Negative Resistance

RF/Photonics Lab
November 2019
Jared Alves

Negative Resistance

Arguably the most fundamental equation in electrical engineering is Ohm’s Law (V = I*R) which states that voltage is proportional to the product of current and resistance. From this equation, it is apparent that increasing a voltage across an element will increase the current through that element assuming the resistance is fixed. With a resistor, electrical energy is dissipated in the form of thermal energy (heat) due to the voltage drop between the terminals of the device. This is in direct contrast to the concept of negative resistance, which causes electrical power to be produced instead of dissipated.

Generally, negative resistance refers to negative differential resistance, as negative static resistance is not typically used.  Static resistance is the standard V/I ratio while differential resistance takes the derivative dV/dI. The following image shows an I-V curve with several slopes. The inverse of B yields a static resistance, and the inverse of line C is differential resistance (both evaluated at the point A). If the differential curve has a negative slope, this indicates negative differential resistance.

negres1

Even when differential resistance is negative, static resistance remains positive. This is because only the AC component of the current flows in the reverse direction. A device would consume DC power but dissipate AC power. This is because the current decreases as the voltage increases, leading to

negres2

A tunnel diode is a semiconductor device that exhibits negative resistance due to a quantum mechanical effect called “tunneling”.

Doppler Effect

RF/Photonics Lab
November 2019
Michael Benker

 

Doppler Effect

                The Doppler Effect is an important principle in communications, optics, RADAR systems and other systems that deal with the propagation of signals through space. The Doppler Effect can be summarized as the resultant change to a signal’s propagation due to movement either by the source or receiving end of the signal. As the distance between two objects changes, so does the frequency. If, for instance, a signal is being propagated towards an object that is moving towards the source, the returning signal will be of a higher frequency.

260px-Dopplerfrequenz

The Doppler Effect is also applied to rotation of an object in optics and RADAR backscatter scenarios. A rotating target of a radar or optical system will return a set of frequencies which reflect the distances of each point on the target. If one side of the target is moving closer while the other side is moving away, there will be both a higher and lower frequency component to the return signal.

Dopplereffectsourcemovingrightatmach0.7

034/100 Loaded Q and External Q

100 ADS Design Examples Based on the Textbook: RF and Microwave Circuit Design
November 2019
Michael Benker

Example 4.2-3: Analyze the parallel resonator that is attached to a 50 Ohm source and load as shown.

034.0

This problem is specifically asking to define the Q factor related to this circuit. The Q factor is a ratio of energy stored (by an inductor or capacitor) to the power dissipated in a resistor. The Q factor varies with frequency since the effect of a capacitor or inductor also vary with frequency. For a series resonant circuit, the “unloaded” Q factor is defined by the following function: Qu = X / R = 1/(wRC) = wL/R

The unloaded Q factor of a parallel resonant circuit: Qu = R / X = R/(wL) = wRC

Overall, the Q factor is a measure of loss in the resonant circuit. A higher Q corresponds to lower loss, while a lower Q indicated higher loss. An “unloaded” Q factor means that the resonator is not connected to a source or load. The above circuit can no longer apply the “unloaded” Q factor formulas due to the presence of a source and a load. There are two further Q factor formulas that need to be considered: loaded Q factor and external Q factor. The loaded Q factor includes the source resistance and load resistance with the resistance of the circuit. The external Q factor refers to only the source resistance and load resistance together.

For the above circuit, the loaded Q factor for the parallel resonator is defined as:

Loaded Q = (Rs + R + Rl)/(wL) = (Source resistance + R + load resistance) / (wL)

The external Q factor for the source resistance and load resistance is:

External Q = (Rs + Rl)/(wL) = (Source resistance + load resistance)/(wL)

The relationship between the different types of Q factors are:

1/(Loaded Q) = 1/(External Q) + 1/(Unloaded Q)

033/100 Parallel Resonant Circuits

100 ADS Design Examples Based on the Textbook: RF and Microwave Circuit Design
November 2019
Michael Benker

Example 4.2-2: Analyze a rearrangement of the RLC components into a parallel configuration.

As observable by the following figures, the resonant frequency and impedance value remain the same for the parallel RLC circuit. What may be understood by this is that the reactance of the inductor cancels out the reactance of the capacitor at this frequency of 505 MHz.

The input admittance of a parallel resonant circuit is: Y = (1/R) + jwC + (1/jwL).

The angular frequency, w = 2*pi*f = 1 / (sqrt(L*C)).

033.1033.2

032/100 Series Resonant Circuits

100 ADS Design Examples Based on the Textbook: RF and Microwave Circuit Design
November 2019
Michael Benker

Example 4.2-1: Analyze a one port series RLC circuit with R = 10 Ohms, L = 10 nH and C = 10 pF.

According to the following results, the input impedance at resonance is 10 Ohms, which is the value of the resistor.

The input impedance of an RLC series circuit is modeled by the following formula, a rather basic expression: Z = R + jwL + 1/(jwC)

The power delivered to the resonator is: P = |I|^2 * Z / 2.

032.1032.2

Skin Effect

RF/Photonics Lab
Jared Alves
November 2019

Skin Effect

                The skin effect is an important characteristic of alternating current within conductors. With direct current, charges are distributed evenly when flowing through a conductor. However, due to the Skin Effect as the frequency of the conduction current is increased, the charges distribute in greater quantities towards the surface of the conductor. In other words, the current density (J) decreases with greater depth in the conductor.   As shown, the current density is per area.

skin1

The skin depth of the conductor is the length from the surface of the conductor inward in which the majority of the charge is contained at frequencies higher than DC.

skin2

As shown in the equation above, skin depth is inversely proportional to frequency so at higher frequency values, the effective resistance of the conductor increases which reduces the cross-sectional area, as shown below.

skin3

The figure demonstrates that the conductor becomes more “hollow” at higher frequencies as the electric charges avoids traveling through the center. This is because the back EMF is strongest towards the center of the conductor. Maxwell’s equations explain that magnetic field strength is proportional to current and therefore as current intensity changes, so does magnetic field strength. The changing magnetic field creates an electric field opposing this change in intensity which causes the counter EMF effect. This creates an almost “Faraday cage” effect with the electrons at the center of conductor as the electric field cannot penetrate as deep into the conductor with increasing frequency.

skin4

The skin depth is technically defined as the length from the surface to the inside of a conductor in which J (current density) decays to 1/e of Js (current density at the surface). The imaginary part of the above equation shows that for each skin depth of penetration, the current density phase is delayed by 1 radian.

Interferometry – Introduction

RF/Photonics Lab
Jared Alves
November 2019

Interferometry – Introduction

                Interferometry is a family of techniques in which waves are superimposed for measurement purposes. These waves tend to be radio, sound or optical waves. Various measurements can be obtained using interferometry that portray characteristics of the medium through which the waves propagate or properties of the waves themselves. In terms of optics, two light beams can be split to create an interference pattern when the waves combine (superimpose). This superposition can lead to a diminished wave, an increased wave or a wave completely reduced in amplitude. In an easily realizable physical sense, tossing a stone into a pond creates concentric waves that radiate away from where the stone was tossed. If two stones are thrown near each other, their waves would interfere with each other creating the same effect described previously. Constructive interference is the superposition of waves that results in a larger amplitude whereas destructive interference diminishes the resultant amplitude. Normally, the interference is either partially constructive or partially destructive, unless the waves are perfectly out of phase. The following image displays total constructive and destructive interference.

interferrometry1

A simple way to explain the operation of an interferometer is that it converts a phase difference to an intensity. When two waves of the same frequency are added together, the result depends only on the phase difference between them, as explained previously.

interferrometry2The image above shows a Michelson interferometer which uses two beams of light to measure small displacements, refractive index changes and surface irregularities.  The beams are split using a mirror that is not completely reflective and angled so that one beam is reflected, and one is not. The two beams travel in separate paths which combine to produce interference. Whether the waves combine destructively or constructively depends on distancing between the mirrors. Because the device shows the difference in path lengths, it is a differential device. Generally, one leg length is kept constant for control purposes.

023/100 Distributed Bias Feed Design

100 ADS Design Examples Based on the Textbook: RF and Microwave Circuit Design
November 2019
Michael Benker

Example 2.11-3: Calculate the physical line length of the λ/4 sections of 80 Ω and 20 Ω microstrip lines at a frequency of 2 GHz. Create a schematic of a distributed bias feed network.

A high impedance microstrip line of λ/4 can be used to replace the lumped inductor from problem 022/100 Example 2.11-2E. Likewise a quarter wave impedance line of a low impedance can replace the lumped shunt capacitor. The 80 Ohm and 20 Ohm transmission lines can be made using LineCalc at 2 GHz. The taper, tee and end-effect element are used to simulate the circuit most correctly and to remove discontinuities between the models.

023.1023.2

The return loss null occurs at 1.84 GHz, indicating that the system could be optimized better to adjust center frequency. The high impedance line length is now adjusted to center the frequency to 2 GHz:

023.3023.4

 

 

022/100 Microstrip Bias Feed Networks

100 ADS Design Examples Based on the Textbook: RF and Microwave Circuit Design
November 2019
Michael Benker

Example 2.11-2E: Design a lumped element biased feed network.

Bias feed networks are an important application of high impedance and low impedance microstrip transmission lines. The voltage bias may be needed for a device that is connected to the microstrip line, such as a transistor, MMIC amplifier or diode. The inductor in the circuit below is used as an “RF Choke”, which is used in tandem with a shunt or bypass capacitor for a “bias decoupling network.” Lumped elements are typically used for frequencies below 200 MHz.

The following is a typical bias feed network, followed by a simulation:

022.1022.2

 

021/100 Distributed Inductance and Capacitance

100 ADS Design Examples Based on the Textbook: RF and Microwave Circuit Design
November 2019
Michael Benker

Example 2.11-2D: Convert the lumped element capacitors and inductors to distributed elements.

This is the schematic that needs to be changed into distributed element microstrip lines:

021.1

The following formulas are needed to calculate the inductive and capacitive line lengths to simulate this schematic using microstrip lines.

Inductive line length: (frequency)*(wavelength)*(Inductance)/(impedance of line)

Capacitive line length: (frequency)*(wavelength)*(Capacitance)*(impedance of line)

In order to know what at which frequency the inductance or capacitance are calculated, let’s run the simulation of the above circuit:

021.2

This circuit is centered at 10 GHz, since the circuit behaves as a terminated open-circuited transmission line with an open-parallel resonance at 180 degrees, or twice the length of a quarter wave line.

The above circuit is then modeled as follows:

021.3

021.4

020/100 Open-Circuited Transmission Line with Termination

100 ADS Design Examples Based on the Textbook: RF and Microwave Circuit Design
November 2019
Michael Benker

Example 2.11-2C: Calculate the input impedance of a quarter wave open-circuited microstrip transmission line using termination with end effects.

An open circuit microstrip line generates a capacitive end effect due to radiation. This radiation is observable in the results from the following simulation. Note that the impedance at 180 degrees is more capacitive than was the open circuit transmission line with out any termination.

020.1020.2019.3

019/100 Open-Circuit Transmission Line

100 ADS Design Examples Based on the Textbook: RF and Microwave Circuit Design
November 2019
Michael Benker

Example 2.11-2B: Calculate the input impedance of a quarter wave open-circuited microstrip transmission line for a given length of time.

The reactance of a lossless open circuit transmission line can be modeled as being equal to the characteristic impedance multiplied by the cotangent of the electrical length of the transmission line in degrees.

X = Z * cot(Θ)

To construct this circuit, a termination of 1 MOhms is used to simulate an open circuit. As the electrical length in degrees varies with frequency (the wavelength), a static electrical length of a transmission line varied over many frequencies will suffice to demonstrate the reactance of a varying electrical length transmission line. The following circuit was created with a transmission line optimized for 10 GHz, similar to the Short-circuited Transmission Line:

019.1019.2

The results above are consistent with the theoretical model of an open circuit transmission line being modeled by a cotangent relationship. At the optimized frequency (at which the transmission line length is quarter-wave) it can be observed that the impedance of the line is measured to be zero. At a half-wave length and other multiples of a half wave length, the transmission line generates high levels of resonance.

019.3

018/100 Short-circuited Transmission Line

100 ADS Design Examples Based on the Textbook: RF and Microwave Circuit Design
November 2019
Michael Benker

Example 2.11-2A: Calculate the input impedance of a short-circuited microstrip transmission line for a given electrical length of the line.

This circuit was built with a quarter-wave microstrip synthesized for 10 GHz with given substrate (conductivity of gold) using the LineCalc tool.

018.1018.2

The following results conclude that a short-circuited quarter-wave transmission line has high impedance, similar to an open circuit. A short circuited transmission line that is not a quarter-wave transmission line will not have high impedance as demonstrated by frequencies far outside of the range of optimization (10 GHz). This phenomena is is consistent with electromagnetic theory on transmission lines.

018.3018.4

Theoretical relationship between transmission line length (short-circuited) and it’s imaginary impedance component:

018.5