A voltmeter is used to measure Vo in the circuit in Fig....

A voltmeter is used to measure $V_{o}$ in the circuit in Fig. $2.122 .$ The voltmeter model consists of an ideal voltmeter in parallel with a $100-\mathrm{k} \Omega$ resistor. Let $V_{s}=40 \mathrm{V}, R_{s}=10 \mathrm{k} \Omega,$ and $R_{1}=20 \mathrm{k} \Omega$. Calculate $V_{o}$ with and without the voltmeter when (a) $R_{2}=1 \mathrm{k} \Omega$ (b) $R_{2}=10 \mathrm{k} \Omega$ (c) $R_{2}=100 \mathrm{k} \Omega$

A voltmeter is used to measure $V_{o}$ in the circuit in Fig. $2.122 .$ The voltmeter model consists of an ideal voltmeter in parallel with a $100-\mathrm{k} \Omega$ resistor. Let $V_{s}=40 \mathrm{V}, R_{s}=10 \mathrm{k} \Omega,$ and $R_{1}=20 \mathrm{k} \Omega$. Calculate
$V_{o}$ with and without the voltmeter when
(a) $R_{2}=1 \mathrm{k} \Omega$
(b) $R_{2}=10 \mathrm{k} \Omega$
(c) $R_{2}=100 \mathrm{k} \Omega$

Key Concepts

Voltage Divider Rule

This fundamental principle in circuit analysis allows you to determine the voltage across a particular resistor in a series circuit. It works by taking the ratio of a resistor’s value to the total series resistance and multiplying by the total voltage applied. This concept is essential when calculating voltages in any network where resistors are connected in series and is used extensively in analyzing circuits where the voltage division is affected by other components or measurement instruments.

Instrument Loading Effect

The loading effect refers to the phenomenon where the finite input resistance of a measuring instrument, such as a voltmeter, alters the circuit conditions when connected. An ideal voltmeter would not affect the circuit, but real voltmeters have internal resistances that, when placed in parallel with the circuit element under test, divert some current. This change results in a measurement error, an important consideration when evaluating practical circuits.

Equivalent Resistance in Parallel Networks

Calculating the equivalent resistance of resistors connected in parallel is a key concept in circuit analysis. When the voltmeter, which includes an internal resistor, is connected in parallel with a circuit component, it is necessary to compute their equivalent resistance to understand how the circuit’s performance is altered. This concept is central to accurately determining the effect that additional parallel resistive paths have on the overall circuit behavior.

Circuit Analysis Under Measurement Conditions

This concept involves examining how the act of measurement itself influences circuit parameters. In practical circuit analysis, linking measurement devices alters the original circuit, primarily through their input resistance, which can affect voltage readings and current distributions. Understanding this interaction is critical for designing circuits that require accurate measurements and for interpreting observed discrepancies between ideal and measured values.

Transcript

00:01 In this problem of dc circuits we have given a figure let me show you first.

00:06 So here is the figure and we have given that a volt meter is used so here is the voltmeter so this is called voltmeter so this is used to measure the voltage v knot in the circuit.

00:21 Now the volt motor model consists of an ideal voltmeter in parallel with 100 kilo -oom so this is the 100 kilombo oom register and now we have given that vs so this is vs is equals to 40 volt so this is 40 volt and r s so this is r s is equals to 10 kilo om so this is 10 kilo om and also we have given that r1 is equals to 20 kilo oom so these are the data and now we have to calculate v0 width and without the voltmeter so first we have to calculate v -0 with and without when r2 is equals to so r2 is equal to 1 kilo -oom so here this is the v -0 so we have to find the value of v0 so v0 is with the voltmeter in place so this would be rm so rm is equal to 100 so we can say here rm is equals to 100 kilo -oom so we have the formula which is v -0 is equals to since these two are 2 and r this is or we can say rm are in parallel and these two r s and r 1 are in series and then again here these two are in parallel so v -0 is equal to this would be r2 with this would be rm and divided with say this would be r1 plus r s and the combination of the parallel so plus combination is r2 plus rm and this is multiplied with the vs so this is with the voltmeter in place and without so this is the data without volt meter so without volt meter v0 is equals to we have the formula which is so this would be r1, r2 and rs are in series.

02:36 So this would be, we have to find the resist, this is the voltage between r2.

02:40 So this would be r2 divide with r1 plus r2 plus rs multiplied with vs.

02:47 So now in first case putting all these values, so here with voltmeter, we have to put r2 is equal to this term.

02:57 So we are getting v0 is equals to this would be here r2 parallel with rm.

03:06 So r2 is equals 1 kilo -oom and r1 is equal to 100.

03:09 So this would be 100 divided with 1001 and this would be in kilo -oom.

03:16 So putting these values together...

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