Alternating Current: Basic Circuits for Applications

Abstract

In this chapter, we take advantage of the simplicity and effectiveness of the symbolic method to solve some simple circuits under sinusoidal excitation. We address both significant examples for the understanding of physical phenomena and circuits of general interest for practical applications. We start with a closer inspection of circuits consisting of a resistance, an inductance and a capacitance (RLC circuits), similar to the circuit used in the previous chapter to introduce the symbolic method. RLC circuits have special importance since they show resonant behavior, a phenomenon ubiquitous in physics and of relevance in fields ranging from elementary mechanics to sub nuclear interactions.

Problems

Problem 1

Compute the transfer function of a low-pass RL filter taking into account the inductor ohmic resistance $R_L$.

Problem 2

Compute the transfer function of a high-pass RL filter taking into account the inductor ohmic resistance $R_L$.

Problem 3

Derive relations (6.14), (6.15) and (6.16).

Problem 4

Compute the parameters of the Thévenin’s equivalent circuit of a series RLC circuit seen from the inductor terminals at the resonance frequency. [A. $Z_{eq}=Q_s^2R+j\omega _0L$; $V_{eq}=Q_sV_g$.]

Problem 5

Compute the equilibrium conditions for the bridge circuit shown in the figure. [A. $R_1R_4=(R_3+R)R_2; R_1L_4=R_2 L_3$.]

Problem 6

Using the series–parallel transformation, compute the equilibrium conditions for the bridge circuit shown in the figure. [A. $R_2/R_4=R_4/R_3+L_4/L_3; \omega ^2=R_4 R_3/L_4 L_3$.]

Problem 7

The Heaviside bridge , shown in the figure, can be used to measure the mutual induction coefficient between the two windings of a transformer. Show that at the equilibrium the following relation holds: $M=(R_1L_3-R_2L_4)/(R_1+R_2)$, independently of the generator frequency.

Problem 8

Compute the complex attenuation of the circuit in the figure, using for example Thévenin’s theorem. Show that it is a bandpass filter with transmission maximum at $\omega _0=1/RC$. Compute the angular frequencies corresponding to the attenuation equal to its peak value divided by $\sqrt{2}$. [A. $A(\omega )=1/(3+j(\omega RC-1/\omega RC)), \omega _{1,2}=\omega _0(\sqrt{13}\pm 3)/{2}$.]

Problem 9

Use the loop method to obtain the equation system describing the circuit shown in the figure. Perform the analysis of its coefficient determinant to show that the circuit is resonant for two values of the angular frequency $\omega$. Show that for these frequencies the impedance seen by the generator is null. [A. $\omega _{1,2}^2=(3\pm \sqrt{5})/(2LC)$.]

Problem 10

Compute the attenuation $A(\omega )$ of the two-port circuit shown in the figure and show that it is always real but divergent at two frequency values. [A. $A(\omega )={1}/[3-(\omega ^2 LC+\frac{1}{\omega ^2 LC})],\omega _{1,2}^2=(3\pm \sqrt{5})/(2LC)$.]

Problem 11

Show that the circuit of previous problem has two resonances at the angular frequencies $\omega _{1,2}^2=(3\pm \sqrt{5})/(2LC)$ for which the impedance seen at the input is null. Show also that this impedance diverges at $\omega =1/\sqrt{LC}$.

Problem 12

Compute the input impedance $Z_{eq}$ of the network shown in the figure choosing the value of the inductance L to cancel its reactance under the hypothesis $R_L\ll 1/\omega C_2$. [A. $Z_{eq}=R_L(1+C_2/C_1)^2; L=(1/C_1+1/C_2)/\omega ^2$.]

Problem 13

* In this problem, the reader can verify that different definitions of resonance given in the literature are not equivalent.

Solve the RLC parallel circuit taking into account the ohmic resistance of the inductor. The diagram of the circuit is obtained by adding a resistance r in series to the inductance L in the circuit shown in the text in Fig. 6.2. Then, compute the transfer function $H(\omega )$ given by the ratio of the output voltage, measured across the capacitor, and the generator voltage. Show that the amplitude of the transfer function has a maximum at the angular frequency $\omega _\mathrm{max}$ given by:

$$\begin{aligned} \omega _\mathrm{max}=\omega _o\sqrt{\sqrt{1+2\frac{r}{R}+2\left( \frac{r}{\omega _oL}\right) ^2}-\left( \frac{r}{\omega _oL}\right) ^2} \end{aligned}$$

where $\omega _o=1/\sqrt{LC}$ is the resonance frequency we obtained with $r=0$. Adopting a McLaurin series expansion in the small parameter r, to second order the previous expression becomes:

$$\begin{aligned} \omega _\mathrm{max}\simeq \omega _o\left( 1+\frac{r}{2R}-\frac{3}{8}\frac{r^2}{R^2}+\cdots \right) \end{aligned}$$

The reader is warned that the calculation of $\omega _\mathrm{max}$ is laborious and requires an appropriate amount of attention in the various steps that lead to the solution.

Show that the admittance of the circuit, as seen by the voltage generator, vanishes at an angular frequency $\omega _X$, different by $\omega _\mathrm{max}$:

$$\begin{aligned} \omega _{X}=\sqrt{\frac{1}{LC}-\left( \frac{r}{L}\right) ^2}=\sqrt{\omega _o^2-\left( \frac{r}{L}\right) ^2}\simeq \omega _o\left( 1-\frac{1}{2}\frac{C}{L}r^2+\cdots \right) \end{aligned}$$

Finally show that the circuit has a normal mode of oscillation at the angular frequency $\omega _{n}$:

$$\begin{aligned} \omega _n=\omega _o\sqrt{1-\frac{L}{4CR^2}+\frac{r}{2R}-\frac{Cr^2}{4L}} \end{aligned}$$

Problem 14

* With reference to the previous problem, evaluate the effect of the resistance r of the inductor on the value of the merit factor Q of the circuit from the width of the attenuation peak. Show that Q is approximatively given by:

$$\begin{aligned} Q=\frac{1}{\omega _o}\frac{R}{L}\simeq \frac{1}{\omega _o}\frac{R}{L+RCr} \end{aligned}$$

Hint: replace the series of r and L with the parallel of $r_{eq}$ and $L_{eq}$ adopting a series-parallel transformation. Both $r_{eq}$ and $L_{eq}$ are frequency dependent:

$$\begin{aligned} r_{eq}=r\left( 1+\frac{\omega ^2L^2}{r^2}\right) \quad L_{eq}=L\left( 1+\frac{r^2}{\omega ^2L^2}\right) \end{aligned}$$

Compute Q in the approximations $r_{eq}\simeq \omega _o^2L^2/r$ and $L_{eq}\simeq L$, both verified at sufficiently high frequency.