Thevenin Equivalent Simulator — Maximum Power Transfer

Thevenin's theorem guarantees that any linear two-terminal network composed of resistors and independent sources (voltage or current) can be replaced, when viewed from a chosen terminal pair, by an open-circuit voltage $V_{th}$ in series with an equivalent internal resistance $R_{th}$. $V_{th}$ is the voltage measured with the terminals open, and $R_{th}$ is the resistance seen from the terminals when every independent internal source is "zeroed" (voltage sources shorted, current sources opened).

With a load $R_L$ and a wiring resistance $R_w$ added in series, the total resistance is $R_{total}=R_{th}+R_w+R_L$. The current is $I_L=V_{th}/R_{total}$, the load voltage is $V_L=I_L R_L$, and the load power is $P_L=I_L^2 R_L=V_{th}^2 R_L/(R_{th}+R_w+R_L)^2$. Setting $dP_L/dR_L=0$ yields $R_L=R_{th}+R_w$, the maximum power transfer condition. At that match $P_{\max}=V_{th}^2/(4(R_{th}+R_w))$ and the efficiency $\eta=R_L/(R_{th}+R_w+R_L)$ equals exactly 50%.

With the default values $V_{th}=12$ V, $R_{th}=5$ Ω, $R_L=10$ Ω, $R_w=0$ Ω, the simulator reports $I_L=0.800$ A, $V_L=8.000$ V, $P_L=6.40$ W and $\eta=66.7\%$. At the matched point $R_L=5$ Ω the load power reaches $P_{\max}=12^2/(4\cdot 5)=7.200$ W — sliding R_L to 5 confirms the peak.

Audio power stages: Speakers are sold with nominal impedances (4 Ω, 8 Ω, 16 Ω) chosen to match the output stage of the power amplifier. A mismatch reduces output power and can cause frequency-dependent distortion. Output transformers in tube amplifiers act as impedance converters that match low-impedance speakers to high-impedance plate circuits.

RF and antenna matching: Wireless transmitters insert a matching network (L, π or T) between the output stage and the antenna so that the 50 Ω source impedance is presented with the same load impedance. A mismatch produces reflected power, raises the SWR (standing-wave ratio), and at worst destroys the power amplifier.

Sensor front-ends: Piezoelectric sensors, photodiodes, thermopiles and similar low-signal sources have high output impedance. Rather than maximum power, they are read with a very large load (high-impedance op-amp input) to extract maximum voltage, or with noise matching when SNR matters more than efficiency.

Battery internal resistance measurement: EV and laptop battery packs are diagnosed by internal resistance, a key aging indicator. Connect a known load briefly, measure the voltage drop and compute $R_{th}=(V_{open}-V_{loaded})/I$. BMS firmware tracks the rising R_th as an early warning of cell degradation or thermal anomalies.

The most common error is confusing maximum power transfer with maximum efficiency. At the matched point efficiency is exactly 50% — the other half is dissipated as heat in R_th. Efficiency-first systems such as power grids use R_L much larger than R_th to push η above 95%. Signal-level circuits such as audio outputs or sensor amplifiers match and accept the 50% trade. Compare R_L = 5 and R_L = 100 in the simulator and the trade-off is unmistakable.

Next, forgetting the wiring resistance R_w when matching. Long cables or thin leads can have R_w of the same order as R_th, so the effective source impedance becomes R_th + R_w and the matched load shifts to R_L = R_th + R_w. The V_th measured at the source appears at the load only after a voltage drop along the wire, and signal quality drops with it. Always re-evaluate matching with the wiring included.

Finally, missing the frequency dependence of a Thevenin equivalent. Real circuits contain capacitance and inductance, so the Thevenin impedance Z_th becomes frequency-dependent. A pure-resistance approximation is fine at DC or low frequency, but RF circuits and high-speed digital traces need a matching design that accounts for frequency response — antenna matching networks and PCB trace impedance design are typical examples.