Brandy’s Gain Formula
Gi = ZPn/Zi
Brandy’s Gain Formula applies to
passive circuits. It states that the circuit
gain from an input to a circuit node n, is the parallel combination (ZP) of the
Impedances connected to node n divided by the connecting impedance (Zi).
ZPn is the impedance from node n to ground, or simply the
node impedance. Using this concept, the
passive circuit gain Gi is the destination node
impedance divided by the connecting impedance.
This is replacement for the Legacy Potential Divider equation.
Brandy’s Gain Formula is the corner
stone of K9 Analysis. Most of the K9
concepts come from Brandy. If you are
familiar with Legacy circuit analysis, you may find the concepts strange. In K9 analysis, circuits propagate signals
from inputs (voltages) to outputs (voltages).
The circuit operation is described by the input to output gain. Instead of voltage and current, K9 uses gain.
Let’s look at a generic passive
circuit. The schematic is shown below.
V1 to Vm are ideal voltage source inputs.
Zi is the Impedance that
connects the Input Voltage Source to node n.
ZPn is the parallel combination
of impedances connected to node n, the output node impedance.
ZPn = Z1
// Z2 // Z3 // … //Zm // Z0
Gi is the gain from input i to
node n.
The nodes in the
schematic are named 0 to n. 0 is
reserved for the ground node. The ground
node is more than just ground in K9 Design and Analysis. Ground is considered an input. The Ground Gain (Go) is the gain from the
ground node (0).
This schematic does
not look like a typical circuit schematic.
There are no component values, just a lot of Zs. K9 prefers not to distinguish between
components. Everything is an Impedance
(Z). Z can be a resistor or a two
terminal combination of resistors, capacitors, and inductors. The magnitude of the Impedance is the
important parameter. For reactive
components, the impedance magnitude will vary with frequency. Brandy prefers to
hide this complexity. (Pun intended). Whenever you see a Z, you may interpret
it as a poor resistor symbol.
The gain equations
don’t care what the component is.
What’s important is how the components are connected. The // operator denotes a parallel
combination. The standard product over
sum formula is too messy.
Brandy’s Gain
Formula introduces the concept of a Node Impedance. The Node Impedance is the Impedance between
the Node and Ground. The Node Impedance
is the parallel combination of all the components connected at the Node, ZPn. The ZP term
greatly simplifies gain equations. The
example below illustrates how simple a circuit equation can be if you use ZP
terms.
K9 tries to avoid
component values. The gain equation is
what is important. If you use component
values in the analysis, you can easily hide where the output value came from. A gain equation will tell you how a component
value affects the output. The intent of
K9 analysis is to make equations simple.
The Gain equation
allows a more accurate result. You can
use actual component impedance rather than ideal approximations. You can easily
include stray Impedances, even when they are not explicitly shown on the
schematic. K9 analysis is not only
simpler, but better.
Let’s analyze
the above circuit.
Legacy
Analysis
Legacy Analysis views the circuit as a Potential Divider consisting of a Series Impedance (R1) and an Output Impedance (R2) connected to ground. The gain is R2 / (R1 + R2). Using superposition, the circuit equation is:
V(out) =
V1 * (Z2//Z3//…//Zm//Z0) / (Z1 +(Z2//Z3//…//Zm//Z0))
+ V2 *
(Z1//Z3//…//Zm//Z0) / (Z2 + (Z1//Z3//…//Zm//Z0))
+ V3 *
(Z1//Z2//Z4//…//Zm//Z0) / (Z3 + (Z1//Z2//…//Zm//Z0))
.
.
+ Vm * (Z1//Z2//…//Zm-1//Z0) / (Zm
+ (Z1//Z2//…//Zm-1//Z0))
The equation above uses a different parallel
impedance combination for each input. This makes evaluation tedious.
Brandy’s Gain Formula provides a more compact equation.
K9
Analysis
K9 analysis first
calculates the output Node Impedance, ZPout and then
uses Superposition and Brandy’s Gain formula to yield the equation below.
ZPout =
Z1//Z2//Z3//…//Zn//Z0
V(out) =
V1 * ZPout / Z1
+ V2 *
ZPout / Z2
+ V3 *
ZPout / Z3
.
.
+ Vn * ZPout / Zn
The Legacy Analysis produces a compact result
for circuits with a single input. For multiple inputs Brandy’s Gain
Formula will produce more compact results.
In the above equation the Ground Gain (G0) is
not included. If you include it, you may
notice that the sum of the gains is equal to one. Daisy extents this concept to linear circuits.
Brandy’s Gain Formula is used in Shadow's Design Procedure to simplify the design of
passive summing circuits.
K9 Analysis uses Brandy’s Gain formula to construct a
Signal Flow
Graphs (SFG). With Brandy’s Gain Formula, the SFG is easy to construct
and resembles the circuit schematic.
Brandy’s
Formula is the cornerstone of K9 Analysis.
The other sections build on Brandy’s Gain Formula.
Brandy is no longer
with us. I ask that you honor her by
keeping the name. Whenever you see a ZP
term, think of Brandy.
The proof of Brandy’s Gain Formula is
left as a reader exercise. If you need a hint, I can be contacted at k9analysis@netzero.net.