# Understanding Resistor Behavior at High Frequencies

### Key Takeaways

• At DC and low frequency, resistor behavior is dependent on physical parameters and resistivity.

• Resistors act as a combination of resistance, inductance, and capacitance at high frequency.

• The parasitic inductance is associated with the length of a resistor. The parasitic capacitance is due to the end connecting terminals that act as plates. Engineers can study the behavior of electronics endlessly and still be surprised by their unpredictable behavior sometimes. In electronics, behavioral changes—such as those seen in lumped elements like resistors, capacitors, inductors, and active elements—are common. For example, how a resistor behaves at high frequencies is different from how it behaves at low frequencies.

To avoid any surprises, it is  important to analyze the high frequency behavior of passive elements and active elements when designing RF and microwave circuits. In this article, we will focus the discussion on resistor behavior at high frequency.

## Resistor Behavior At High Frequencies

The most common lumped elements in electronics circuits are resistors, capacitors, and inductors. The resistance property of resistors limits the free flow of current through the circuit. The resistance can be mathematically expressed with the following equation, where is the resistivity of the material, l is the length of the material, and a is the cross-sectional area of the material. At DC and low frequency, resistor behavior is dependent on the physical parameters and the resistivity, which is the property of the material and frequency independent.

At high frequency, resistors are frequency-dependent elements that showcase different behavior at different frequencies. The equation above becomes obsolete, as the parasitic capacitance and inductance of the resistor are active at high frequency. In fact, each resistor is associated with inductance and capacitance due to non-idealities in the materials, shape, and size of the resistor.

The following qualities are responsible for altering the behavior of resistors at high frequency:

• Physical dimensions

• Properties of resistor materials

• Connecting wires

## Parasitics and Resistors

Resistors act as a combination of resistance, inductance, and capacitance at high frequency. Parasitic inductance is associated with the length of the resistor. The parasitic capacitance is due to the end connecting terminals that act as plates. Resonant frequency is associated with parasitic capacitance and inductance.

It is the parasitic inductance (L) and capacitance (C) that make the resistor frequency-dependent. If L and C are the parasitic inductance and capacitance of the resistor, then equation 2 gives the resonant frequency and equation 3 gives the effective impedance of the resistor at frequency f. ### Parasitic Inductance and Capacitance

Resistors show parasitic inductance due to the conductivity of the material with which it is made of. The effect of inductive reactance will be less at DC and low-frequency AC. The parasitic capacitance effect is also seen at high frequency. The parasitic effects become active in high frequency AC applications. At the resonant frequency, the parasitic effects are null. When the operating frequency is less than the resonant frequency, the parasitic capacitance is dominant. As the operating frequency crosses the resonant frequency, the parasitic effect is more inductive.

At high frequency, the parasitic inductance and capacitance in resistors cause unwanted couplings between various circuit blocks and delayed circuit responses. The parasitic inductance can be self-inductance or mutual inductance, depending on the components in the vicinity of the resistor. Self-inductance is capable of distorting the signals whereas mutual inductance introduces noises in the circuit. Depending on the parasitics present in the resistor, the time constants L/R and RC determine the response time.

## Skin Effect

Skin effect is experienced by resistors at high frequency. At low frequency, the distribution of the current is uniform throughout the resistor. As the frequency increases, the current distribution becomes non-uniform. At high frequency, the current in resistors concentrates on the surface of the resistor. The current is confined only to the surface at RF frequency.

When designing RF circuits, engineers should always consider resistor behavior at high frequency. Cadence’s software can help model RF circuits with the parasitic effects and skin effects in resistors and other lumped passive elements.