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3.7 Impedance in the Frequency Domain

The important feature of the frequency domain is that the only waveforms that can exist are sine waves. We can only describe the behavior of ideal circuit elements in the frequency domain by how they interact with sine waves: sine waves of current and sine waves of voltage. These sine waves have three and only three features: the frequency, the amplitude, and the phase associated with each wave.

Rather than describe the phase in cycles or degrees, it is more common to use radians. There are 2 x π radians in one cycle, so a radian is about 57 degrees. The frequency in radians per second is referred to as the angular frequency. The Greek letter omega (ω) is used to denote the angular frequency. ω is related to the frequency, by:

Equation 3-9 where:

• ω = the angular frequency, in radians/sec

• f = the sine-wave frequency, in Hertz

We can apply sine-wave voltages across a circuit element and look at the sine waves of current through it. When we do this, we will still use the same basic definition of impedance (that is, the ratio of the voltage to the current) except that we will be taking the ratio of two sine waves, a voltage sine wave and a current sine wave.

It is important to keep in mind that all the basic building-block circuit elements and all the interconnects are linear devices. If a voltage sine wave of 1 MHz, for example, is applied across any of the four ideal circuit elements, the only sine-wave-frequency components that will be present in the current waveform will be a sine wave at 1 MHz. The amplitude of the current sine wave will be some number of Amps and it will have some phase shift with respect to the voltage wave, but it will have exactly the same frequency. This is illustrated in Figure 3-7. Figure 3-7. The sine-wave current through and voltage across an ideal circuit element will have exactly the same frequency but different amplitudes and some phase shift.

When we take the ratio of two sine waves, we need to account for the ratio of the amplitudes and the phase shift between the two waves.

What does it mean to take the ratio of two sine waves, the voltage and the current? The ratio of two sine waves is not a sine wave. It is a pair of numbers that contains information about the ratio of the amplitudes and the phase shift, at each frequency value. The magnitude of the ratio is just the ratio of the amplitudes of the two sine waves:

Equation 3-10 The ratio of the voltage amplitude to the current amplitude will have units of Ohms. We refer to this ratio as the magnitude of the impedance. The phase of the ratio is the phase shift between the two waves. The phase shift has units of degrees or radians. In the frequency domain, the impedance of a circuit element or combination of circuit elements would be of the form: at 20 MHz, the magnitude of the impedance is 15 Ohms and the phase of the impedance is 25 degrees. This means the impedance is 15 Ohms and the voltage wave is leading the current wave by 25 degrees.

The impedance of any circuit element is two numbers, a magnitude and a phase, at every frequency value. Both the magnitude of the impedance and the phase of the impedance may be frequency dependent. The ratio of the amplitudes may vary with frequency or the phase may vary with frequency. When we describe the impedance, we need to specify at what frequency we are describing the impedance.

In the frequency domain, impedance can also be described with complex numbers. For example, the impedance of a circuit can be described as having a real component and an imaginary component. The use of real and imaginary components allows the powerful formalism of complex numbers to be applied, which dramatically simplifies the calculations of impedance in large circuits. Exactly the same information is contained in the magnitude and the phase information. These are two different and equivalent ways of describing impedance.

With this new idea of working in the frequency domain, and dealing only with sine waves of current and voltage, we can take another look at impedance.

We apply a sine wave of current through a resistor and we get a sine wave of voltage across it that is simply R times the current wave:

Equation 3-11 We can describe the sine wave of current in terms of sine and cosine waves or in terms of complex exponential notation.

When we take the ratio of the voltage to the current for a resistor, we find that it is simply the value of the resistance:

Equation 3-12 The impedance is independent of frequency and the phase shift is zero. The impedance of an ideal resistor is flat with frequency. This is basically the same result we saw in the time domain, still pretty boring.

When we look at an ideal capacitor in the frequency domain, we will apply a sine-wave voltage across the ends. The current through the capacitor is the derivative of the voltage, which is a cosine wave:

Equation 3-13 This says the current amplitude will increase with frequency, even if the voltage amplitude stays constant. The higher the frequency, the larger the amplitude of the current through the capacitor. This suggests the impedance of a capacitor will decrease with increasing frequency. The impedance of a capacitor is calculated from:

Equation 3-14 Here is where it gets confusing. This ratio is easily described using complex math, but most of the insight can also be gained from sine and cosine waves. The magnitude of the impedance of a capacitor is just 1/ωC. All the important information is here. As the angular frequency increases, the impedance of a capacitor decreases. This says that even though the value of the capacitance is constant with frequency, the impedance gets smaller with higher frequency. We see this is reasonable because the current through the capacitor will increase with higher frequency and hence its impedance will be less.

The phase of the impedance is the phase shift between a sine and cosine wave, which is –90 degrees. When described in complex notation, the –90 degree phase shift is represented by the complex number, –i. In complex notation, the impedance of a capacitor is –i/ωC. For most of the following discussion, the phase adds more confusion than value and will generally be ignored.

A real decoupling capacitor has a capacitance of 10 nF. What is its impedance at 1 GHz? First, we assume this capacitor is an ideal capacitor. A 10-nF ideal capacitor will have an impedance of 1/(2 π × 1 GHz × 10 nF) = 1/(6 × 109 × 10 x 10-9) = 1/60 ~ 0.016 Ohms. This is a very small impedance. If the real decoupling capacitor behaved like an ideal capacitor, its impedance would be about 10 milliOhms at 1 GHz. Of course, at lower frequency, its impedance would be higher. At 1 Hz, its impedance would be about 16 MegaOhms.

Let's use this same frequency-domain analysis with an inductor. When we apply a sine wave current through an inductor, the voltage generated is:

Equation 3-15 This says that for a fixed current amplitude, the voltage across an inductor gets larger at higher frequency. It takes a higher voltage to push the same current amplitude through an inductor. This would hint that the impedance of an inductor increases with frequency.

Using the basic definition of impedance, the impedance of an inductor in the frequency domain can be derived as:

Equation 3-16 The magnitude of the impedance increases with frequency, even though the value of the inductance is constant with frequency. It is a natural consequence of the behavior of an inductor that it is harder to shove AC current through it with increasing frequency.

The phase of the impedance of an inductor is the phase shift between the voltage and the current which is +90 degrees. In complex notation, a +90 degree phase shift is i. The complex impedance of an inductor is Z = iωL.

In a real decoupling capacitor, there is inductance associated with the intrinsic shape of the capacitor and its board-attach footprint. A rough estimate for this intrinsic inductance is 2 nH. We really have to work hard to get it any lower than this. What is the impedance of just the series inductance of the real capacitor that we will model as an ideal inductor of 2 nH, at a frequency of 1 GHz?

The impedance is Z = 2 × π × 1 GHz × 2 nH = 12 Ohms. When it is in series with the power and ground distribution and we want a low impedance, for example less than 0.1 Ohms, 12 Ohms is a lot. How does this compare with the impedance of the ideal-capacitor component of the real decoupling capacitor? In the last problem, the impedance of the ideal capacitor element at 1 GHz was 0.01 Ohm. The impedance of the ideal inductor component is more than 1000 times higher than this and will clearly dominate the high-frequency behavior of a real capacitor.

We see that for both the ideal capacitor and inductor, the impedance in the frequency domain has a very simple form and is easily described. This is one of the powers of the frequency domain and why we will often turn to it to help solve problems.

The value of the resistance, capacitance, and inductance of ideal resistors, capacitors, and inductors are all constant with frequency. For the case of an ideal resistor, the impedance is also constant with frequency. However, for a capacitor, its impedance will decrease with frequency, and for an inductor, its impedance will increase with frequency.

It is important to keep straight that for an ideal capacitor or inductor, even though its value of capacitance and inductance is absolutely constant with frequency, its impedance will vary with frequency.