## Input Impedance

### The Input Impedance Spectrum

The concept of impedance (borrowed from electrical engineering) was introduced previously as a ratio of sinusoidal pressure to flow.  The definition implies (requires) that we can only divide one sinusoid by another when the 2 oscillations are of the same frequency.  A more complicated wave, however, can always be represented as a sum of sinusoidal oscillations simply by including multiple frequencies.  When working with the cardiovascular system, we most typically have a fundamental frequency, due to the heart rate, and additional frequencies or harmonics that are integer multiples of the fundamental.  A heart rate of 60, for example, corresponds to a fundamental frequency of 1 Hz, but pressure and flow signals in that case would likely include significant signal strength at 2 Hz, 3 Hz, 4 Hz, .. etc.

I've indicated repeatedly that input impedance only makes sense if sinusoids of differing frequency don't interact with each other; it isn't appropriate if we can input a 2 Hz pressure wave into a system and get anything but a 2 Hz flow wave out ( and vice versa ).  This stipulation is one aspect of linearity and we know that nonlinearity of the circulation is always present so that the input impedance can only be an approximation for its represention; it will be adequate for some purposes but not others.  More explicitly, we can always determine the frequency spectra of the pressure and flow signals, we can always compute a ratio (the impedance), but the ratio doesn't necessarily characterize anything about the circulation unless the latter is linear.

Although recorded pressure and flow signals might contain only a few harmonics, the possibility of inputs at other frequencies remains, i.e. frequencies not present in a specific recording.  The first paragraph above alluded to a signal containing sinusoids at 1 Hz, 2, 3 , etc., but nothing in between these values.    If we had the option, a full characterization of input impedance would be accomplished by inputting a wide range of individual sinusoids, varying both frequency and amplitude.  This also would allow us to test for linearity.  We would want to make sure that inputting a pressure sinusoid of a given frequency results only in a flow of the same frequency and that doubling the amplitude of the pressure sinusoid doubles the amplitude of the corresponding flow sinusoid (etc.).  Inputting frequencies as pristine individuals would allow us to compute the pressure-to-flow ratio for each and plot the result at each and every frequency of interest.  With this thought experiment in mind, the input impedance is not a single (complex) number but a spectrum; a function of frequency.  Alas that the procedure just outlined is not a practical approach for obtaining the spectrum, although we do have the option of pacing the heart to generate a range of input frequencies.  In the article that follows, however, any frequency of interest can be computed through the physical concepts and mathematics of a model.  However obtained, the input impedance spectrum is another example of a transfer function.  Once available, we are be able to compute the pressure signal from any flow signal by applying the transfer function frequency by frequency to the input (if the system is linear).

Obviously time-varying pressure and flow cannot be presupposed to be sinusoidal.  In the circulation, however, typical signals include only a few harmonics -- a few frequencies that can be extracted from the whole using frequency analysis techniques. If we record pressure and flow at a specific location, we see that both fluctuate over time; I've indicated these in the past as $$\large p(t)$$ and $$\large q(t)$$ respectively.  The clinical literature tends to refer to these as "phasic" pressure (and flow rate).   Within a straight conduit, both values also vary with a readily defined axial position, e.g. $$\large z$$ as measured along the conduit from the inlet or outlet; it would be proper to indicate this with symbols $$\large p(z,t)$$ and $$\large q(z,t)$$.

When these pressure flow signals are expressed as a sum of their frequency components, they are no long functions of time but of frequency.  A mathematical process called the Fourier transform converts (transforms) the time domain signals to the frequency domain so that a spectrum is obtained; you could also say that the Fourier transform extracts the individual frequency components from the time-domain signals.  (Remember that your Doppler ultrasound machine does this multiple times per second, converting time domain sound signals to the frequency domain to obtain the Doppler shift frequencies.)   To make it clear that this has occurred in the text, I'll write $$\large P(j\omega)$$ as corresponding to $$\large p(t)$$ and $$\large Q(j\omega)$$ as the transform of $$\large q(t)$$ to indicate the Fourier transform has occurred. I'll leave out the $$\large j\omega$$ part when it gets tedious.  As suggested by the above hemodynamic thought experimental, the input impedance spectrum is the result of determining pressure and flow signals ($$\large p(t)$$ and $$\large q(t)$$  ) at the same location in the circulation, performing the Fourier transform procedure on both signals (extracting the magnitude and phase of the sinusoidal oscillations for the relevant frequencies), and dividing each pressure oscillation by the corresponding flow oscillation:

$$\Large Z(j\omega) = \frac{P(j\omega)}{Q(j\omega)}$$

The input impedance embodies everything about the system downstream of the measurement, i.e. everything downstream of the input site, with the previously indicated stipulation that the system is linear.  We'll soon be well acquainted with the fact that the location of the measurements, the input site, has a dramatic effect on the spectrum.  So more appropriately:

$$\Large Z(z, j\omega) = \frac{P(z, j\omega)}{Q(z, j\omega)}$$

The above introductory figure compares the input impedance spectrum (blue) to the characteristic impedance (red) of an elastic conduit with the distal end clamped.   For the model spectrum shown, there is a certain amount of viscosity included within the conduit fluid and we'll study what effect(s) this is producing shortly.  Both the modulus and phase of the input impedance exhibit regular oscillations above and below values of the characteristic impedance; the latter is NOT dependent on measurement location but is characteristic of the conduit as long as the physical properties are not changing along the length (diameter, wall material and thickness, fluid density and viscosity, etc.). This article is all about the nature of the input impedance and why it exhibits the features you see.

### Measurement Location and Frequency Relationship

The next section shows a sequence  of single frequency wave transmission videos to help understand how the input impedance spectrum is related to wave reflection and the measured pressure and flow waves.  For this model we have the contrived / unlikely scenario of zero viscosity (no friction) and a clamp on the far right end of a conduit causing an infinite impedance wave reflection site at the termination; the pressure doubles there for every frequency and the flow goes to zero.   Two measurement sites are placed in the conduit, magenta (upstream) and blue (downstream).  Those 2 sites will not be moved for the next few figures; we're trying to find out how the location of the measurements affects the observations.

The first input wave is at a relatively low frequency of $$1/3$$ Hz and it so happens that the upstream (magenta) measurement has been placed at a pressure node - the closest pressure node for this reflection site and frequency (since there would be others farther away from the reflection site).   We have an anti-node for the flow wave at this location and the impedance is the pressure sinusoid divided the flow sinusoid, $$\large \frac{P}{Q} = 0$$.

Next is the impedance spectrum plot for this system.  Magenta corresponds to the upstream measurement site and the vertical black line shows where we are in the spectrum for this $$1/3$$ Hz wave; the impedance modulus is zero.  The spectrum indicates that if a lower frequency wave had been input, we'd be farther to the left on the spectrum and the impedance at the upstream site would have been some value greater than zero.  In fact, that's exactly what we have for the downstream (blue) site which is closer to the reflection site than $$1/4$$ wavelength.

The above figure introduces the fact that the input impedance spectrum is entirely, completely dependent on the location of measurement / observation.  The  2 spectra were obtained from the same system but at different measurement locations.  We see that both exhibit regular maxima and minima, but at different frequency spacing.  This whole section is really just about why this spacing interval occurs.

In the next recording, a slightly higher frequency has been entered ($$1/2$$ Hz) and blue site is now at a pressure node.

The input impedance is zero at $$1/2$$ Hz (black vertical line) for the blue site (below).  The upstream (magenta) site is now farther than $$1/4$$ wavelength from the reflection site and the impedance is growing with increasing frequency.

At $$2/3$$ Hz (below) the upstream magenta site is at a flow node and pressure anti-node.  The impedance ratio, $$\frac{P}{Q}$$, at this location is infinite, but I hasten to add that this can occur only because our system has zero viscosity; the impedance could not actually go that high (nor can it go all the way to zero.

The input impedance spectrum at $$2/3$$ Hz (black vertical line) shows the modulus spike haring off for infinity; the value at the blue site is growing as frequency increases, but actually is less than the characteristic impedance (red) at this particular frequency.

And finally as you see in the next example at 2 Hz, we happen to have both measurement sites placed at a flow node (pressure anti-node).  However it isn't the same flow node in terms of how far the 2 measurement sites are from reflection site.  The blue site is one wavelength from the reflection site (2 half wavelengths), and the magenta site is 3 half wavelengths from the reflection site.  Both sites exhibit (infinite) peaks at this frequency, but a different number of peaks intervene between the 2 Hz frequency and 0.

What we're observing is the duality between distance from a reflection site and input frequency as it appears in the input impedance spectrum.  For distance, however, the thing that matters is wavelength.   We're going to see a peak in the impedance spectrum whenever the measurement site is a multiple of the half wavelength from a closed reflection site.  The wavelength, $$\lambda$$, can be expressed in terms of the frequency, $$f$$, and wave speed, $$c$$:

$$\Large \lambda = \frac{c}{f}$$

Peaks in the impedance spectrum occur when the distance, $$d$$, from a closed reflection site is a multiple of the half wavelength, $$\lambda/2$$:

$$\Large d = \frac{n \; \lambda}{2} = \frac{n \; c}{2f_{\text{max}}}$$

where n = 1, 2, 3, etc.  Rearranging:

$$\Large f_{\text{max}} = \frac{n \;c}{2d}$$

where I've used $$f_{\text{max}}$$ to stand for a "max" frequency i.e. frequency peak.  This expresses what we've seen in the experiment, that peaks occur at regular intervals in the spectrum and the peaks occur closer together (in frequency) as the measurement site is moved farther away from the reflection site.   We see also that the minima occur between the maxima.

$$\Large f_{\text{min}} = \frac{n \; c}{4d} \; : \large n = 1, 3, 5, ..$$

As math folks like to do, we can put both formula into a similar format:

$$\Large f_{\text{max}} = \frac{2n \; c}{4d} \; : \large n = 0, 1, 2, 3, ...$$

$$\Large f_{\text{min}} = \frac{(2n +1) \; c}{4d} \; : \large n = 0, 1, 2, 3, ...$$

We'll write this one more time in a different way (hopefully simplifying the concept).  These formula show that the frequency maxima and minima are obtained by multiplying a frequency by a number.  I'm going to call the frequency, $$\large \Delta f = \frac{c}{2d}$$, the change in frequency between 2 adjacent maxima, OR minima, on the impedance spectrum.  I don't know how they do this in the literature, but it clearly sets apart this important aspect of the spectrum.  There is an associated time as well, $$1/\Delta f$$, which is the round trip time of the wave, i.e. starting at the measurement location and traveling to and from the reflection site.  Here are the formulas in alternate form:

$$\Large f_{\text{max}} =n \Delta f \; : \large n = 0, 1, 2, 3, ...$$

$$\Large f_{\text{min}} = \frac{2n+1}{2} \Delta f \; : \large n = 0, 1, 2, 3, ...$$

The next 2 figures show the impedance spectra measured at the same locations as above; the upper figure is for a closed (clamped) reflection site, the lower is an open ("to air") site.  The above formula apply to the open reflection too, but are reversed;  frequency maxima for the open reflection correspond to minima for the closed reflection and vice versa.

Above: Input impedance spectrum for a closed reflection termination in the inviscid system. Below: open reflection termination.

The closed reflection case exhibits infinite impedance at 0 frequency (hey, the end is clamped) while the open reflection case has 0 impedance there since there's no viscosity -- no friction to resist the flow.  It so happens that impedance peaks intermittently coincide in frequency for the clamped end, i.e. every 3rd peak of the magenta site with every 2nd peak of the blue.  In this instance, no peaks coincide for the open termination model, but coincident impedance minima occur at the same frequency values as the peaks did for the closed termination case.  In comparison with the clamped end, the open termination situation essentially exchanges maxima for minima and vice versa.

While we're at it, note how the impedance phase oscillates.  For this bizarre no-viscosity example, the phase is pegged at either $$\pi/2$$ or $$-\pi/2$$ radians; the measured pressure and flow are always $$90^{\circ}$$ out of phase with each other, true at every location and for every frequency.  Note also that the phase exhibits a positive-negative crossover both at maxima and minima of the modulus.  It crosses negative to positive at a modulus minimum and positive to negative at a modulus maximum (try saying that 10 times fast); this is true for both open and closed reflection sites.

So now here are the rules about this that we have to get a handle on (at for this simplistic situation).

• In the presence of reflected waves, the input impedance spectrum exhibits (regular) oscillations with frequency.
• Peaks in the spectrum (and troughs) occur at intervals that are related to how far the measurement site is from the reflection site.

### And Now, with Viscosity ...

I'll recap a video to remind you what the 2 Hz wave transmission looks like in real time when there's a clamp on the distal end (right side).

When viscosity is added, the wave attenuates as it travels -- in either direction.  The reflected wave cannot be larger than the incident wave (unless there's an infusion of energy from somewhere).  Unlike the above zero viscosity model, the reflected wave can never fully cancel the antegrade wave and we have relative nodes and anti-nodes as we saw in the last article (next video).

(Note different y-axis scaling between the videos and the following figures.)  The next figure is a freeze frame from the no-viscosity video ...

.. and the next one is a freeze with viscosity ($$\mu = 0.01$$, like water).

In the presence of viscosity, the effect of reflected waves diminishes with distance from the reflection site, due to attenuation, and you'll readily imagine that the reflected wave would be negligible far enough away from the site. Study the wave envelope carefully to contemplate how the effect of the reflected wave varies with distance from the reflection site, yet still results in relative nodes and anti-nodes.

You'll notice also a nuance about the measurement locations.  Although they haven't been changed between the viscosity and no-viscosity models, the measurement sites don't exactly coincide with the pressure anti-nodes anymore in the viscous model; they seem to have moved a little if the distance from the reflection site is measured in terms of wavelength.  This is because the addition of viscosity affects the wave speed -- the phase velocity -- slowing the waves down a little so that the round trip time takes a little longer.  The next plot shows the phase velocity as a function of frequency, and with the phase velocity of the no-viscosity model set as the y-axis maximum (~ 176 cm/sec).   This way you get a feel for the degree to which phase velocity was affected by a relatively low viscosity liquid.

Now here's the input impedance spectra for both models, without (above) and with (below) viscosity added; plots are aligned in frequency for vertical comparisons:

With viscosity added, we no longer have peaks in the impedance spectrum that range from $$0$$ to $$\infty$$, since the reflected waves are not of sufficient magnitude to fully cancel the antegrade waves so that neither pressure or flow can be identically $$0$$.  We still have regularly occurring maxima and minima and they're still spaced each half wavelength.  You'll notice that the location (frequency) of the peaks doesn't exactly coincide between the 2 models due to the slower wave speed with viscosity (compare peak frequency at the far right peak for example).  We'd also find with the viscous model that peaks don't occur at perfectly regular intervals anymore; the wave speed varies a little with frequency so the round trip time to the reflection site changes a little too.

### Reflection Coefficient

It will come as no surprise now that the reflection coefficient is also a function of both position and frequency; we can expect to see a spectrum that varies with axial location.   Here's a recap of the wave transmission freeze frame to remind you of the viscous system we're looking at and positions of measuring sites upstream (magenta vertical line) and downstream (blue).

This system has fluid viscosity $$\large \mu = 0.01$$, the same as water.  Although the freeze frame above shows a single frequency wave (2 Hz), we can look at the reflection coefficient spectrum, $$\large \Gamma(j,\omega)$$, as a function of frequency at these sites:

If you're starting to get the hang of this, the modulus of $$\large \Gamma$$  (upper half of figure) makes good sense.  For a given location, the reflection coefficient tapers off with frequency.  This is part of your daily experience doing echocardiography; lower frequency ultrasound exhibits better penetration so the magnitude ratio ($$\large |\Gamma|$$) of the reflected to incident wave decreases with frequency.  Nor is it any surprise that the magnitude of wave reflections is less for the site that's farther from the reflection site (magenta).  The waves (any / all frequencies) have a longer round trip distance and are more attenuated by the friction, so $$\large \frac{P_r}{P_a} = \Gamma$$ is less if we're farther away.  This is another phenomenon you're fully familiar with and it's the reason there are TGC (time-gain compensation) controls on an ultrasound machine; we need to boost the reflected waves more that traveled farther to adequately reconstruct an image.

The phase of $$\large \Gamma$$ is also shown and it will help you to think about the freeze frame figure to visualize this result.  When we divide 2 complex numbers, $$\large \frac{P_r}{P_a} = \Gamma$$ for example, the resulting phase angle is the difference between the phases of the numerator and denominator, i.e.  $$\large \angle \Gamma = \angle P_r - \angle P_a$$.   From either/any measurement site, inputting a higher frequency wave results in more wave crests and troughs between the measurement site and the reflection site.  If wave speed was completely independent of frequency, we'd simply see a linear increase in the phase difference between the antegrade and reflected waves.  Think of counting how many degrees or radians of change the frozen wave exhibits over (twice) the distance to the reflection site.  The fact that the phase plot exhibits the saw-toothed appearance is due simply to the convention that we confine the reported angle to $$\large \pm \pi$$ radians.  Remember that if we're lucky enough to actually be able to determine both the antegrade and retrograde waves (modulus and phase) at a given site, we still don't know how many wave peaks there are between us and the reflection site unless we know the whole picture from the wave transmission video (spatial aliasing).  Measuring from a site that's farther away from the reflection site (magenta site) simply means that the wave undergoes a greater phase shift for the round trip.  We see a greater rate of phase shift with frequency on the phase plot (sawtooth pattern is more compressed, ratcheting through more cycles of $$2\pi$$ radians as frequency changes).

For the next figures we have the same system as above but with greater viscosity, $$\large \mu = 0.04$$ (similar to blood, not that the demos are designed for you to understand the circulation, yet).  With greater viscosity, waves are more attenuated and the magnitudes of the reflected waves are smaller, other things being equal.  Here's a freeze frame from the wave propagation at 2 Hz (notice less oscillation in the wave envelope compared to $$\large \mu = 0.01$$:

And here's the reflection coefficient spectrum at the 2 measurement locations.

One more widget in the thought process.  For the next figure we go back to $$\large mu = 0.01$$, but the stiffness of the wall material (Young's modulus of elasticity) has been increased by a factor of $$5$$.  As we've seen this increases the wave velocity so that the reflection site is effectively closer to the measurement site - fewer wavelengths between the sites at any frequency you choose.  Just as we saw above when comparing a near measurement site to a far one, the sawtooth pattern of the phase plot becomes more stretched out; a faster phase velocity is just like placing the measurement location closer to the reflection site.   However there's also an interaction between the tube compliance (wave velocity) and the viscosity (friction).  As the wall material stiffens, $$\large |\Gamma|$$ increases at any given location or frequency.

This is something we would have inferred from the math pertaining to transmission lines where we saw previously that:

$$\Large \gamma = \sqrt{\frac{Z_L}{Z_T}}$$

where $$\large Z_L$$ is the longitudinal impedance and $$\large Z_T$$ is the transverse impedance.  Increasing the wall stiffness increases $$\large Z_T$$ but doesn't affect $$\large Z_L$$, so $$\large \gamma$$ becomes smaller as we increase the wall stiffness.  With $$\large \gamma$$ as an exponent in the wave equation, e.g.

$$\Large P_a(z) = P_a(0) \; e^{-\gamma \; z}$$

this translates into less change in the wave (both modulus and phase) per unit of distance.  Both the antegrade and retrograde waves are attenuated less in a stiffer tube, so reflected waves are more pronounced and the reflection coefficient is greater (other things remaining equal).  Note that it gets trickier if we change something like tube diameter which affects tube compliance, but also friction to a much greater degree.  Increasing tube diameter increases reflected waves; the compliance of the tube is increased (slower wave velocity) but the attenuation is greatly decreased because pressure loss is inversely related to the 4th power of the radius.

### Apparent Wave Velocity

We looked at traveling waves in the last article and tried to get a handle on how to measure how fast they travel.  The result was, it's durn difficult when reflected waves are present!! As you might expect, the apparent wave velocity will depend on the extent to which reflected waves are present and so will depend on proximity of the measurement location to a reflection site, frequency, viscous effects, etc... all the aspects that have had an impact on the reflection coefficient and input impedance.  Here again is a freeze frame from the 2 Hz wave propagation showing the upstream (magenta) and downstream (blue) measurement locations($$\large \mu = 0.01$$):

And next are the Apparent Wave Velocity spectra at the 2 measurement locations: