Symbolic Circuit Analysis in MatLab

This document describes a A MATLAB® tool for deriving and solving circuit equations symbolically. It is split up into several segments.

- Downloading SCAM
- Defining a circuit for SCAM: Defining the netlist
- Solving a DC Circuit
- Example 1: Voltages
- Example 2: Currents
- Example 3: Getting MNA Equations
- Example 4: A more complex circuit

- Generating a transfer function
- Finding currents in wires
- Example 7: Finding the current in a wire

Before using the program you must first download it and save it on your computer where MATLAB can access it. Download the file now. You can also get the MATLAB file along with the example netlists in a zip file. Note: you must have the symbolic toolbox to run this code.

The SCAM program cannot simply read a schematic diagram so we need to develop a method for representing a circuit textually. This can be done using a device called a netlist that defines the interconnection between circuit elements. If you have used SPICE (Simulation Program with Integrated Circuit Emphasis) this is a familiar concept. A good review is given by Jan Van Der Spiegel at the University of Pennsylvania. The process is easily demonstrated by example.

Let us use SCAM to define the circuit below:

We start by defining the nodes. The only restriction here is that the nodes must be labeled such that ground is node 0, and the other nodes are named consecutively starting at 1. The choice of which number to assign to which node is entirely arbitrary.

SCAM requires a text file with one line for each component in the circuit. This circuit has 4 components (3 resistors and 1voltage source), and will require 6 lines to define it. Each type of component has its own format for its corresponding lines in a file. These are shown below. The labels N1, N2, etc... correspond to the nodes in the circuit.

Component TypeSymbolSCAMDescriptionResistor R1 N1 N2 1000

- R1 is between nodes
N1andN2, and has a value of 1000 Ohms- The value of the component must be written out (no abbreviations like kOhm) as a number.
- The name of the component is R
x, wherexcan be any combination of letters and numbers. R1, Rabc, Ra1 are all valid names.Capacitor C1 N1 N2 1E-6

- Similar comments to the resistor
Inductor L1 N1 N2 1E-3

- Similar comments to the resistor
Voltage Source V1 N1 N2 12

- Similar comments to the resistor
- Node
N1is connected to the positive node,N2to the negative node.- The current through the source is one of the unknowns, it is defined as shown below:
Current Source I1 N1 N2 1

- Similar comments to the resistor
- Current flows out of node
N1and into nodeN2.

Op Amp O1 N1 N2 N3

- Similar comments to the resistor but with three nodes as shown.

Let's define the current through the voltage source to give a final circuit diagram:

The netlist file is:

V1 1 0 12 R1 1 2 1000 R2 2 0 2000 R3 2 0 2000

## Example 1: Voltages

Let's use the circuit we have been examining

We will create a text file containing the netlist:

V1 1 0 12 R1 1 2 1000 R2 2 0 2000 R3 2 0 2000and save it in the directory seen by MATLAB. I edited such a file (using the MATLAB editor) and saved it in my SCAM directory as

Example1.cir. If you are running MATLAB, you should see the file if you use thedircommand:>> dir . .. scam.asv scam.m Example1.cirTo run the program, assign the filename of the circuit to be analyzed to the variable

fname, and then call the program. The output from the MATLAB window is shown below:>> fname='Example1.cir'; >> scam Started -- please be patient. Done! Elapsed time = 0.651 seconds. Netlist V1 1 0 12 R1 1 2 1000 R2 2 0 2000 R3 2 0 2000 Solved variables: [ v_1] [ v_2] [ I_V1]The netlist is displayed, followed by the unknown variables whose solution is determined. Let's look at the value of v_2 (the voltage at node 2):

>> v_2

v_2 = 1/(R2*R3+R1*R3+R1*R2)*R2*R3*V1or I_V1 (the current through the voltage source)

>> I_V1

I_V1 = (-R3-R2)/(R2*R3+R1*R3+R1*R2)*V1In addition to the unknowns, several other variables are created in the workspace (this is why the SCAM program is a

scriptinstead of afunction). The important variables created, in addition to the unknowns, are a value corresponding to each of the elements. We can examine the value of any element, for example V1 or R2>> V1

V1 = 12

>> R1

R1 = 1000We can use these values to get numeric values for the unknowns:

>> eval(v_2)

ans = 6

>> eval(I_V1)

ans = -0.0060which shows that the voltage at node 2 is 6 volts, and the current through V1 is 6 mA.

## Example 2: Currents

What happens if we are interested in the current through R2 instead of just node voltages, from node 2 to node 0. We know that the current through R2 is just the voltage drop across R2, divided by the value of the resistance. We can do the solution either symbolically or numerically.

>> v_2/R2

ans = 1/2000/(R2*R3+R1*R3+R1*R2)*R2*R3*V1

>> eval(ans)

ans = 0.0030The current through R1, from node 1 to node 2, is just the voltage across R1 divided by its value.

>> (v_1-v_2)/R1

ans = 1/1000*V1-1/1000/(R2*R3+R1*R3+R1*R2)*R2*R3*V1

>> eval(ans)

ans = 0.0060Other quantities can be similarly determined. For example the ratio of v_2 to V1:

>> v_2/V1

ans = 1/12/(R2*R3+R1*R3+R1*R2)*R2*R3*V1

>> eval(ans)

ans = 0.5000

## Example 3: Generating MNA Equations

The SCAM program also defines the

A,XandZmatrices from the MNA method. (Note that the SCAM program uses uppercase variables forXandZ).>> A

A = [ 1/R1, -1/R1, 1]

[ -1/R1, 1/R1+1/R2+1/R3, 0]

[ 1, 0, 0]

>> X

X = [ v_1]

[ v_2]

[ I_V1]

>> Z

Z = [ 0]

[ 0]

[ V1]We can use these variables to recreate the circuit equations. To get the left side of the equations we just multiply A*X:

>> A*X ans = [ 1/R1*v_1-1/R1*v_2+I_V1] [ -1/R1*v_1+(1/R1+1/R2+1/R3)*v_2] [ v_1]or, in a slightly easier to read form:

>> pretty(A*X) [ v_1 v_2 ] [ --- - --- + I_V1 ] [ R1 R1 ] [ ] [ v_1 / 1 1 1 \ ] [- --- + |---- + ---- + ----| v_2] [ R1 \ R1 R2 R3 / ] [ ] [ v_1 ]The left side of the equation is given by Z:

Z = [ 0] [ 0] [ V1]Using the information above, we can get any of the MNA equations. To get the equations for node 2, simply take the 2nd row of the right and left sides of the equations:

## Example 4: A More Complex Circuit

We can also apply the program to more complex circuits, such as the following (Example 3 from the MNA Examples page with values given to each component) (with nodes already labeled, and the currents through the voltage sources also labeled for clarity):

The netlist for this circuit is given by

Vg 1 0 4 Vx 3 2 6 R1 1 2 1 R2 2 0 4 R3 3 0 2 It 1 2 1I entered this into a file and named it

Example4.cir. To analyze the circuit we proceed as before, by setting thefnamevariable, and starting the program:>> fname='Example4.cir'; >> scam Started -- please be patient. Done! Elapsed time = 2.544 seconds. Netlist Vg 1 0 4 Vx 3 2 6 R1 1 2 1 R2 2 0 4 R3 3 0 2 It 1 2 1 Solved variables: [ v_1] [ v_2] [ v_3] [ I_Vg] [ I_Vx]We can solve for the voltage at node 2 either symbolically or numerically:

>> v_2 v_2 = R2*(It*R1*R3+Vg*R3-Vx*R1)/(R2*R3+R1*R3+R1*R2) >> eval(v_2) ans = 1.1429We can find the current through R1 (symbolically or numerically):

>> (v_1-v_2)/R1 ans = Vg-R2*(It*R1*R3+Vg*R3-Vx*R1)/(R2*R3+R1*R3+R1*R2) >> eval(ans) ans = 2.8571We can find the MNA equation for node 2:

>> q=A*X q = [ 1/R1*v_1-1/R1*v_2+I_Vg] [ -1/R1*v_1+(1/R1+1/R2)*v_2-I_Vx] [ 1/R3*v_3+I_Vx] [ v_1] [ -v_2+v_3] >> pretty(q(2)) v_1 / 1 1 \ - --- + |---- + ----| v_2 - I_Vx R1 \ R1 R2 / >> Z(2) ans = Itor, in a cleaner format

## Example 5: Op Amps w/ resistors

The next example shows an op-amp with 2 resistors in the standard inverting configuration along with its netlist (

Example5.cir).

CircuitNetlist

Vin 3 0 Symbolic R1 1 3 Symbolic R2 2 1 Symbolic OAmp 0 1 2Since we don't have values for the components in the circuit, they are declared to be "Symbolic". We can now solve this circuit to determine the gain between Vin and node 2.

>> fname='Example5.cir'; >> scam Started -- please be patient. Done! Elapsed time = 1.633 seconds. Netlist Vin 3 0 Symbolic R1 1 3 Symbolic R2 2 1 Symbolic OAmp 0 1 2 Solved variables: [ v_1] [ v_2] [ v_3] [ I_Vin] [ I_OAmp] >> v_2/Vin ans = -1/R1*R2Some notes on this circuit:

- If a value is not given for an element, the value can be declared symbolic (as above). However this is not necessary, the user must simply ensure that the value is not a number -- so instead of "Symbolic", the value of R1 could have been "zyx" or even left blank.
- Because of the way we have used MNA to handle op-amps, the circuit above would give the exact same results even if the input terminals were switched. In practice this would not work. It is the responsibility of the user to make sure negative feedback exists.

## Example 6: Converting the symbolic transfer function to a MATLAB object.

Often we would like to take our results and use them in MATLAB for other calculations. this is especially true when working with transfer functions. The example below shows how this can be accomplished. This circuit is from the MNA with Capacitors and Inductors page with values given to each component, with nodes already labeled, and the currents through the voltage sources also labeled for clarity. The netlist is called

Example6.cir.

CircuitNetlist

Vin 3 0 Symbolic R2 3 2 1000 R1 1 0 1000 C1 1 0 1E-6 C2 2 1 10E-6 L1 1 0 0.001Now let's solve and get the transfer function symbolically

>> fname='Example6.cir'; >> scam Started -- please be patient. Done! Elapsed time = 1.583 seconds. Netlist Vin 3 0 Symbolic R2 3 2 1000 R1 1 0 1000 C1 1 0 1E-6 C2 2 1 10E-6 L1 1 0 0.001 Solved variables: [ v_1] [ v_2] [ v_3] [ I_Vin] >> v_2/Vin %Find transfer function between input and node 2 ans = (s*L1+s^2*C1*R1*L1+s^2*C2*R1*L1+R1)/(s*L1+s^2*L1*C2*R2+s^2*C1*R1*L1+s^3*C1*R1*L1*C2*R2+s^2*C2*R1*L1+R1+R1*s*C2*R2) >> collect(ans) %Collect similar terms ans = ((C2*R1*L1+C1*R1*L1)*s^2+s*L1+R1)/(s^3*C1*R1*L1*C2*R2+(C2*R1*L1+L1*C2*R2+C1*R1*L1)*s^2+(L1+R1*C2*R2)*s+R1) >> pretty(ans) %Pretty print it 2 / 3 ((C2 R1 L1 + C1 R1 L1) s + s L1 + R1) / (s C1 R1 L1 C2 R2 / 2 + (C2 R1 L1 + L1 C2 R2 + C1 R1 L1) s + (L1 + R1 C2 R2) s + R1)We can also get a numerical transfer function

>> eval(v_2/Vin) ans = (1/1000*s+11/1000000*s^2+1000)/(10001/1000*s+21/1000000*s^2+1/100000000*s^3+1000)While this answer is correct, it is not in a very convenient form, and we can't do any actual simulation with it. However we can easily convert the expression to a MATLAB transfer function object. First we separate out the numerator and denominator, and then convert them to MATLAB polynomials.

>> [n,d]=numden(eval(v_2/Vin))

n = 100000*s+1100*s^2+100000000000

d = 1000100000*s+2100*s^2+s^3+100000000000

>> mySys=tf(sym2poly(n),sym2poly(d))

Transfer function:

1100 s^2 + 100000 s + 1e011

--------------------------------

s^3 + 2100 s^2 + 1e009 s + 1e011We are now free to perform any of the MATLAB function relating to polynomials. Shown below are a step response and a Bode plot.

Step ResponseBode Plot>> step(mySys) >> bode(mySys)

## Example 7: Finding the current in a wire

Now consider the case of finding the current through a wire. In particular, consider the previous circuit:

We would like to find the current shown. How do we do this. One way would be to find each current into and out of node 1 and solve for the unknown current. Another way is to introduce a voltage source of zero volts (i.e., a short circuit), and its concomitant node. The netlist is

Example7.cir.

CircuitNetlist

Vin 3 0 Symbolic R2 3 2 1000 R1 1 0 1000 C1 4 0 1E-6 C2 2 4 10E-6 L1 1 0 0.001 Vsc 4 1 0This voltage source has no effect on the circuit, but forces the computation of the current I_Vsc.

>> fname='Example7.cir'; >> scam Started -- please be patient. Done! Elapsed time = 3.014 seconds. Netlist Vin 3 0 Symbolic R2 3 2 1000 R1 1 0 1000 C1 4 0 1E-6 C2 2 4 10E-6 L1 1 0 0.001 Vsc 4 1 0 Solved variables: [ v_1] [ v_2] [ v_3] [ v_4] [ I_Vin] [ I_Vsc] >> eval(I_Vsc) ans = -1/100000*(-1/1000*s-1000)*s*Vin/(10001/1000*s+21/1000000*s^2+1/100000000*s^3+1000) >> simplify(ans) ans = (s+1000000)*s*Vin/(1000100000*s+2100*s^2+s^3+100000000000) >> pretty(ans) (s + 1000000) s Vin ------------------------------------------ 2 3 1000100000 s + 2100 s + s + 100000000000

This is the end of my description of SCAM. We have come a long way from some simple circuit theory, to a description of Modified Nodal Analysis (along with an algorithm for applying MNA - including reactive elements and op-amps), to this page describing a MATLAB tool that performs MNA on a circuit, given a netlist.

I hope you find SCAM to be a useful tool. Please contact me with any comments.