Introduction
I assume that you have some background in Electronics, and that you are familiar with circuit analysis. You may have found that Electronics can be difficult. There are many different approaches and here is yet another. The information presented here is just an aid. It is not intended to provide an introduction to Electronics. On this page, I’ll introduce some of the concepts used in K9 analysis. K9 was designed for op-amp circuits. The concepts can be used in discrete circuits. The concepts are not applicable to high frequency circuits where propagation time is relevant.
There are many different approaches to Circuit Analysis:
Linear Circuit analysis uses mathematically sound procedures. This
includes Mesh analysis
and Nodal analysis
. You model the circuit via a set of equations. The procedure
guarantees that the equations are valid and can be solved. There are many
ways to solve the equations. If you have reactive components (capacitors) you
may need calculus.
Electronics uses an informal analysis technique. Disclosing circuit operation is more important than being mathematically correct. It’s OK to take some liberties with math as long as you verify the answers.
To avoid simultaneous equations, Electronics uses a set of circuit tricks to reduce a circuit to a simpler form. The procedures are not general, and may only apply to a specific circuit. If you’ve learned the tricks, analog circuit analysis can be simple. A problem occurs when the circuit becomes large and the tricks interact.
In Electronics, Op-amp circuits are analyzed via an “elegant” analysis procedure. I have always considered this Op-Amp Analysis procedure to be a comedy of errors. The test of any procedure is whether you can apply it to a new circuit. This one fails. I have also found that some Electronics concepts are very misleading, see the Legacy page.
K9 analysis is a combination of the above. K9 tries to avoid the math of formal analysis. The VSA procedure uses Nodal analysis to get a solid base. The node equations are well hidden. K9 uses equations to describe circuit behavior and Impedances to avoid the need for calculus.
K9 tries to avoid circuit tricks. There is no need to reduce circuits. The procedures are general and do not get complicated for large circuits. An equation may become large, but is always easy to derive.
K9 uses a divide and conquer approach. Instead of looking at the entire complete circuit, K9 prefers to look at a single point (node) in the circuit. The voltage at this node is dependent upon the voltages of other nodes that it is connected to. The dependency is modeled via gain. K9 considers each dependency individually and uses Superposition to combine the effects. Brandy provides the gain formula for passive circuits and Plato provides the gain formula for single op-amp circuits.
The equations used in K9 assume that there is no circuit interaction. This keeps the analysis simple. Corrections are applied to handle any circuit interaction. The corrections are simple and can handle all circuit interactions. Feedback is not an issue.
Let’s look at some of the concepts.
A Node is a point in the circuit. The Node can be an input, an output, an internal node, or Ground. In K9, Ground is an input. The node that an equation defines is the output, and the nodes that the equation depends on are inputs.
In K9, a node “n” has a node impedance (ZPn) which is the parallel combination (ZP) of all the impedances (Z) that connect at the node. This greatly simplifies equations. It also allows you to add loads at nodes after the analysis. Just include the load in the node impedance. The node impedance may not be the actual circuit impedance from the node to ground. Circuit interaction can alter the node impedance. You may need to apply correction factors to get the real impedances. K9 prefers to ignore circuit interaction. Why complicate the analysis?
K9 considers one output at a time. The output can be any circuit variable, voltage is preferred. The output is dependent upon a set of other variables, which may be inputs or other circuit variables.
Circuit operation is described via an output equation. K9 assumes that the output equation is a Linear equation. If the output is Vout and the circuit has two inputs, V1 and V2, then
Vout = g1 * V1 + g2 * V2
In Math terminology, g1 and g2 are constants. For circuits, g1 and g2 are gains. An equation can be rearranged into many forms. K9 only uses the SFG format.
Output = ∑ (gain * input)
The output value is the sum of gains times input values. A circuit output voltage is dependent on a set of inputs. Each input contributes to the output. The gain determines how much is contributed. The output is the sum of the input contributions.
The term Linear has mayor mathematical implications. One is that Superposition applies. You can consider one input at a time. A circuit creates an output by summing the contributions from inputs. Any linear circuit is hence a summing circuit. No specific schematic is implied.
Linear imposes some circuit restrictions. You can’t have an Analog multiplier or other non-linear circuit. If the circuit is non-linear, you can use small signal analysis. This makes the circuit linear about a bias point, but only for small signals.
The analysis contains a procedure to transform the circuit schematic into a Signal Flow Graph. Mason’s Gain Formula is used for circuit equations. No circuit reduction is needed. With practice, you can visualize the circuit schematic as a signal flow graph and write the circuit equation by inspection.
If a circuit has multiple inputs, Electronics uses Superposition to create multiple circuits. Each circuit has one active input and the other inputs are set to a zero value. This requires you to analyze a different circuit for each input. K9 uses only one circuit. Superposition is implied in the circuit equation. Multiple inputs pose no difficulty.
Modeling components as Impedances, avoids the need for calculus and complex math. It allows stray component values to be included. Just add them at the end. Circuit equations come from the circuit topology. There is no need to complicate things. (pun intended)
K9 uses a unified approach. Most op-amp circuits are a subset of the General Summing Circuit. The analysis of the General Summing circuit replaces the separate procedures used in Legacy Design. There is only one circuit to study, one analysis equation and one design procedure.
Isn’t a unified procedure complicated? K9 is actually much simpler. Op-amp circuit analysis uses Rf/RI for an inverting amplifier gain and 1+Rf/RI for non-inverting amplifier gain. The K9 procedure uses Rf/Ri for all circuit gains. In K9, the difference between positive and negative gain is the op-amp terminal that the input connects to.
The K9 procedure is equivalent to each of the Legacy procedures for simple circuits and far simpler for complicated circuits. Circuits with multiple inputs are considered difficult. If the circuit has both positive and negative gains, many Legacy design procedures use two op-amps. The K9 single op-amp procedure is simple and works for all cases.
The K9 procedures contain some simple checks to detect mistakes. This makes the procedure bigger. The check is a great aid for students and teachers. Correct solutions are easier to grade.
Don’t separate procedures provide a better design? The K9 design procedure is actually better from a circuit performance point of view. Shadow designs the best circuit. When compared with a Legacy design, Shadow’s circuit may have an extra resistor which is only needed to cancel bias current error. Ozzie can identify this component.
Legacy texts would like you to believe that op-amp circuits are difficult, that there is a mayor difference between inverting and non-inverting circuits, and that you need to consider circuit concepts such as “Virtual Ground”. K9 hide this. There are no circuit tricks in K9.
Legacy texts would like you to believe that equations are difficult. K9 only uses simple not interactive equations. You don’t have to solve the equations. The Analysis works for all linear circuits. There is no need to reduce circuits. An equation may become large, but is always easy to derive.
Legacy texts would like you to believe that multiple input circuits are difficult to design. The K9 Design allows any number of inputs. Multiple positive and negative inputs pose no problem. There is no need for an extra op-amp.
The output equation contains only relevant terms. If a resistor is connected across an ideal voltage source, the node impedance will mask the resistor. This operation is totally hidden.
K9 hides some of the signal flow in a circuit. The VSA procedure ignores all signal flow via the ground node. Components connected to ground only appear in the node impedance. They do not appear in output equations. Sounds crazy, but works and is actually mathematically sound.
K9 hides the Node Equations used for analysis. K9 hides that positive inputs interact. K9 hides that analog design and analysis is difficult.
K9 uses a voltage approach. Inputs are ideal voltage sources. The output is a voltage. This works well for op-amp circuits. It does not work for discrete analog circuits. There is an extension to include current; under construction.
K9 prefers to describe circuits by their behavior. Gain is the main variable. K9 only considers the local circuit gain without any circuit interaction. Simple Corrections are applied when interaction is present.
The VSA procedure uses component gains. Each circuit component allows signal flow. The gain of the signal flow may vary with directions. For a passive circuit, the gain is simply the destination node impedance divided by the connecting impedance (Brandy’s Gain Formula). For active circuit, the gain is determined by the component model. An amplifier simply has a gain of A. You only need the gain value when evaluation the result.
K9 prefers to describe circuits via gain equations. Component values are rarely used. The equation tells you what components effect the output equation. If a component is missing from the equation, then this component does not affect this output. You can remove components from a circuit by setting the component value to infinite. You can replace components with a short by setting the component value to zero. A single equation can be used for many circuits. Plato's Gain formula is the equation for many single op-amp circuits.
By using a different approach, some new circuit concepts and equations are discovered.
The most radical new concept is Daisy’s theorem. It states that the sum of all circuit gains is equal to one.
K9 uses Plato's Rf/Ri gain formula for single op-amp circuits; works for both positive and negative inputs.
K9 uses Impedance instead of Resistors and Capacitors to provide a simpler and better analysis. It’s simpler because no complex math is needed. It’s better because a more accurate analysis is possible. Stray components can be added in the evaluation.
K9 introduces the concept of Node Impedance which is the parallel combination of all the impedances connected to a node. This greatly simplifies equations. It even allows you to add loads after the analysis.
K9 VSA provides a simple Analysis procedure for transforming a circuit schematic into a Signal Flow Graph. Circuit equations are easy to derive. No simultaneous equations to solve. No calculus is needed. You write the equation by inspection.
K9 introduces the concept of Ground Gain. Ground is a zero voltage input and like other inputs has gain. Minimizing ground gain is one way to improve circuit performance.
K9 Analysis came about via a strange set of events and lots of trial and error. I’ll try to list the events in chronological order.
The first event occurred in college. I was a co-op student and not able to take courses in a normal sequence. I took control systems before electronics. In control systems we initially learned to solve problems via block diagram reduction, the bag of tricks approach. We then learned Signal Flow Graph (SFG) analysis and Mason’s Gain Formula. With Mason’s formula there was no need for reduction.
Later I took the Electronics class. The instructor did not present a clear explanation of series feedback, parallel feedback, and other circuit tricks. Instead of learning the tricks, I decided that it was simpler to model the circuit as a control system. I am still not comfortable with circuit tricks.
I continued to use SFGs in my subsequent employment. A problem with SFGs is that they represent equations and may not illustrate circuit operation. After lots of experimentation, I developed the technique described here. It allows the SFG to provide a simple circuit description that most can understand. The key component to the transformation is Brandy’s gain formula.
My first mayor design was a telephone conference bridge. The design required an analog summing circuit for each port. To optimize stability the sign of signals was altered. Each port had a different set of signs. In the design I noticed that for each port, if you considered ground an input, than the sum of the gains was always equal to one. Later I discovered that for any passive circuit, the sum is also one. This became Daisy’s theorem.
In the mid 80s, I taught a night class at a local community college. The course was targeted to Electronic Technology students and included op-amp circuits. The text presented a fragmented approach using lots of circuit tricks. I noticed that all the single op-amp circuits varied only in the input and ground connections. They are a subset of the General Summing Amplifier circuit. A unified analysis is possible and the gain equation is easy to derive via SFGs.
The gain equation showed that op-amp gain is simply Rf/Ri (Plato’s Gain Formula). For analysis you need to apply a fudge factor to positive gains, and for design (Shadow’s design procedure) you need to add a ground resistor. I presented the techniques to the class. The students loved it.
I shared the discovery with fellow workers and tried to share it with text authors. Most did not reply. The ones that replied stated that a unified theory is not desirable. Students need to learn the concepts of the individual circuits.
A poorly edited K9 design procedure was published in EDN. Refined versions have been published on the web.
There is a map of the web site. The information is provided in a partially organized manner. Just follow the links or use your browser button to navigate.
The Web-pages were created with MS Word. Some pages may have problems if viewed with MS Internet Explorer. I recommend Firefox.
My thesis adviser stressed that every student has an obligation to extend the knowledge base a tiny amount. This is my contribution. You need to do the same. With your help and the Dogs help, we can make Analog Circuits Dog-Gone-Simple
K9 is dedicated to the dogs. They provided support. When things went bad, I could always count on them. Without the dogs, K9 would not exist. I ask that you honor them by keeping their name with the formula or procedure.