I'm doing this exercise (Exercise 7.21 in Fundamentals of Physics II by R. Shankar). It wants me to evaluate a circuit with an AC source series with DC source series with two parallel impedances. The details are unimportant for the question. The book says to use superposition which I was able to use to get the right answer. But, I don't understand why I cannot just write out the KVL equations with all the complex voltage, real voltage, currents, and impedances and take the real part at the end. This was how it was taught in the book. The only problematic part is that the AC and DC sources are in series. But mathematically, I don't see any problems with just adding them. For this circuit then, $$-V_C-V_R+I_2Z_2 = 0$$ $$-I_2Z_2+I_3Z_3=0$$ where V_C is complex, V_R is real, and all the currents and impedances may be complex.
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- 1\$\begingroup\$ Please include the schematic, and indicate on the schematic what the symbols (\$V_C\$, \$V_R\$, etc.) refer to. \$\endgroup\$The Photon– The Photon2024-10-02 17:51:42 +00:00Commented Oct 2, 2024 at 17:51
- \$\begingroup\$ Sorry, I'm not a professional. I guess I can draw it on my whiteboard and put a photo here. I don't have a working knowledge of schematic drawing software. \$\endgroup\$procommania– procommania2024-10-02 18:11:59 +00:00Commented Oct 2, 2024 at 18:11
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7 I don't understand why I cannot just write out the KVL equations with all the complex voltage, real voltage, currents, and impedances and take the real part at the end.
In an AC analysis there's no way express, for example, \$v(t) = 10\ V+ 0.25 \cos(2\pi 100 t + \pi/4)\ V\$. An AC analysis can only deal with one frequency at a time.
When you're doing AC analysis, you solve for every frequency component separately.
That means the \$V_R\$ when you're solving the DC component of the circuit is not the same as \$V_R\$ when you're solving for the AC component at 100 Hz. And that is not the same as when you're solving for the AC component at 101 Hz. You can't just say that a 10 V DC voltage plus a 0.25 V AC voltage with phase \$\pi/4\$ is "\$10 + 0.25\angle\pi/4\$" in an AC analysis because the 10 V component and \$0.25\angle\pi/4\$ parts are different frequency components.
- \$\begingroup\$ Thank you. I think as a rule it sort of makes sense. But, there is nothing mathematically that prevents us from adding all sorts of complex numbers (just use Euler's identity and add the real and complex parts). Is it correct to say that there is no physical interpretation of why that would spit out gibberish? My logic was that it's the real part that matters. Regardless of frequency, the real parts of the voltages (i.e. the cos part added with V_R) can be added with no problem and KVL will still work. \$\endgroup\$procommania– procommania2024-10-02 18:10:30 +00:00Commented Oct 2, 2024 at 18:10
- \$\begingroup\$ RE "Is it correct to say that there is no physical interpretation of why that would spit out gibberish?". That's backwards. Sure you can add any two complex numbers with the same units. You should not think it will produce anything but gibberish unless there is a physical reason for the sum to be meaningful. You should assume gibberish is the default, and a meaningful result is only possible if there is a sensible physical interpretation. \$\endgroup\$The Photon– The Photon2024-10-02 18:21:50 +00:00Commented Oct 2, 2024 at 18:21
- \$\begingroup\$ RE "the real parts of the voltages ... can be added with no problem and KVL will still work", Yes you'll still be able to formulate equations, but those equations won't describe the circuit you want --- they'll be equations for a different circuit where all those voltages represent signals at the same frequency. \$\endgroup\$The Photon– The Photon2024-10-02 18:23:01 +00:00Commented Oct 2, 2024 at 18:23
- \$\begingroup\$ @procommania, think about what would happen if your circuit had two AC sources, one at 100 Hz and one at 150 Hz. Would you just add all the components together? What value would you use for \$Z_C\$? \$\endgroup\$The Photon– The Photon2024-10-02 18:26:53 +00:00Commented Oct 2, 2024 at 18:26
- \$\begingroup\$ I see. Resistors are fine since they're not frequency dependent; the real part of the voltage when added is still valid; and ohm's law still works in the reals. But, capacitors and inductors are frequency dependent, so we must have a singular frequency in order to work with the complex form ohm's law. Superposition of cosines of different frequencies cannot be represented by one complex number. Thanks for the explanation! \$\endgroup\$procommania– procommania2024-10-02 18:37:38 +00:00Commented Oct 2, 2024 at 18:37