RLC Steady State Consider an RLC series circuit connected to an AC voltage sin(o

Question

RLC Steady State Consider an RLC series circuit connected to an AC voltage sin(omega d t) where is the voltage amplitude and omega d is the driving angular frequency. It was shown in class that when Kirchhoff's loop rule is used on this circuit the following differential equation results: where i is the current in the circuit and q is the charge on the capacitor. When this equation is differentiated with respect to time and the relation i = dq/dt is used, we have the second order equation: L d2i/dt2 + R di/dt + 1/Ci = omega d cos(omega) dt In this exercise you will use calculus and algebra to find the steady-state solution for the current i and show that the results are the same as those obtained using phasors. To receive credit you must type your solution using an equation editor. Solve the problems below by hand first, then type them in Microsoft Word using an equation editor (or you may use other suitable software). Those of you who have had (or are taking) a course in differential equations will recognize this procedure as the method of undetermined coefficients. The steady-state solution of (2) is a particular solution you will find using i = A sin (omega d t) + B cos(omega d t) where A and B are coefficients to be determined {not by initial conditions but by the requirement that i satisfy Equation (2)). Substitute (3) into Equation (2), differentiating appropriately. Present this result as Equation 4. Collect the sum of all the coefficients of sin (omega d t) on the left of Equation 4 and set this equal to zero. (Why?) Next, collect the sum of all the coefficients of cos(omega d t) on the left of Equation 4 and set this equal to (Why?) The operations of step 2 above give you two equations in the two unknown coefficients A and B. Put these equations in standard form (recall Experiment EM-8) and label these as Equations 5 and 6. Solve (5) and (6) using Cramer's rule. (Hint: First show that Delta = omega 2d Z2 where Z is the circuit's impedance.) Show that and where X = XL - XC. (Note the order of terms in X.) Substitute these results back in Equation (3) above and call this result Equation 7. (XL and XC are, of course, the inductive reactance of the inductor and the capacitive reactance of the capacitor, respectively.) Next define a phase angle using the diagram shown at the right. Show that your Equation 7 from step 4 can then be written as , as stated in class. (Hint: Use the diagram and an appropriate trigonometric identity.) Finally, verify that the formulas for current amplitude I and phase angle are the same as those found using the method of phasors.

Explanation / Answer

Transient Response Circuits

A circuit that has an AC source (or a number of AC sources) of the type:

(i.e. v(t) = 120 cos (t + 45) V to V = 120 45 V)

Transient Response Circuits:

Any circuit with time variant circuit elements (L and/or C) that has a DC source of the type:

The two general types we have covered are:

The Natural Response circuits are those that have energy storing elements that have charged up to an initial value (L or C), then we find the rate at which a circuit parameter (i(t), or v(t)) naturally decays to a final value.

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The Step Response circuits are those that have energy storing elements that may (or may not) have charged up to an initial value (L or C), then we find the rate at which a circuit parameter changes value (i(t), or v(t)) to a final value (usually non-zero).

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We have covered three types of Transient Response circuits so far:

NOTE:

To find voltage across an inductor, first find the current as a function of time (across the inductor) then use:

vL(t) = L diL(t)/dt to find the voltage.

To find current across a capacitor, first find the voltage as a function of time (across the capacitor) then use:

iC(t) = C dvC(t)/dt to find the current.

RLC circuits are resolved by finding a second order ordinary differential equations characteristic roots.

Although frequencies are associated with RLC circuits, impedances have no meaning, because:

RLC Transient Response circuits are not the same as Steady State AC circuits!

The frequencies derived from the roots of the defining equation are simply the natural frequencies associated with the particular circuit - don't confuse those with the driving frequency () of an AC circuit!

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