The “surge current” is determined by the properties of the transmission line itself. Specifically, the transmission line’s characteristic impedance will determine how much current initially flows. The characteristic impedance depends upon the geometry of the transmission line and the materials used to construct it. We will see below that the characteristic impedance also affects many other aspects of transmission line behaviour, such as whether reflections occur and how much power is delivered to the load.

Circuit theory is only valid when the time scales of interest are much longer than the time it takes for signals to travel through the circuit. Therefore, circuit theory will fail in situations where:

• The cables are very long, or
• The signals are changing very quickly (i.e. they are of very high frequency).

The behaviour of systems with long cables or high frequency signals is described by transmission line theory.

Transmission lines

A “transmission line” is any two conductors that carry a signal and its return path. Examples of transmission lines include:

• The two wires in Figure 1 that connect the source to the load.
• A printed circuit board trace and the ground plane beneath it.
• A length of coaxial cable (which has an inner conductor for the signal and an outer shield).

The term “transmission line” in the context of electronics usually means that high frequency effects are being considered. A pair of conductors that we intend to analyse using regular circuit theory (such as Kirchhoff’s laws) would not usually be called a transmission line.

Introducing the structure of a transmission line

Consider a length of coaxial cable as an example of a transmission line. Imagine dividing the cable into many short segments of length . Within each segment, there will be an electric field and a magnetic field. Consequently, each segment of the cable will behave electrically like a capacitor and an inductor (see Figure 2).

The combination of many inductive and capacitive elements leads to the equivalent circuit model shown in Figure 3. The inductors and capacitors are distributed along the length of the line. Note that has units of H/m and has units of F/m, which is necessary so that and have units of H and F respectively.

Intuitive description of transmission line behaviour

Before jumping into detailed mathematical analysis, let’s consider a few simple cases. Suppose that we have a transmission line with and . These are realistic numbers for a coaxial cable that has characteristic impedance To simplify our initial analysis, we’ll replace the differential elements with small segments of length 1 cm. This leads to the circuit model shown in Figure 4.

You might have expected that so many inductors and capacitors could trigger some messy resonance effect, but in fact the behaviour shown in Figure 5 is quite simple. The transmission line simply introduces a delay in the signal. You can imagine a wave travelling down the line with a certain propagation speed. The load doesn’t see any voltage until the wave reaches it.

Another insightful case is when the load is disconnected, as shown in Figure 6. In this simulation, the voltage source is a 1 V pulse (square wave) that turns on at with a rise time of .

The simulation result is shown in Figure 7. You can understand these results by thinking of a forward travelling wave originating at the source that is subsequently reflected at the open-circuit load.

We can notice several important details:

• The long time (steady-state) behaviour is as predicted by circuit theory.
• The rise in is delayed due to the propagation time. From this plot, you can see the time taken for the forward travelling wave to propagate down the line.
• When the forward travelling wave reaches the open circuit at the end, it is reflected back towards the source. It therefore takes two propagation times for the voltage at the source to reach its final value. In other words, the predictions of circuit theory take effect after a time delay of twice the propagation time.

The small oscillations visible in Figure 7 arise because the network of 16 inductors and 16 capacitors is not a perfect representation of a transmission line. We introduced an error when we chose for this simulation. For increased accuracy, we need to take the limit of and use differential equations in the analysis.

Transmission line theory

In general, a transmission line also incorporates some series and shunt resistances. These are modelled using a differential series resistance (units of Ω/m) and a shunt conductance (units of S/m). The circuit is shown in Figure 8.

The characteristic impedance of a lossy transmission line is given by

The telegrapher’s equations describe the behaviour of a transmission line. The equations in the AC phasor domain are

where is the AC voltage phasor at position along the line, and is the AC current phasor.

Equation says that voltage drops according to Ohm’s law as we move along the line based on the series impedance . Equation says that some current leaks away from the line based on the shunt admittance .

Differentiating Eq. with respect to and substituting Eq. into it gives the second-order differential equation

If your math skills are particularly sharp, then you will recognise that an equation like this will have two solutions of the form where

Physically, these correspond to a forward travelling wave and a reverse travelling wave. The general solution is a linear combination of the forward and reverse waves,

where and are the amplitudes of the forward and reverse travelling waves, respectively. The negative sign in Eq. arises because the reverse wave’s current is travelling in the opposite direction.

Lossless transmission lines

In high frequency electronics, it is common that and . In this case, we can neglect and and consider the transmission line to be lossless.

Taking , the characteristic impedance simplifies to

Notice that for a lossless line, is a real number.

Furthermore, the solution to the telegrapher’s equations becomes

A simple interpretation of these equations can be obtained by making a connection to AC circuit theory. Recall that AC circuit analysis is based on complex exponentials of the form . The physically meaningful quantity is the real part of , which is a sinusoid (with a phase shift if is complex). Often, we omit the factor, and just write . We call this the phasor representation of the voltage.

Considering only the forward travelling wave and writing the time dependence explicitly, we can reduce Eq. to

where is called the propagation constant. We can clearly see the time and spatial dependence of the voltage. The wave repeats in time with period

and in space with wavelength

Studying the electromagnetism of transmission lines (out of scope for this subject) reveals another important result:

where is the permittivity of the dielectric material between the conductors (relative to a vacuum), and c is the speed of light in a vacuum.

Substituting into Eq. gives an important equation for the wavelength of an electrical signal in a transmission line

Here is the speed of light, is the electrical frequency in Hz, and is the relative permittivity of the dielectric material used to make the transmission line.

We can also use these equations to determine the propagation speed of a signal, which we call the phase velocity. Given sinusoid of the form we imagine a point of constant phase (for instance, the peak of the sinusoid), and track that point of constant phase as it appears to move down the line. Hence, for , differentiating with respect to time gives , and hence

This important result states that signals propagate down a transmission line at the speed of light divided by a scaling factor that depends on the dielectric constant of the insulator that is used to manufacture the transmission line.

Reflections due to mismatched impedances

When a transmission line with characteristic impedance is connected to a load or another transmission line with a different characteristic impedance, the forward travelling wave will be partially reflected back towards the source.

Suppose that we inject a forward travelling wave (we no longer bother writing the always-present factor). Whenever this wave reaches a junction to a different impedance, the mismatch will generate a reflected wave , where the reflection coefficient is

Here is the load impedance (or the characteristic impedance of the next transmission line) and is the characteristic impedance of the transmission line.

The total voltage and current at any point along the line is the sum of the forward and reflected waves,

The reflection coefficient is a complex number, since the reflected wave can experience a phase shift. The magnitude of is limited to the range . A few important cases are as follows.

Impedance matching

We learned in the previous section that wave reflections are generated when there is a mismatch in impedance (e.g. when the characteristic impedance of the line does not match the impedance of the load).

If you’re not able to change the impedances, then an alternative is to implement a matching network that transforms the impedance as seen looking into the load (Figure 9).

To avoid dissipating power, the matching network should be made from reactive components (inductors and capacitors).

The L network

A simple matching network is shown in Figure 10. This network consists of two reactive elements and in something like a sidewise “L”.

The operation of the network can be understood as illustrated in the figure. The parallel combination has a smaller impedance than alone. A properly chosen can make the impedance small enough that the real part of the parallel combination matches the transmission line. However, it will leave some imaginary component, which is cancelled by .

Designing the matching network means choosing whether and are inductors or capacitors and what their values should be.

We will illustrate the analysis for the case of Figure 10 (a). Let’s begin by considering the input impedance looking into the matching network,

A match is achieved when the real and imaginary parts of both sides are equal. Taking real and imaginary parts produces two separate equations, from which we can solve for the two unknowns and .

Specifically, rationalising the denominator,

If we assume that and are real, then the real part of this equation gives

We can choose to be positive (an inductor) or negative (a capacitor).

Meanwhile, taking the imaginary part of the equation gives

The next step is to determine the capacitor or inductor values. The reactance of a capacitor is and the reactance of an inductor is . For a given frequency , we therefore obtain the component values.

Q factor of the matching network

A matching network will only achieve a good match at a specific frequency, as shown in Figure 11.

We characterise the bandwidth of the match by the Q factor, defined as

where is the centre frequency and is the bandwidth.

The T and Pi networks

Higher Q matches can be achieved using the T and Pi networks shown in Figure 12. The reactances can be capacitor or inductors in various combinations.

Design equations for these networks are given various references, for example, NXP Application Note AN267 and RF Electronics: Design and Simulation by C. K. Kikkert.

Noise

Noise is the random perturbation of voltages and currents which occurs in practical circuits. As an intuitive picture, consider the current flowing in a resistor, as shown in Figure 13. Even if the voltage is perfectly constant, there is still some noise. Each individual electron experiences thermal agitation and follows a somewhat random path. On average the current is given by Ohm’s law but at any given instant the signal may randomly be higher or lower due to thermal perturbations of electrons.

We must clarify the meaning of “randomness” when it comes to noise. Noise is random in the sense that its instantaneous value cannot be predicted and hence cannot be compensated for by subtracting it away. However, noise can be characterised by its average power level, defined as

where is the noise voltage. We use because power is proportional to voltage squared. The units of are , i.e. we have omitted the factor of that would convert to units of watts. This is because the most important quantity is the signal-to-noise ratio, defined as

A factor of would cancel out, and hence it is convenient to simply define noise power and signal power as voltage squared. Furthermore, it is convenient to define measurements in a way that is independent of the load impedance.

All circuits are affected by noise, but the effects are particularly significant when the signal amplitude is small, and hence the SNR is small.

In practical measurements, the averaging time (the value of ) merely needs to be large enough to capture several cycles of the lowest frequency component that is present. This can be achieved by choosing a value of , measuring the power, then increasing and making sure that the measured power is not changing.

Noise spectrum

It is common to analyse noise in the frequency domain. We define the power spectral density (PSD) of the noise via the conceptual arrangement shown in Figure 14.

Consider a bank of ideal bandpass filters each having a passband width of 1 Hz. These separate the signal into separate channels, and the power delivered by each channel is measured. The result from each power meter gives the amount of power in a 1 Hz bandwidth, or equivalently, power per Hz. In this case, “power” has units of . Therefore power spectral density has units of . We use the notation to refer to the power spectral density.

In practice, the PSD is calculated using digital signal processing techniques (the discrete Fourier transform). However, the calculation method can be considered as an implementation detail. The conceptual arrangement of Figure 14 is sufficient to understand the meaning of the PSD.

Johnson-Nyquist noise (thermal noise of resistors)

Resistors generate thermal noise with a completely flat PSD:

where is the power spectral density of the noise, is the Boltzmann constant, is absolute temperature, and is the resistance.

A flat PSD is called “white noise” because it contains all frequencies, in analogy with the colour white that is composed of all wavelengths. In practice, there is an upper limit of frequency; the noise cannot continue to infinitely high frequencies or else it would contain infinite power. Nevertheless, thermal noise can be considered as “white” across the practical measurement bandwidths of electronic circuits.

From Eq. (), we see that resistor thermal noise is proportional to temperature and also proportional to resistance. More generally, this effect applies to any circuit element that dissipates power. If the input impedance of any network is , then define and use Eq. () to evaluate the PSD of the noise.

As an alternative notation, we can also write the PSD as

where the bar indicates an average. This notation makes it explicit that we are considering a source whose mean-squared value is with units .

1/f noise (flicker noise)

Another type of noise is called flicker noise or noise because of the frequency dependence in its PSD:

Such a frequency dependence is called “pink noise.” It occurs in many electronic devices.

In the case of MOSFETs, the flicker noise is characterised by

where is a constant that depends upon the manufacturing process, is the width of the MOSFET, is the length of the MOSFET, and is the oxide capacitance per unit area.

Generally the flicker noise will be most significant at lower frequencies, but since it decays at higher frequencies, there is a crossover point where the thermal noise becomes the most significant. This crossover point is called the corner, as shown in Figure 15. The corner frequency is typically tens or hundreds of MHz in modern CMOS processes.

Noise voltages

By analogy with root-mean-square quantities, it is common to take the square root of power spectral density, i.e.

The units of the noise voltage is . The unit has no profound meaning, it simply arises because we have taken the RMS value of a power spectral density. The power spectral density (units ) can be converted to “volts” by taking the square root, which gives units of .

Such values are often given in datasheets. For example, the INA826 instrumentation amplifier lists its noise as 18 .

Incorporating noise in a circuit model

Consider a resistor , which can represent either a physical resistor or the input resistance of some downstream circuit stage. The effects of thermal noise can be modelled by the addition of a voltage source or current source as shown in Figure 16. Either a voltage or current source can be used; they are Thevenin and Norton equivalents. Sometimes one form leads to simpler analysis. The choice between them can be made based upon analytic convenience.

One should be careful in circuit analysis because the source amplitudes are written in terms of the power spectral density with units of or . For example, the Norton equivalent circuit transformation is given by (with factor instead of the normal ). The factor cancels with the in the numerator of the voltage spectral density.

More formally, if a power spectral density is passed through a network with transfer function then the result is a new PSD given by

where we have substituted . Notice that the transfer function magnitude is squared because has units of volts squared (per hertz).

Op-amp circuits at high frequencies

Appropriately chosen op-amps can be used up to hundreds of MHz. For example, the Texas Instruments LMH6609 has a bandwidth of 280 MHz (at and and costs approx. AUD $5 in small quantities).

Let’s consider the process to adapt an op-amp design to work at high frequencies.

A typical low frequency op-amp circuit is shown in Figure 17. This design uses dual power supplies so that the DC level of the signal can be 0 V. The capacitors and provide power supply decoupling (to minimise and ripple).

We notice some features of this type of op-amp circuit:

• The input and output are directly coupled (i.e. DC coupled). Therefore the DC level of the signal is meaningful.
• The input impedance is very large.
• The output impedance is low.

Now, let’s consider how to modify this design for high frequency applications. Generally in high frequency signals it is common to AC couple between stages, so the DC level has no significance. Hence, there is no need to use dual power supplies because the DC level can be arbitrarily chosen to be the midpoint of the supply rails.

Figure 18 shows a high frequency version of the non-inverting amplifier. Notice the following features:

• The input and output are AC coupled. Therefore the DC level of the signal has no meaning.
• The signal is level-shifted by and the two resistors to lie at the midpoint of the voltage supply. A weakness of this design is that power supply ripply is strongly coupled through . You will improve on this design in tutorial questions.
• The input impedance is determined by a termination resistor Therefore the amplifier stage can have an input impedance of , which is a standard value for RF electronics. In this design is DC coupled to permit the previous stage to sink a DC current if required by its particular design.
• The output impedance is primarily determined by (assuming ). This permits the amplifier to have an output impedance of .

The capacitor is very important because it AC couples the gain setting resistor . This means that the DC gain and AC gain are different. Notice that the DC gain must be 1 or else the bias point of the input would be amplified at the output and would saturate the op-amp. Adding means that the amplifier can have a high gain for the signal while maintaining unity gain at DC.