# Circuit Designer’s Notebook: Effective Capacitance vs Frequency

by Richard Fiore, Director, RF Applications Engineering – American Technical Ceramics Corp.

It is generally assumed that the capacitance value selected from a vendor’s catalog is constant over frequency. This is essentially true for applications with applied frequencies that are well below the capacitor’0s self-resonant frequency. However, as the operating frequency approaches the capacitor’s self-resonant frequency, the capacitance value will appear to increase, resulting in an effective capacitance (C_{E}) that is larger than the nominal capacitance. This article will address the details of effective capacitance as a function of the application operating frequency. In order to illustrate this phenomenon, a simplified lumped element model of a capacitor connected to a frequency source operating in a network will be considered, as depicted in **Figure 1**.

This model has been selected because the effective capacitance is largely a function of the net reactance developed between the capacitor and its parasitic series inductance (L_{S}). The equivalent series resistance “ESR”shown in this illustration does not have a significant effect on the effective capacitance.

**Effective Capacitance:**

The nominal capacitance value (C_{O}) is established by a measurement performed at 1 MHz. In typical RF applications, the applied frequency is generally much higher than the 1 MHz measurement frequency, hence at these frequencies, the inductive reactance (X_{L}) associated with the parasitic series inductance (L_{S}) becomes significantly large as compared to the capacitive reactance (X_{C}). **Figure 2** illustrates that there is a disproportionate increase in X_{L} as compared to X_{C} with increasing frequencies. This results in an effective capacitance that is greater than the nominal capacitance. Finally, at the capacitor’s series resonant frequency, the two reactances are equal and opposite, yielding a net reactance of zero. The expression for C_{E} becomes undefined at this frequency.

As illustrated in **Figure 1**, the physical capacitor can be represented as C_{O} in series with L_{S}. The impedance of the series combination of C_{O} and L_{S} can then be set equal to C_{E}, which may be referred to as an “ideal equivalent” capacitor. This will yield the following equation:

j (ω LS – 1/ω C0) = – j 1/ω CE

ω^{2} LS – 1/C0 = – 1/CE

The relationship between the operating frequency F0 and the effective capacitance CE can then be stated as:

CE = C0/(1 – ω^{2} LS C0)

CE = C0/[1 – (2π F0)^{2} LS C0]

Where:

C_{E} = Effective Capacitance at the application frequency, (F_{0})

C_{0} = Nominal Capacitance at 1 MHz

L_{S} = Parasitic Inductance, (H)

F_{0} = Operating Frequency, (Hz)

From this relationship, it can be seen that as the applied frequency increases, the denominator becomes smaller, thereby yielding a larger effective capacitance. At the capacitor’s series resonant frequency, the denominator goes to zero and the expression becomes undefined. The relationship of C_{E} vs frequency is a hyperbolic function as illustrated in **Figure 3**.

Example:

Consider an ATC 100A series 100pF capacitor.

Calculate the effective capacitance (C_{E}) at 10MHz, 100MHz, 500MHz, 900MHz, 950MHz.

Solution: Calculate by using the relationship:

C_{E} = C_{0}/[1– (2π F_{0}) 2 L_{S} C_{0}]. Refer to **Table 1**.

**Application Considerations: **

Impedance matching and minimum drift applications such as filters and oscillators require special attention regarding C_{E}. For applications below the capacitor’s self-resonant frequency, the net impedance will be capacitive (-j), whereas for applied frequencies above resonance, the net impedance will be inductive (+j). Operating above series resonance will correspondingly place the impedance of the capacitor on the inductive side of the Smith chart (+j). When designing for these applications, both C_{E} and the sign of the net impedance at the operating frequency must be carefully considered.

In contrast, the majority of coupling, bypass and DC blocking applications are usually not sensitive to the sign of the impedance and can be capacitive or inductive, as long as the magnitude of the impedance is low at the applied frequency. The effective capacitance will be very large and the net impedance will be very low when operating close to resonance. At resonance, the net impedance will be equal the magnitude of ESR and the capacitance will be undefined.

(474)