Electronics: A Guide for Physicists (DRAFT -- in development)

Let’s assume that if we apply the branch circuit analysis method, we end up with $N_1$ independent junction law equations and $N_2$ loop law equations. Each junction law equation will have the form

where $a_{rn}$ and $b_{rn}$ have values of +1, -1, or 0 depending on whether the currents $i_n$ and $I_{s_n}$ are entering, exiting, or not present at a junction and $r$ is an index counting over KCL equations.

We will also end up with $N_2$ independent loop law equations. All loops chosen for KVL analysis must be absent known current sources $I_{s_n}\text{.}$ Since the current is already known on circuit legs containing circuit sources, so this restriction will not cost us anything. Each of these loop law equations will have the form

The variables $m_{rn}$ have values of either +1, or -1 depending on the battery orientation relative to the direction travelled around the loop, or a value of 0 for batteries not present in a given loop. Variable $c_{rn}$ has units of resistance (since each nonzero term in these sums represents a voltage change described by Ohm’s Law). Here, the index $r$ counts over KVL equations.

These $N_1$ instances of (2.6.1) and $N_2$ instances of (2.6.2) can be re-expressed as a matrix equation $\mathbf{A}\mathbf{x}=\mathbf{v}\text{.}$ First, let’s rearrange these equations so that all unknowns are on the left side of the equation and all of the source terms are on the right

We can express these equations in the form $\mathbf{A}\mathbf{x}=\mathbf{v}$ if we let

Now, every element in matrix $\mathbf{A}$ is either a unitless number or a function of resistances only. More to the point, there is no voltage or current dependence in $\mathbf{A}\text{.}$ If we multiply both sides of our matrix equation by ${\mathbf{A}}^{-1}\text{,}$ we get

The solution that is most important to us is $i_{\text{ext}}$ as we wish to examine the relationship between output current and voltage. If we actually perform the calculations necessary to invert $\mathbf{A}\text{,}$ use matrix multiplication to find an expression for $i_{\text{ext}}\text{,}$ and then group source terms and rename constants multiplying each source ($v_{\text{ext}},V_s,I_s$), the resulting solution is of the form

where all ${\alpha }_n$ and ${\beta }_n$ terms are functions of resistances only and are thus constants. This result demonstrates that $i_{\text{ext}}$ is a linear function of $v_{\text{ext}}$ with slope ${\alpha }_0$ and a y-intercept that equals the two summations on the right side of the equation. So, we have now shown that any circuit comprised of solely linear circuit elements is itself a linear circuit.

Now that the linearity of our complicated circuit has been demonstrated, it follows that any simplified circuit that follows an identical linear relationship will have behavior of $i_{\text{ext}}$ and $v_{\text{ext}}$ that is identical to the original circuit, regardless of the load that is connected across the output terminals. One such simplified circuit is the Thevenin equivalent circuit of Figure 2.6.2. The Thevenin equivalent circuit will reproduce the output i-v relationship of our complicated circuit if

In a similar way, we can see that the Norton equivalent circuit can also demonstrate identical output i-v behavior when

If one wishes to find Thevenin and Norton equivalent parameters analytically, it is often easier to follow this procedure:

Sometimes, one does not have a circuit schematic but only has the physical circuit. In this case, a digital multimeter can be used to physically perform the following procedure: