# 8. Complete Nodal Analysis > [!figures] Figure I: LTspice Circuit Used for Simulation and Measurement > ![[Pasted image 20261001015813.png|center|400]] > [!table] LTspice Node Voltages > > | Node | Voltage [V] | > | --- | ---: | > | A | 5.000 | > | B | 3.055 | > | C | 2.515 | > | D | 1.326 | > | E | 0 | <div style="page-break-after: always;"></div> > [!table] LTspice Component Currents > > | Component | Current Magnitude [mA] | > | --- | ---: | > | $R_1$ | 8.841 | > | $R_2$ | 3.602 | > | $R_3$ | 3.602 | > | $R_4$ | 5.239 | > | $R_5$ | 8.841 | > | $V_s$ | 8.841 | - The LTspice simulation agreed with the hand analysis. - Negative current signs in LTspice indicate that the simulator's reference direction is opposite the selected conventional-current direction. > [!figures] Figure II: LTspice DC Operating-Point Results > ![[Pasted image 20261001015843.png|center|500]] > [!table] Measured Node Voltages > > | Node | Voltage [V] | > | --- | ---: | > | A | 4.993 | > | B | 3.120 | > | C | 2.711 | > | D | 1.370 | > | E | 0 | > [!table] Measured Component and Source Currents > > | Component | Current [mA] | > | --- | ---: | > | $R_1$ | 8.87 | > | $R_2$ | 3.57 | > | $R_3$ | 3.55 | > | $R_4$ | 1.73 | > | $R_5$ | 8.84 | > | $V_s$ | 8.84 | > [!calculation] Node B Voltage Percent Error > > $\%\text{ error}= > \left|\frac{3.120-3.055}{3.055}\right|\cdot100 > =2.13\%$ > [!calculation] Node D Voltage Percent Error > > $\%\text{ error}= > \left|\frac{1.370-1.326}{1.326}\right|\cdot100 > =3.31\%$ # 9. Equivalent Circuits > [!figures] Figure III: Thevenin and Norton Equivalent Circuits > ![[Pasted image 20261001015945.png|center|300]] > ![[Pasted image 20261001015952.png|center|300]] > [!table] Calculated Equivalent-Circuit Parameters > > | Parameter | Value | > | --- | ---: | > | $V_{Th}$ | $2.357\text{ V}$ | > | $R_{Th}$ | $326.34\Omega$ | > | $I_N$ | $7.27\text{ mA}$ | $ I_N=\frac{V_{Th}}{R_{Th}} $ $ I_N=\frac{2.357}{326.34} =0.00722\text{ A} \approx7.22\text{ mA} $ > [!figures] Figure IV: Thevenin and Norton LTspice Data > ![[Pasted image 20261001020053.png|center|300]] > ![[Pasted image 20261001020101.png|center|300]] $ I_{Load}=\frac{V_{Th}}{R_{Th}+R_L} $ $ I_{Load} = \frac{2.357}{326.34+330} \approx3.59\text{ mA} $ $ V_{Load}=I_{Load}R_L $ $ V_{Load} = (3.59\text{ mA})(330\Omega) \approx1.19\text{ V} $ $ V_{R_3}=V_C-V_D $ $ V_{R_3}=2.515-1.326=1.189\text{ V} $ $ I_{R_3}=3.602\text{ mA} $ The equivalent circuit therefore reproduces the original LTspice load behavior closely. > [!table] Measured Open-Circuit Voltage and Short-Circuit Current > > | Measurement | Measured Value | Theoretical Value | Percent Error | > | --- | ---: | ---: | ---: | > | $V_{OC}$ | $1.199\text{ V}$ | $2.357\text{ V}$ | 49.13% | > | $I_{SC}$ | $3.401\text{ mA}$ | $7.27\text{ mA}$ | 53.22% | > [!calculation] Open-Circuit Voltage Percent Error > > $\%\text{ error}= > \left|\frac{1.199-2.357}{2.357}\right|\cdot100 > =49.13\%$ > [!calculation] Short-Circuit Current Percent Error > > $\%\text{ error}= > \left|\frac{3.401-7.27}{7.27}\right|\cdot100 > =53.22\%$ - The measured $V_{OC}$ and $I_{SC}$ had significantly larger errors than expected. - These errors suggest that the physical open-circuit and short-circuit measurement setups did not exactly reproduce the required terminal conditions. > [!table] Original vs Equivalent Circuit Load Measurements > > | Load [$\Omega$] | $V_{Load}$ Original [V] | $V_{Load}$ Equivalent [V] | Voltage Error | $I_{Load}$ Original [mA] | $I_{Load}$ Equivalent [mA] | Current Error | > | ---: | ---: | ---: | ---: | ---: | ---: | ---: | > | 100 | 0.514 | 0.751 | 46.11% | 5.355 | 7.155 | 33.61% | > | 1,000 | 1.744 | 2.15 | 23.28% | 1.744 | 1.925 | 10.38% | > | 10,000 | 2.115 | 2.753 | 30.17% | 0.223 | 0.215 | 3.59% | - The equivalent and original circuits showed the same general loading behavior. - The physical measurements contained noticeable differences, especially for the $100\Omega$ load. - The current measurements for the $10k\Omega$ load were particularly close, with only $3.59\%$ error.