1.2: Passive components at high frequency

RF inductors and capacitors also have loss and parasitic elements. With inductors there is both series resistance and shunt capacitance mainly from interwinding capacitance, while with capacitors there will be shunt resistance and series inductance. Even the pads and leads of components change the behavior of components. In this section, some examples of the effects of non-ideal behavior will be shown. It will also be possible to play with those examples to see how changing the parasitic elements effects performance. Then the concept of \(Q\) will be introduced as a way to quantify the performance of components.

AC resistance

At high frequencies, the current is limited to a small layer near the surface by the skin effect, given by:

\[ \delta = \sqrt{\frac{2\rho}{\omega\mu_r\mu_0}} \],

where \(\rho\) is the resistivity, \(\omega\) is the angular frequency, \(\mu_r\) is the relative magnetic permeability (unity for most materials), and \(\mu_0\) is the magnetic permeability of free space. For frequencies in the GHz range (standard WiFi or bluetooth), the skin depth is already around ~10 µm. The resistance of a conductor of length \(L\) and cross section \(A\) is given by

\[R = \rho\frac{L}{A}\]

But, even if the conductor is very thick (a few mm in the case of a rectangular waveguide), the majority of the current flows within the skin depth. Thus, \(A\) must be calculated using the skin depth rather than the thickness.

AC resistance calculation \(\PageIndex{1}\)

Consider a 1 mm diameter copper conductor of length 1 m. What is the DC resistance and the resistance at 10 GHz?

Solution

Copper has a conductivity of \(1.7\times 10^{-8}\) Ωm. The area of the wire is \(A=\pi r^2 = 7.9\times 10^{-7}\) m\(^2\), so the DC resistance is 0.02 Ω.

At 10 GHz, the skin depth is \(\delta = \sqrt{\frac{2\rho}{\omega\mu_r\mu_0}} =\) 930 nm. So the area in which the current flows is \(A=\pi (\delta^2 + 2r\delta)=2.9\times 10^{-9}\) m\(^2\). Thus, the resistance at 10 GHz is 5.8 Ω, nearly 270 times more than the DC resistance.

Lead inductance

Modern high frequency electronics make use of surface mount components to minimize the effect of wire inductance. The inductance due to a wire is given by

\[L = 2l\left[2.3\log\left(\frac{4l}{d}\right)-0.75\right] \label{eq:wireInductance}\].

where \(l\) and \(d\) are the length and diameter of the wire. For those interested, this result is derived on pages 302-305 Vol 4, Num. 2 of the Bulletin of the Bureau of Standards, "The self and mutual inductances of linear conductors," Edward B. Rosa. Equation \ref{eq:wireInductance} provides an excellent illustration of why through hole components are so unfavorable for high frequency designs. This is shown in the figure \(\PageIndex{1}\), where the reactance of an ideal 1 pF capacitor with 1 cm leads is plotted.

Figure \(\PageIndex{1}\): Reactance of an ideal capacitor with 1 cm leads. Note that the behavior is only capacitive for values below about 200 MHz.

Notice that the reactance is negative, as expected up to a few hundred MHz, but for higher frequencies, the capacitor behaves like an inductor. Hence, when modeling high frequency circuits the parasitic reactances and resistances cannot automatically be neglected.

Introduction to \(Q\)

As illustrated in Figure \(\PageIndex{1}\), the reactance of component is only close to the ideal value below the (self) resonance frequency of the \(LC\) circuit created between the ideal capacitor and the parasitic inductance. This is generally true, so practical inductor or capacitor is limited to operation below the self-resonant frequency determined by the inductance and capacitance itself resonating with its reactive parasitics. The impact of loss is quantified by the \(Q\) factor (the quality factor). \(Q\) is loosely related to bandwidth in general and the strict relationship is based on the response of a series or parallel connection of a resistor (\(R\)), an inductor (\(L\)), and a capacitor (\(C\)). The response of an \(RLC\) network is described by a second-order differential equation with the conclusion that the \(3\text{ dB}\) fractional bandwidth of the response (i.e., when the power response is at its half-power level below its peak response) is \(1/Q\).

Figure \(\PageIndex{2}\): Transfer characteristic of a resonant circuit. (The transfer function is \(V/I\) for the parallel resonant circuit of Figure \(\PageIndex{3}\)(a) and \(I/V\) for the series resonant circuit of Figure \(\PageIndex{3}\)(b).)

Figure \(\PageIndex{3}\): Second-order resonant circuits.

The fractional bandwidth is\(\Delta f /f_{0}\) where \(f_{0} = f_{r}\) is the resonant frequency at the center of the band and \(\Delta f\) is the \(3\text{ dB}\) bandwidth. The described relationship between \(Q\) and bandwidth is not true for any network other than a second-order circuit, but as a guiding principle, networks with higher \(Q\)s will have narrower bandwidths.

Definition of \(Q\)

The \(Q\) factor of a component at frequency \(f\) is defined as the ratio of \(2πf\) times the maximum energy stored to the energy lost per cycle. In a lumped-element resonant circuit, stored energy is transferred between an inductor, which stores magnetic energy, and a capacitor, which stores electric energy, and back again every period. Distributed resonators function the same way, exchanging energy stored in electric and magnetic forms, but with the energy stored spatially. The quality factor is

\[\label{eq:1}Q=2\pi f_{0}\left(\frac{\text{average energy stored in the resonator at }f_{0}}{\text{power lost in the resonator}}\right) \]

where \(f_{0}\) is the resonant frequency.

A simple response is shown in Figure \(\PageIndex{2}\) for a parallel resonant circuit with elements \(L,\: C,\) and \(G = 1/R\) (see Figure \(\PageIndex{3}\)(a)),

\[\label{eq:2}Q=\omega_{r}C/G=1/(\omega_{r}LG) \]

where \(f_{r} =\omega_{r}/(2π)\) is the resonant frequency and is the frequency at which the maximum amount of energy is stored in a resonator. The conductance, \(G\), describes the energy lost in a cycle. For a series resonant circuit (Figure \(\PageIndex{3}\)(b)) with \(L,\: C,\) and \(R\) elements,

\[\label{eq:3}Q=\omega_{r}L/R=1/(\omega_{r}CR) \]

These second-order resonant circuits have a bandpass transfer characteristic (see Figure \(\PageIndex{2}\)) with \(Q\) being the inverse of the fractional bandwidth of the resonator. The fractional bandwidth, \(\Delta f /f\), is measured at the half-power points as shown in Figure \(\PageIndex{2}\). (\(\Delta f\) is also referred to as the two-sided \(−3\text{ dB}\) and sometimes as the fullwidth half maximum (FWHM) bandwidth

Figure \(\PageIndex{4}\): Loss elements of practical inductors and capacitors: (a) an inductor has a series resistance \(R\); and (b) for a capacitor, the dominant loss mechanism is a shunt conductance \(G = 1/R\).

bandwith.) Then

\[\label{eq:4}Q=f_{r}/\Delta f \]

Thus the \(Q\) is a measure of the sharpness of the bandpass frequency response. The determination of \(Q\) using the measurement of bandwidth together with Equation \(\eqref{eq:4}\) is often not very precise, so another definition that uses the much more sensitive phase change at resonance is preferred when measurements are being used. With \(\phi\) being the phase (in radians) of the transfer characteristic, the definition of \(Q\) is now

\[\label{eq:5}Q=\frac{\omega_{r}}{2}\left|\frac{d\phi}{d\omega}\right| \]

Equation \(\eqref{eq:5}\) is another equivalent definition of \(Q\) for parallel \(RLC\) or series \(RLC\) resonant circuits. It is meaningful to talk about the \(Q\) of circuits other than three-element \(RLC\) circuits, and then its meaning is always a ratio of the energy stored to the energy dissipated per cycle. The \(Q\) of these structures can no longer be determined by bandwidth or by the rate of phase change.

\(Q\) of Lumped Elements

\(Q\) is also used to characterize the loss of lumped inductors and capacitors. Inductors have a series resistance \(R\), and the main loss mechanism of a capacitor is a shunt conductance \(G\) (see Figure \(\PageIndex{3}\)).

The \(Q\) of an inductor at frequency \(f =\omega /(2π)\) with a series resistance \(R\) and inductance \(L\) is

\[\label{eq:6}Q_{\text{I}}=\frac{\omega L}{R} \]

Since \(R\) is approximately constant with respect to frequency for an inductor, the \(Q\) will vary with frequency.

The \(Q\) of a capacitor with a shunt conductance \(G\) and capacitance \(C\) is

\[\label{eq:7}Q_{\text{C}}=\frac{\omega C}{G} \]

\(G\) is due mainly to relaxation loss mechanisms of the dielectric of a capacitor and so varies linearly with frequency but also it is usually very small. Thus the \(Q\) of a capacitor is almost constant with respect to frequency. For microwave components \(Q_{\text{C}} ≫ Q_{\text{I}}\), and both are smaller than the \(Q\) of most transmission line networks. Thus, if the length of a transmission line is not too long for an application, transmission line networks are preferred. If lumped elements must be used, the use of inductors should be minimized.

Loaded \(Q\) Factor

The \(Q\) of a component as defined in the previous section is called the unloaded \(Q,\: Q_{U}\). However, if a component is to be measured or used in any way, it is necessary to couple energy in and out of it. The \(Q\) is reduced and thus the resonator bandwidth is increased by the power lost to the external circuit so that the loaded \(Q\), the \(Q\) that is measured, is

\[\begin{align}Q_{L}&=2\pi f_{0}\left(\frac{\text{average energy stored in the resonator at }f_{0}}{\text{power lost in the resonator and to the external circuit}}\right) \nonumber \\ \label{eq:8} &=\frac{1}{1/Q + 1/Q_{X}}\end{align} \]

and

\[\label{eq:9}Q_{X}=\left(\frac{1}{Q_{L}}-\frac{1}{Q_{U}}\right)^{-1} \]

where \(Q_{X}\) is called the external \(Q\). \(Q_{L}\) accounts for the power extracted from the resonant circuit. If the loading is kept very small, \(Q_{L} ≈ Q_{U}\).

Summary of the Properties of \(Q\)

In summary:

1. \(Q\) is properly defined and related to the energy stored in a resonator for a second-order network, one with two reactive elements of opposite types.
2. \(Q\) is not well defined for networks with three or more reactive elements.
3. It is only used (as defined or some approximation of it) for guiding the design.

Some example component equivalent circuits

In the following, example equivalent circuits are given for resistors, inductors and capacitors. Not every example of an equivalent circuit focuses on the \(Q\), but, from the circuit diagrams it is clear that all of these circuits are damped oscillators.

An equivalent circuit for a resistor is shown in Figure \(\PageIndex{5}\).

Figure \(\PageIndex{5}\): A non-ideal resistor is replaced by a series of ideal inductances and capacitances.

Exercise \(\PageIndex{1}\)

The equivalent circuit shown in Figure \(\PageIndex{5}\) is modeled in the code below, with a plot to show the real and imaginary components of the effective impedance. The values for the parasitic inductance and capacitance are set quite high so that the effect is easy to see. These values are free to be modified. What range of parasitic inductance and capacitance are allowed for a resistor to function at 2.4 GHz and 5 GHz (WiFi carrier frequencies)?

Interactive Element

from matplotlib import pylab as plt
import numpy as np
from scipy.constants import *
#######values that you can play with####
f = np.logspace(7, 11, 200) #frequency range (10 MHz - 100 GHz)
R = 1e3 #nominal value of the resistor

L = 1e-9 #parasitic inductance
C = 1e-12 #parasitic capacitance

########where calculations are performed#####
X_l = 2j*2*pi*f*L
X_c = -1j/(2*pi*f*C)
Z1 = X_l + R
Z = Z1*X_c/(Z1 + X_c)

##########plot everything##########
plt.figure()
plt.semilogx(f, Z.imag, color='xkcd:fuchsia')
plt.semilogx(f, Z.real, color='xkcd:brick')
plt.xlabel("Frequency (Hz)")
plt.ylabel("Impedance ($\Omega$)")

Hello world!

Answer

First adjust the model range to cover 1 - 10 GHz. You should find that the inductance can be quite high (~1 nF), but the parasitic capacitance needs to be around 1 fF.

The equivalent circuit for a capacitor is shown in Figure \(\PageIndex{6}\).

Figure \(\PageIndex{6}\): A non-ideal capacitor has both an inductive and a resistive component. Note that for an ideal capacitor, the power factor \(cos\phi = 0\), leading to an infinite \(Q\).

Exercise \(\PageIndex{2}\)

Again, the example in Figure \(\PageIndex{6}\) is modeled in the code below. In the current generation of automotive radar, the transmitters and receivers operate between 77-81 GHz. Use the code below to try and estimate the maximum allowed values for the parasitic inductors and resistors.

Interactive Element

from matplotlib import pylab as plt
import numpy as np
from scipy.constants import *
#######values that you can play with####
f = np.logspace(7, 11, 200) #frequency range (10 MHz - 100 GHz)

Rs = 1e-3
Rd = 100e13
L = 1e-9
C = 10e-12
##### Calculations#####
X_l = 1j*2*pi*f*L
X_c = -1j/(2*pi*f*C)
Z1 = X_l + Rs
Z2 = X_c*Rd/(X_c + Rd)

Z = Z1 + Z2

##### Plotting ######
plt.figure()
plt.semilogx(f, Z.imag, color='xkcd:fuchsia')
plt.semilogx(f, X_c.imag, color = 'xkcd:brick')
plt.xlabel("Frequency (Hz)")
plt.ylabel("Impedance ($\Omega$)")

        hello world
      

Answer

Adjust the simulation range to the range of interest (you can use non-integer values in the logspace function. For instance, a starting value of 10.5 means a starting frequency of ~30 GHz). You should find that the impedance is relatively insensitive to the two resistance values (but the \(Q\) is!), while the parasitic inductance should be lower than 0.1 pH and preferably about 0.01 pH.

An equivalent circuit for an inductor is shown in Figure \(\PageIndex{7}\). An ideal inductor, \(L\), has a parallel parasitic capacitance, \(C_d\), and the wire windings have some resistance \(R_s\). At low frequency, the inductor behaviour dominates as it has the lowest reactance, while at high frequency, the capacitor dominates. Thus, an inductor will show capacitive behavior at high frequencies.

Figure \(\PageIndex{7}\): A non-ideal inductor has both a capacitive and a resistive component. Note that for an ideal inductor \(R_s = 0\), leading to an infinite \(Q\).

Exercise \(\PageIndex{3}\)

Below is a model that calculates the impedance of a non-ideal inductor using the equivalent circuit in Figure \(\PageIndex{7}\). A high frequency inductor (Vishay IFCB-402) has the following properties:

L @ 500 MHz (nH)	Self resonance frequency (MHz)	Max DC resistance (\(\Omega\))	\(Q\) @ 1700 MHz	\(Q\) @ 800 MHz	\(Q\) @ 100 MHz
1.0	12000	0.15	33	21	7

Use the script below to estimate the parasitic capacitance. Is the tabulated DC resistance a good estimate? If not, what value best fits the rest of the data? The inductance has a tolerance of 0.3 nH. Does adjusting the inductance within tolerance give you a better fit to the tabulated values? What is the maximum frequency at which you would use this inductor? Note, you may want to change the frequency range to get a better estimate.

Interactive Element

from matplotlib import pylab as plt
import numpy as np
from scipy.constants import *
#######values that you can play with####
f = np.logspace(7, 11, 200) #frequency range (10 MHz - 100 GHz)

Rs = 0.15

L = 1e-9
C = 10e-12
##### Calculations#####
X_l = 1j*2*pi*f*L
X_c = -1j/(2*pi*f*C)
Z1 = X_l + Rs
Z = Z1*X_c/(X_c + Z1)

Q = X_l/Rs

Q_points = np.asarray([33, 21, 7])
f_points = np.asarray([1700e6, 800e6, 100e6])
f_sr = 1/(2*np.pi*np.sqrt(L*C))


##### Plotting ######
plt.figure()
plt.semilogx(f, Q, color='xkcd:fuchsia', label="Calculated Q")
plt.semilogx(f_points, Q_points, color = 'xkcd:brick', marker = '+', label="Tabulated Q")
plt.xlabel("Frequency (Hz)")
plt.ylabel("Q")
plt.legend()

plt.figure()
plt.semilogx(f, Z.imag, color='xkcd:fuchsia', label="Total reactance")
plt.semilogx(f, X_l.imag, color = 'xkcd:brick', label="Ideal reactance")
plt.xlabel("Frequency (Hz)")
plt.ylabel("Impedance ($\Omega$)")
plt.legend()

628.3185307179587j

 

Answer

You should adjust the DC resistance and inductance so that tabulated and calculated \(Q)\ values are as close to each other as possible. You may have to change the frequency scale so that this is clear to see. Adjust the frequency range to cover only the self resonance frequency and adjust the parasitic capacitance until the self resonance frequency occurs near 12000 MHz (you don't have to be exact). This indicates that the DC resistance is not a good estimate at high frequency. The AC resistance is over twice the DC resistance at 1200 MHz, while at 100 MHz, the AC resistance is insignificant. You should find that the parasitic inductance must be about 0.18 pF. The ideal and calculated inductance start to deviate from each other significantly at 3 GHz.