Detailed explanation of resistance, capacitance and inductance in RF circuits

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Detailed explanation of resistance, capacitance and inductance in RF circuits [Copy link]

Resistors, capacitors and inductors are the most commonly used components in electronic circuits. In low-frequency electronic circuits or DC circuits, the characteristics of these components are very consistent. But what about RF circuits? Today, we will continue to learn the basics of RF circuits based on the introduction of Professor Lei Zhenya's book "Introduction to Microwave Engineering".

In the previous article we introduced the frequency effect of wires in electromagnetic fields.

No.1 Resistors

Resistors are one of the most commonly used basic components in electronic circuits. Their basic function is to convert electrical energy into heat and generate voltage drop.

In electronic circuits, one or more resistors can form a step-down or voltage divider circuit for DC bias of the device, or can be used as a load resistor in a DC or RF circuit to perform certain specific functions.

Generally, there are several types of resistors: high-density carbon dielectric composite resistors, wirewound resistors made of nickel or other materials, metal film resistors made of temperature-stable materials, and thin film chip resistors made of aluminum or beryllium-based materials.

The application of these resistors is related to their constituent materials, structural dimensions, cost price, and electrical performance. Thin film resistors are the most commonly used in RF/microwave electronic circuits, and surface mount components (SMD) are generally used. There are three types of resistors used in monolithic microwave integrated circuits: semiconductor resistors, deposited metal film resistors, and mixtures of metals and dielectrics.

The resistance of a substance is related to the mobility of electrons and holes inside the substance. From the outside, the bulk resistance of a substance is related to the conductivity σ and the volume of the substance L×W×H, that is,

In RF applications, the equivalent circuit of a resistor is relatively complex. It not only has resistance, but also has lead inductance and parasitic capacitance between wires. Its nature is no longer pure resistance, but has both "resistance" and "resistance". The specific equivalent circuit is shown in Figure 2-4. In the figure, Ca represents the charge separation effect, that is, the equivalent capacitance between the plates of the resistor pins; Cb represents the capacitance between the leads; and L is the lead inductance.

For wirewound resistors, their equivalent circuit also needs to consider the inductance L1 caused by the wirewound part and the capacitance C1 between the windings. The capacitance Cb between the leads is generally smaller than the internal winding capacitance and can be ignored. The equivalent circuit is shown in Figure 2-5.

Take a 500Ω metal film resistor as an example (see Figure 2-4 for the equivalent circuit), assuming that the lead lengths at both ends are 2.5cm each, the lead radius is 0.2032mm, the material is copper, and Ca is known to be 5pF. Calculate the lead inductance according to formula (2-3), and find the total impedance versus frequency curve of the equivalent circuit in Figure 2-4, as shown in Figure 2-6.

As can be seen from Figure 2-6, at low frequencies, the impedance is equal to the resistance R, and as the frequency increases to more than 10MHz, the influence of the capacitor Ca begins to dominate, resulting in a decrease in the total impedance; when the frequency reaches about 20GHz, a parallel resonance point appears; after crossing the resonance point, the influence of the lead inductance begins to appear, and the impedance increases and gradually shows an open circuit or finite impedance value. This result shows that a resistor that seems to be independent of frequency is no longer just a resistor when used in the RF/microwave band, and special attention should be paid to it in the application.

The basic structure of a resistor is a rectangular block as shown in the figure above. In microwave integrated circuits, curved rectangular resistors are used to optimize the circuit structure and certain parasitic parameters.

No.2 Capacitor

At low frequencies, capacitors can generally be considered as parallel plate structures, where the size of the plates is much larger than the distance between the plates. The capacitance is defined as

In the formula, A is the plate area, d represents the distance between the plates, and ε=ε0εr is the dielectric constant of the plate as the filling medium.

Ideally, there is no current in the medium between the plates. At RF/microwave frequencies, the actual medium is not an ideal medium, so there is conduction current inside the medium, which also causes losses. More importantly, the charged particles in the medium have a certain mass and inertia. Under the action of the electromagnetic field, it is difficult for them to oscillate synchronously. There is a lag in time, which also causes energy loss.

Therefore, the impedance of the capacitor is composed of the conductance Ge and the susceptance ωC in parallel, that is,

In the formula, the current is caused by conductivity,

Where σd is the conductivity of the medium.

In RF/microwave applications, the lead inductance L, the series resistance Rs of the lead conductor loss, and the dielectric loss resistance Re must also be considered, so the equivalent circuit of the capacitor is shown in the figure.

For example, for a 47pF capacitor, assuming that the dielectric filling between its plates is Al2O3, the loss tangent is 10-4 (assuming it is independent of frequency), the lead length is 1.25cm, and the radius is 0.2032mm, the frequency response curve of its equivalent circuit can be obtained as shown in Figure 2-8.

As can be seen from Figure 2-8, its characteristics have deviated a lot from the ideal capacitor in the high frequency band. It can be imagined that in real situations when the loss tangent itself is still a function of frequency, its characteristic variation will be more serious.

No.3 Inductor

The inductor commonly used in electronic circuits is generally a coil structure, also known as a high-frequency choke at high frequencies. Its structure is generally made of straight wires wound along a columnar structure, as shown in the figure.

The winding of the wire constitutes the main part of the inductance, while the inductance of the wire itself can be ignored. The inductance of the thin solenoid is

In the formula, r is the radius of the solenoid, N is the number of turns, and l is the length of the solenoid. After considering the parasitic bypass capacitance Cs and the series resistance Rs of the lead conductor loss, the equivalent circuit diagram of the inductor is shown in the figure

For example, a copper inductor coil with N=3.5, a coil radius of 1.27mm, a coil length of 1.27mm, and a wire radius of 63.5μm. Assuming it can be regarded as a thin and long solenoid, according to formula (2-10), its inductance part can be calculated as L=61.4nH. Its capacitance Cs can be regarded as the capacitance generated by a flat plate capacitor. The distance between the plates is assumed to be the distance between two turns of the spiral d=l/N=3.6×10-4mm, the plate area A=2alwire=2a(2πrN), lwire is the total length of the wire wound into a coil, and according to formula (2-7), Cs=0.087pF can be obtained. The self-impedance of the wire can be obtained by the formula, that is, 0.034Ω. Therefore, the impedance frequency characteristic curve corresponding to the equivalent circuit shown is shown in the figure.

It can be seen from the above figure that the high-frequency characteristics of this copper inductor coil are completely different from those of an ideal inductor. Its impedance increases rapidly before the resonance point, and after the resonance point, it gradually decreases due to the influence of the parasitic capacitance Cs, which has gradually become dominant.