AC analyses¶

The available AC analyses are performed assuming the circuit to be linear time invariant (LTI), meaning that the parameters of the components within the circuit, and the circuit itself, cannot change over time. This is usually a reasonable assumption to make for circuits containing passive components such as resistors, capacitors and inductors, and op-amps far from saturation, and where only the frequency response or noise spectral density is required to be computed.

The linearity property implies the principle of superposition, such that if you double the circuit’s input voltage, the current through each component will also double. This restriction implies that components with a non-linear relationship between current and voltage cannot be simulated. Components such as diodes and transistors fall within this category, but it also means that certain component properties such as the output swing limit of op-amps cannot be simulated. In certain circumstances it is possible to approximate a linear relationship between a component’s voltage and current around the operating point, allowing the component to be simulated in the LTI regime, and this property is exploited in the case of op-amps.

The small-signal AC response of a circuit input to any of its components or nodes can be computed with the AC small signal analysis. The noise spectral density at a node arising from components and nodes elsewhere in the circuit can be computed using the AC small signal noise analysis.

Implementation¶

Zero uses a modified nodal analysis to compute circuit voltages and currents. Standard nodal analysis only allows voltages to be determined across components by using Kirchoff’s current law (which states that the sum of all currents flowing into or out of each node equals zero). A series of equations representing the inverse impedance of each component are built into a so-called admittance matrix, which is then solved to find the voltage drop across each component. This works well for circuits containing only passive components, but not those containing voltage sources. Ideal voltage sources considered in Zero have by definition no internal resistance (in other words, they can supply infinitely high current), and this makes the matrix representation of such a circuit singular due to the row corresponding to the voltage source effectively stating that 0 = 1! To handle voltage sources as well as other voltage-producing components like op-amps, Zero extends the equations derived from Kirchoff’s current law to include columns representing voltage drops between nodes. The circuit can then be solved for a number of unknown voltages and currents.

Consider the following voltage divider, defined in LISO syntax:

r r1 1k n1 n2
r r2 2k n2 gnd

freq log 1 100 101

uinput n1 0
uoutput n2

The following circuit matrix is generated for this circuit (using zero liso /path/to/script.fil --print-matrix):

╒═══════╤═════════╤═════════╤════════════╤═════════╤═════════╤═══════╕
│       │ i[r1]   │ i[r2]   │ i[input]   │ V[n2]   │ V[n1]   │ RHS   │
╞═══════╪═════════╪═════════╪════════════╪═════════╪═════════╪═══════╡
│ r1    │ 1.00e3  │ ---     │ ---        │ 1       │ -1      │ ---   │
├───────┼─────────┼─────────┼────────────┼─────────┼─────────┼───────┤
│ r2    │ ---     │ 2.00e3  │ ---        │ -1      │ ---     │ ---   │
├───────┼─────────┼─────────┼────────────┼─────────┼─────────┼───────┤
│ input │ ---     │ ---     │ ---        │ ---     │ 1       │ 1     │
├───────┼─────────┼─────────┼────────────┼─────────┼─────────┼───────┤
│ n2    │ 1       │ -1      │ ---        │ ---     │ ---     │ ---   │
├───────┼─────────┼─────────┼────────────┼─────────┼─────────┼───────┤
│ n1    │ -1      │ ---     │ 1          │ ---     │ ---     │ ---   │
╘═══════╧═════════╧═════════╧════════════╧═════════╧═════════╧═══════╛

The entries containing --- represent zero in sparse matrix form. The equations this matrix represents look like this (using zero liso /path/to/script.fil --print-equations):

1.00 × 10 ^ 3 × I[r1] + V[n2] - V[n1] = 0
        2.00 × 10 ^ 3 × I[r2] - V[n2] = 0
                                V[n1] = 1
                        I[r1] - I[r2] = 0
                   - I[r1] + I[input] = 0

In the matrix, this equation is arranged to equal 0 on the right hand side; however, we can rearrange the equation for r1 to read: “voltage drop across r1 equals the voltage difference between nodes n2 and n1”. The equation for r2 is similar, but since one of its nodes is attached to ground (which has 0 potential), the voltage difference between its nodes is simply equal to the potential at n2.

Note

In Zero, the circuit is solved under the assumption that the circuit’s ground is 0, which allows the use of standard linear analysis techniques. One side-effect of this approach is that circuits cannot contain floating loops, i.e. loops without a defined ground connection. This is usually not important, but has an effect on e.g. transformer circuits with intermediate or isolated loops. If reasonable for the application, consider supplying a weak connection to ground using e.g. a large valued resistor.

This property does not affect the ability to include floating voltage sources in circuits, as long as these are contained in grounded loops.

In the row corresponding to the voltage input, there is no voltage drop across the source (as per its definition as an ideal source), and its right hand side equals 1. This means that the circuit is solved with the constraint that the voltage between n1 and ground must always be 1. The solver can adjust all of the other non-zero matrix elements in the left hand side until this condition is met within some level of tolerance.

Available analyses¶

Signals¶

The solution x to the matrix equation Ax = b, where A is the circuit matrix above and b is the right hand side vector, gives the current through each component and voltage at each node.

Noise¶

Noise analysis requires an essentially identical approach to building the circuit matrix, except that the matrix is transposed and the right hand side is given a 1 in the row corresponding to the chosen noise output node instead of the input. This results in the solution x in the matrix equation Ax = b instead providing what amounts to the reverse responses between the component and nodes in the circuit and the chosen noise output node. These reverse responses are as a last step multiplied by the noise at each component and node to infer the noise at the noise output node.