[circuithology] Gyrator made easy - Page 1
Forum section: electronics / projects

author: RoGeorge
author_rank: Super Contributor
timestamp: May 06, 2023, 08:06:41 am
content: Bumped last week into gyrators. Looked up the Wikipedia page and it wasn't clear what a gyrator is, unless one already knows, but then why lookup what one already knows. I mean this:QuoteSource:https://en.wikipedia.org/wiki/GyratorWent places looking for something easier to grasp, took some notes, and though of double checking by writing them down in here. It took an unreasonably long time to type them, and kind of lost any coherence of ideas while editing. Please point out any mistakes or misinterpretations you may see.Head firstLet's build a circuit. Take an amplifier, \$A_{neg}\$. It has to be an inverting amplifier. Controlled by an input voltage, \$v_i\$. And that it outputs a current, \$i_o\$.Its gain, \$g_n\$, will be measured in amps per volt. Because \$A_{neg}\$ is an inverting amplifier, its gain \$g_n\$ will be a negative value.Now, take a second amplifier, \$A_{poz}\$, this time a non-inverting one, with gain \$g_p\$, and connect it in the other direction, like this:That's a gyrator.\[ \star \ \star \ \star \]Next, let's load the output with an impedance, \$Z_o\$.\[ \star \ \star \ \star \]Aye, aye... skipperSkippers can skip to the section called 'Plain English'. If not, to calculate the equivalent input impedance, \$Z_i\$, we know:- any impedance \$Z\$, by definition is \$v/i\$, so \$Z_o\$ and \$Z_i\$ can be written as\[ \begin{array}{ll} Z_o = \frac{v_o}{i_o}, & Z_i = \frac{v_i}{i_i} \end{array} \tag{1.1} \label{impedance} \]- the gain of each amplifier is defined as \$g = i/v\$, for fig.3 this becomes\[ \begin{array}{ll} g_n = \frac{i_o}{v_i}, & g_p = \frac{i_i}{v_o} \end{array} \tag{1.2} \label{gain} \]- \$\eqref{gain}\$ tells the amplifiers \$A_{neg}\$ and \$A_{poz}\$ will output current by the formulas\[ \begin{array}{ll} i_o = g_n v_i, & i_i = g_p v_o \end{array} \tag{1.3} \label{current} \]- from \$\eqref{current}\$, \$v_i\$ can be expressed as\[ v_i = \frac{i_o}{g_n} \tag{1.4} \label{voltage} \]By replacing \$v_i\$ with \eqref{voltage} and \$i_i\$ with \$\eqref{current}\$ in \$\eqref{impedance}\$, we get\[ \begin{align}Z_i &= \frac{v_i}{i_i} \\&= \frac{ \frac{i_o}{g_n} }{ g_p v_o } \\&= \frac{i_o}{g_n g_p v_o} \\&= \frac{1}{g_n g_p} \cdot \frac{i_o}{v_o} \\&= \frac{1}{g_n g_p} \cdot \frac{1}{Z_o} \tag{1.5} \label{gyrator_detail}\end{align} \]Note that \$ \frac{1}{g_n g_p}\$ is a constant value dictated by \$A_{neg}\$ and \$A_{poz}\$. If we make the notation\[ k = \frac{1}{g_n g_p} \tag{1.6} \label{gir_const} \]then \$\eqref{gyrator_detail}\$ becomes:\[ Z_i = k \cdot \frac{1}{Z_o} \tag{1.7} \label{gyrator}\]In conclusion, the impedance\$Z_o\$ connected at the output, will appear at the input as the inverse of \$Z_o\$ multiplied by the constant, \$k\$. And that's what a gyrator does.\[ \star \ \star \ \star \]What does that means?Let's take an example. Let's connect at the output a capacitor \$C\$, that will play the role of \$Z_o\$ in the formula \$\eqref{gyrator}\$. The impedance of any capacitor \$C\$ at a given frequency, \$f\$, is\[ Z_C = \frac{1}{2 \pi f C} \tag{1.8} \label{impC} \]By substituting in \$\eqref{gyrator}\$ the output impedance \$Z_o\$ with \$Z_C\$, the impedance seen at the input will be\[ \begin{align}Z_i &= k \cdot \frac{1}{Z_C} \\&= k \cdot \frac{1}{ \frac{1}{2 \pi f C} } \\&= k \cdot 2 \pi f C \tag{1.9} \label{CasL}\end{align} \]Note that the term \$2 \pi f C\$ in \$\eqref{CasL}\$ reminds of the impedance formula of an inductor \$L\$, which is\[ Z_L = 2 \pi f L \tag{1.10} \label{impL} \]\[ \star \ \star \ \star \]Plain EnglishObs. #1 - From \$\eqref{CasL}\$ and \$\eqref{impL}\$ we can say that a gyrator will make a capacitor at its output to appear as an inductance at its input. In short,a gyrator turns C into L, and L into C.Obs. #2 - The amplifiers \$A_{neg}\$ and \$A_{poz}\$ are connected anti-parallel, so a gyrator can be fed as well from its output to the input,a gyrator is bidirectional. That is particularly interesting when connecting a capacitor at both ends, because it will make the whole thing look like an LC tank.Obs. #3 - The second amplifier, \$A_{poz}\$, is a non-inverting one. Its amps per volt gain \$g_p\$ has a positive value, and \$g_p\$ has the dimension of a \$1/R\$, so\$A_{poz}\$ can be as well just a resistorof value \$R = 1/g_p\$.Obs. #4 - Trying to stay brief here. There is more about a gyrator. It's a passive, linear, lossless, non-reciprocal, two-port electrical network element. That will imply many consequences.\[ \star \ \star \ \star \]Implementation examples[fig.4 - gyrator with opamp as Aneg and R as Apoz]Source:https://en.wikipedia.org/wiki/Gyrator[fig.5 - gyrator with transistor as Aneg and R as Apoz]\[ \star \ \star \ \star \]Application examples[fig.6 - simulated inductor to filter power, LC osc without L, audio EQ, etc.]Bibliography:1. The Internet2. How to gyrate - and why (Elektor Magazine #2/1975)https://archive.org/details/ElektorMagazine/Elektor%5Bnonlinear.ir%5D%201975-02/page/n49/mode/2up

author: moffy
author_rank: Super Contributor
timestamp: May 06, 2023, 12:03:26 pm
content: Gyrators were all the buzz years ago for multiband eq's before the truly digital age with cheap 16 bit ADCs/DACs and DSPs. They replaced the largish inductors but if I remember they had a problem with lowish Q, might be misremembering.

author: T3sl4co1l
author_rank: Super Contributor
timestamp: May 06, 2023, 12:32:28 pm
content: One minor correction -- it's g_m, g being the symbol for conductance (for admittance Y = G + jB, B being "susceptance") and 'm' for mutual (the "trans-" part).Tim

author: T3sl4co1l
author_rank: Super Contributor
timestamp: May 06, 2023, 12:59:54 pm
content: Quote from: moffy on May 06, 2023, 12:03:26 pmYes, this would be the problem. If you look at the transfer function of such an element, you get a shelf function between DC and HF, where the gain is limited and the impedance is flat (resistive). The MF semi-asymptote inbetween is where the reactance lies (L or C as the case may be, L of course being the more practical case), but notice it only spans a decade or two using simple elements (e.g. emitter follower gyrator), and the response between corners will be 1st order so the resistance is only as low as the corners are far from the point in question -- that is if the slope spans two decades, and you measure in the middle, it'll have a Q of 10 there. You can do much better with an op-amp of course, but only at low frequencies -- where gain is high, and you still need to be far from the corner frequencies to get a reasonable Q.For a sort of illustration, take this radio receiver AF filter I designed some time ago:It's a Sallen-Key configuration (using a 6AK5 or 5702, not 6V6, but the 6V6 model is surprisingly close at this operating point), so the LF asymptote is fine (2nd order), but the HF asymptote flattens out. For a gyrator, you get the same sort of plateau-slope-floor function for the input impedance, for the same reason (here: feed-forward into the finite cathode impedance (r_k ~ 1/gm), which is also to say, gain is slightly less than 1.0, or gm is finite).The other problem is, as you approach infinite Q, it very quickly becomes a problem that, any minute phase shift in the feedback loop, can push it over infinity -- that is, the loss resistance becomes negative and you have an oscillator instead.A similar problem is the output impedance of an emitter follower, which is inductive, when the base source impedance is resistive. It's inductive because the gain necessarily drops at high frequencies, due to circuit and device capacitance, or approaching fT (which in turn is limited by internal charge diffusion and distributed RC effects, which depend on distance from terminals to active junction area, so are worse for large-geometry parts -- why power transistors are always slower than small-signal types, and specific measures must be taken to maintain bandwidth e.g. "perforated emitter"). Gain dropping means output impedance rising, which can only mean one thing: inductive reactance! Which is why you get ringing when loading such an amplifier with capacitance. Or indeed oscillation, for an inductive input which gets rotated around as negative emitter resistance.Going back to gyrators and op-amp applications, the real quirky one there is the double-pole element -- which I forget the name of offhand, but you can use the general impedance converter network (five elements and two op-amps to get truly anywhere on the Smith chart) and use that with resistors as a filter, though I forget what the tradeoffs are (more general? better transition or stop band?).Oh yeah, here it is:https://en.wikipedia.org/wiki/Frequency_dependent_negative_resistorYou'll have the same limiting problems with amp gain, Q, and phase shift, of course.Tim

author: coppercone2
author_rank: Super Contributor
timestamp: May 06, 2023, 01:43:09 pm
content: made easy, this is a lie.Try implementing them too, especially faster ones... trouble. My results were over complicated oscillators

author: RoGeorge
author_rank: Super Contributor
timestamp: May 06, 2023, 07:03:32 pm
content: Quote from: coppercone2 on May 06, 2023, 01:43:09 pmIndeed, should have been titled "epic fail to make gyrators look easy".The entire formulas paragraph would look better if entirely removed.Took so long to edit that I didn't have the heart to delete it afterwards.Quote from: T3sl4co1l on May 06, 2023, 12:32:28 pmI didn't understand what to change. The chosen symbols for each of the two amplifier's conductance were \$g_n\$ and \$g_p\$, where n and p are standing fornegative (for the inverter amplifier) andpositive (for the non-inverting amplifier). At first it was \$g_1\$ and \$g_2\$, renamed later as \$g_n\$ and \$g_p\$ as an attempt to make it clear which g is for which of the two amplifiers.Quote from: moffy on May 06, 2023, 12:03:26 pmThe plan was to add such an example as an application, then I've run out of time.

author: T3sl4co1l
author_rank: Super Contributor
timestamp: May 06, 2023, 07:15:05 pm
content: Oh, fair. Thought it a typo, sorryTim

author: moffy
author_rank: Super Contributor
timestamp: May 06, 2023, 11:25:10 pm
content: Quote from: coppercone2 on May 06, 2023, 01:43:09 pmYes, oscillators were a very real possibility.

author: mawyatt
author_rank: Super Contributor
timestamp: May 07, 2023, 12:51:45 am
content: Brings back memories of trying to create Gyrators, NICs (Negative Impedance Converters), Non-Foster Networks and such at microwave frequencies utilizing custom ICs with fast SiGe transistors. We were trying to synthesize various networks, including negative capacitors for very broadband work. The end results were disappointing, excess noise and poor linearity pretty much limited performanceWe did use a Gyrator to synthesize an inductor for a Chua Diode based Chaotic Oscillator which was used as a "Simple Analog Encryptor" with "Seeds" being the Lorentz Attractor coefficients, another "like" Chaotic Oscillator would receive and injection lock onto the transmitted signal and if the "Seeds" were correct, decode the transmission.Fun stuffBest,

author: RoGeorge
author_rank: Super Contributor
timestamp: May 07, 2023, 07:33:39 am
content: Quote from: mawyatt on May 07, 2023, 12:51:45 amWow, that's probably the most elaborated application possible for a gyrator!

author: RoGeorge
author_rank: Super Contributor
timestamp: May 07, 2023, 09:09:42 am
content: Quote from: mawyatt on May 07, 2023, 12:51:45 amThank you for mentioning that, spent the morning reading about it, very interesting subject!(i.e. Antennas with Non-Foster Matching Networks, DOI: 10.2200/S00050ED1V01Y200609ANT002)