Thevenin & Norton Equivalent Networks

Thevenin and Norton equivalents compress a linear network as seen from one port. They answer a precise question: what voltage-current relation will any external load encounter at these two terminals? Everything internal that does not affect that relation can be eliminated.

The two forms #

The Thevenin form is $V_{th}$ in series with $Z_{th}$. The Norton form is $I_N$ in parallel with the same $Z_{th}$:

$$I_N=\frac{V_{th}}{Z_{th}},\qquad V_{th}=I_NZ_{th}.$$

For the declared port polarity and output current direction:

$$V=V_{th}-IZ_{th}.$$

Finding the parameters #

1. Remove the load but keep the rest of the network intact.
2. Solve the open-circuit voltage: $V_{th}=V_{oc}$.
3. Find $Z_{th}$ by one of these valid methods:
   • turn off independent sources and reduce the impedance;
   • apply a test voltage/current and use $Z_{th}=V_t/I_t$;
   • use $Z_{th}=V_{oc}/I_{sc}$ when a short-circuit calculation is safe and meaningful.

Dependent sources must remain active. A test source is the general method because it measures the port slope while preserving their control paths.

Load behavior #

For a resistive load in DC:

$$I_L=\frac{V_{th}}{R_{th}+R_L},\qquad V_L=I_LR_L.$$

For AC steady state, replace resistance by complex impedance. The same theorem applies to any linear network at one frequency.

The strongest verification is to connect two different loads and show that the full network and equivalent produce the same terminal voltage and current for both.

The short-circuit current is safe to calculate algebraically even when physically shorting the actual source would be unsafe. Experimental identification should instead use two or more benign loads and fit the terminal line.