3.5 Series and parallel circuits  (Page 3/3)

Let $R_{\mathrm{eq}}$ denote the equivalent resistance of the parallel combination of $R_2$ and $R_L$ . Using $R_1$ 's v-i relation, the voltage across it is $v_1=\frac{R_1{v}_{\mathrm{out}}}{R_{\mathrm{eq}}}$ . The KVL equation written around the leftmost loop has $v_{\mathrm{in}}=v_1+v_{\mathrm{out}}$ ; substituting for $v_1$ , we find $$v_{\mathrm{in}}=v_{\mathrm{out}}(\frac{R_1}{R_{\mathrm{eq}}}+1)$$ or $$\frac{v_{\mathrm{out}}}{v_{\mathrm{in}}}=\frac{R_{\mathrm{eq}}}{R_1+R_{\mathrm{eq}}}$$

Thus, we have the input-output relationship for our entiresystem having the form of voltage divider, but it does not equal the input-output relation of the circuit without the voltage measurement device. We can notmeasure voltages reliably unless the measurement device has little effect on what we are trying to measure. We should lookmore carefully to determine if any values for the load resistance would lessen its impact on the circuit. Comparing theinput-output relations before and after, what we need is $R_{\mathrm{eq}}\approx R_2$ . As $R_{\mathrm{eq}}=(\frac{1}{R_2}+\frac{1}{R_L})^{-1}$ , the approximation would apply if $\gg (\frac{1}{R_2}, \frac{1}{R_L})$ or $\ll (R_2, R_L)$ . This is the condition we seek:

We want to find the total resistance of the examplecircuit. To apply the series and parallel combination rules, it is best to first determine the circuit's structure: What isin series with what and what is in parallel with what at both small- and large-scale views. We have $R_2$ in parallel with $R_3$ ; this combination is in series with $R_4$ . This series combination is in parallel with $R_1$ . Note that in determining this structure, we started away from the terminals, and worked toward them. In most cases, this approach works well; try itfirst. The total resistance expression mimics the structure: $$R_T=R_1\parallel (R_2\parallel R_3+R_4)$$ $$R_T=\frac{R_1{R}_2{R}_3+R_1{R}_2{R}_4+R_1{R}_3{R}_4}{R_1{R}_2+R_1{R}_3+R_2{R}_3+R_2{R}_4+R_3{R}_4}$$ Such complicated expressions typify circuit "simplifications." A simple check for accuracy is the units: Each component ofthe numerator should have the same units (here $\Omega ^3$ ) as well as in the denominator( $\Omega ^2$ ). The entire expression is to have units of resistance; thus,the ratio of the numerator's and denominator's units should be ohms. Checking units does not guarantee accuracy, but cancatch many errors.

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Another valuable lesson emerges from this example concerning the difference between cascading systems and cascading circuits. Insystem theory, systems can be cascaded without changing the input-output relation of intermediate systems. In cascadingcircuits, this ideal is rarely true unless the circuits are so designed . Design is in the hands of the engineer; he or she must recognize what have come to be known asloading effects. In our simple circuit, you might think that making the resistance $R_L$ large enough would do the trick. Because the resistors $R_1$ and $R_2$ can have virtually any value, you can never make the resistance of your voltage measurement device big enough. Said another way, a circuit cannot be designed in isolation that will work in cascade with all other circuits . Electrical engineers deal with this situation through the notion of specifications : Under what conditions will the circuit perform as designed? Thus, you will find thatoscilloscopes and voltmeters have their internal resistances clearly stated, enabling you to determine whether the voltageyou measure closely equals what was present before they were attached to your circuit. Furthermore, since our resistorcircuit functions as an attenuator, with the attenuation (a fancy word for gains less than one) depending only on the ratioof the two resistor values $\frac{R_2}{R_1+R_2}=(1+\frac{R_1}{R_2})^{-1}$ , we can select any values for the two resistances we want to achieve the desired attenuation. Thedesigner of this circuit must thus specify not only what the attenuation is, but also the resistance values employed so thatintegrators—people who put systems together from component systems—can combine systems together and have achance of the combination working.

[link] summarizes the series and parallel combination results. Theseresults are easy to remember and very useful. Keep in mind that for series combinations, voltage and resistance are the keyquantities, while for parallel combinations current and conductance are more important. In series combinations, thecurrents through each element are the same; in parallel ones, the voltages are the same.

Series and parallel combination rules

Series combination rule

Parallel combination rule