# Circuit Theory/1Source Excitement

Phasors:

:   Called the **\"Steady State Solution\"** solution
:   Sometimes the **\"Steady State Solution\"** resulted in a complete
    solution, sometimes a partial solution.
:   Full **\"Steady State Solutions\"** involved integrals and
    derivatives
:   Partial **\"Steady State Solutions\"** involved non-homogeneous
    ODE
    differential equations
:   The partial **\"Steady State Solution\"** is called the
    **\"Particular solution\"**.
:   The **Laplace Solution** is a complete solution that is identical to
    **Phasor/Complex Frequency** when eliminating operators in a
    differential equation (s or jω), but translates the driving function
    (current or voltage source) into the **Laplace domain** differently.
    The **Laplace Solution** can translate any driving function.
    **Phasors** can only translate sinusoidals. The **Convolution
    Integral** (coming up) is easier than either.

Now we are going to finish solving the some of the incomplete,
particular **\"Steady State Solutions.\"**

:   The non-homogeneous ODE equation is turned **homogeneous** by
    setting the driving function (voltage or current source) to zero and
    then using either the \"Method of undetermined coefficients\" or
    \"Variation of parameters\" to find the overall solution. The method
    illustrated here is the **\"Method of undetermined coefficients\"**.
:   This is called the **\"Transient\"** or **Homogeneous Solution**.
:   The **Homogeneous Solution** is found in the time domain, not phasor
    domain
:   Reduce circuit to thevenin or norton equivalent and solve for a
    single component (or group or components)
:   Add **Particular** + **Homogeneous Solution**, then evaluate
    **constants**.

Click on the numeric solutions to see where they came from. Clicking on
particular solutions or non-diff-eq solutions will bounce back to
phasor examples.

+-------------+-------------+-------------+-------------+-------------+
| Calculus    | diff eq     | LR          | LR Solution | Similar RC  |
+=============+=============+=============+=============+=============+
| ${\        | $\texts     | !          | [           | ![          |
| textstyle d | tyle \surd$ |  | $V_s(t) = 1 |  |
|  \over dt}$ |             |             | 20 \sqrt{2} |             |
| ](Circuit_T |             |             |  cos(377t+1 |             |
| heory/Phaso |             |             | 20^\circ)$\ |             |
| rs/Examples |             |             | $V_s(t) =   |             |
| /Example_7  |             |             | R * i(t) +  |             |
| "wikilink") |             |             | L * {d \ove |             |
|             |             |             | r dt}i(t)$\ |             |
|             |             |             | $i_{        |             |
|             |             |             | S_P}(t) = 1 |             |
|             |             |             | 5.9 cos(377 |             |
|             |             |             | t + 1.73)$] |             |
|             |             |             | (Circuit_Th |             |
|             |             |             | eory/Phasor |             |
|             |             |             | s/Examples/ |             |
|             |             |             | Example_7 " |             |
|             |             |             | wikilink")\ |             |
|             |             |             | \           |             |
|             |             |             | $i         |             |
|             |             |             | _{S_H}(t) = |             |
|             |             |             |   2.5781*e^ |             |
|             |             |             | {-\frac{t}{ |             |
|             |             |             | 0.001}}$ |             |
+-------------+-------------+-------------+-------------+-------------+
| ${\        |             | !          | [           | ![          |
| textstyle d |             |  | $I_s(t) = 1 |  |
|  \over dt}$ |             |             | 20 \sqrt{2} |             |
| ](Circuit_T |             |             |  cos(377t+1 |             |
| heory/Phaso |             |             | 20^\circ)$\ |             |
| rs/Examples |             |             | $V_s(t) =   |             |
| /Example_8  |             |             | R * i(t) +  |             |
| "wikilink") |             |             | L * {d \ove |             |
|             |             |             | r dt}i(t)$\ |             |
|             |             |             | $V_s = 1    |             |
|             |             |             | 814cos(377t |             |
|             |             |             | +2.45)$](Ci |             |
|             |             |             | rcuit_Theor |             |
|             |             |             | y/Phasors/E |             |
|             |             |             | xamples/Exa |             |
|             |             |             | mple_8 "wik |             |
|             |             |             | ilink")\<br |             |
|             |             |             | \\\>\<br    |             |
|             |             |             | \\\>already |             |
|             |             |             | solved \... |             |
|             |             |             | no          |             |
|             |             |             | d           |             |
|             |             |             | ifferential |             |
|             |             |             | equation!   |             |
+-------------+-------------+-------------+-------------+-------------+
| ${\texts   |             | !          | [           | ![          |
| tyle \int}$ |             |  | $V_s(t) = 1 |  |
| ](Circuit_T |             |             | 20 \sqrt{2} |             |
| heory/Phaso |             |             |  cos(377t+1 |             |
| rs/Examples |             |             | 20^\circ)$\ |             |
| /Example_9  |             |             | ${          |             |
| "wikilink") |             |             | dI_s(t) \ov |             |
|             |             |             | er dt}= \fr |             |
|             |             |             | ac{1}{R}{dV |             |
|             |             |             | _s \over dt |             |
|             |             |             | } + \frac{V |             |
|             |             |             | _s}{L}$\<br |             |
|             |             |             | \\\>$i_s    |             |
|             |             |             | (t) = 48.1  |             |
|             |             |             | cos(377t +  |             |
|             |             |             | 0.844)$](Ci |             |
|             |             |             | rcuit_Theor |             |
|             |             |             | y/Phasors/E |             |
|             |             |             | xamples/Exa |             |
|             |             |             | mple_9 "wik |             |
|             |             |             | ilink")\<br |             |
|             |             |             | \\\>\<br    |             |
|             |             |             | \\\>already |             |
|             |             |             | solved \... |             |
|             |             |             | no          |             |
|             |             |             | d           |             |
|             |             |             | ifferential |             |
|             |             |             | equation!   |             |
+-------------+-------------+-------------+-------------+-------------+
| [${\textst  | $\texts     | ![          | [           | ![          |
| yle \int}$] | tyle \surd$ | ](PILR.xcf) | $I_s(t) = 1 | ](SVRC.xcf) |
| (Circuit_Th |             |             | 20 \sqrt{2} |             |
| eory/Phasor |             |             |  cos(377t+1 |             |
| s/Examples/ |             |             | 20^\circ)$\ |             |
| Example_10  |             |             | ${          |             |
| "wikilink") |             |             | dI_s(t) \ov |             |
|             |             |             | er dt}= \fr |             |
|             |             |             | ac{1}{R}{dV |             |
|             |             |             | _s \over dt |             |
|             |             |             | } + \frac{V |             |
|             |             |             | _s}{L}$\<br |             |
|             |             |             | \           |             |
|             |             |             | \\>$V_{S_P} |             |
|             |             |             | (t) = 599 \ |             |
|             |             |             | cos(377t +  |             |
|             |             |             | 3.30)$](Cir |             |
|             |             |             | cuit_Theory |             |
|             |             |             | /Phasors/Ex |             |
|             |             |             | amples/Exam |             |
|             |             |             | ple_10 "wik |             |
|             |             |             | ilink")\<br |             |
|             |             |             | \\\>\<br    |             |
|             |             |             | \\\>$V_{   |             |
|             |             |             | S_H}(t) = - |             |
|             |             |             |  258 * e^{\ |             |
|             |             |             | frac{-t}{0. |             |
|             |             |             | 0001}}$ |             |
+-------------+-------------+-------------+-------------+-------------+

