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

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