The voltage across the 30-k Ωresistor in Fig. 25.33 is...

Key Concepts

Measurement Accuracy in Circuit Analysis

Accurate circuit measurements require consideration of the instrument’s impact on the circuit, particularly when the measuring device does not have an infinitely high impedance. Recognizing and compensating for such 'loading effects' is crucial to obtain reliable voltage readings and correctly interpret circuit behavior.

Non-Ideal Voltmeter Loading

Real voltmeters have a finite input impedance rather than an infinite one, and when connected to a circuit, they draw a small amount of current. This results in a parallel connection with the component under test, effectively changing the circuit’s resistance and hence the voltage that is actually measured. This concept is critical to understand measurement errors introduced by the measuring device.

Parallel Resistance Calculation

When a voltmeter is connected across a resistor, the resistor and the voltmeter input impedance form a parallel combination. Calculating the effective resistance of this parallel network is essential for determining the actual voltage across the resistor as seen by the voltmeter, which can differ from the voltage that would be present without the meter connected.

Voltage Divider Principle

The voltage divider rule allows for the calculation of voltage drops across resistors in series circuits. When a measuring instrument affects the circuit by creating a parallel network with one of the resistors, the voltage divider must be reanalyzed with the modified resistor value. This principle is key to understanding how the measured voltage deviates from the ideal value under non-ideal measurement conditions.

Transcript

00:01 So the goal of this problem is to calculate the voltage that different volt meters will read when attached to the 30k resistor.

00:08 And we know that volt meters have to be attached in parallel with the resistor in order to get a reading.

00:14 And part a, the internal resistance of the volt meter is 50 ,000 oms.

00:25 And since we know they're connected in parallel to the 30k, let's find the resistance between those two.

00:31 So in order to do that, we use the equation for two parallel resistors, so r1 times r2, divided by r1 plus r2.

00:44 So r1 will be the 30k resistor, r2, the 50k, divided by r1 plus r2 is 80k.

01:00 So their total resistance is 1 .88 times 10 to the 4.

01:07 Fourth.

01:08 Now let's look at the other side of the circuit where this 40k and this 40k resistor are.

01:13 And they're also in parallel with each other.

01:16 So let's find their combined resistance.

01:18 So rp is equal to 40k times 40k divided by 40k plus 40k is again 80k.

01:34 So their combined resistance is simply two times 10 to the fourth oms.

01:42 So now we have two resistors that are now in series.

01:45 With one another.

01:47 So we can find the r equivalents as 1 .88 times 10 to the fourth.

01:58 It's the first resistor plus two times 10 to the fourth.

02:05 So the combined resistance of the entire circuit is 3 .88 times 10 to the 4th.

02:16 And now we can find the current of the entire circuit since we know the voltage of the battery, which is 100, volts and we know that current i is equal to the voltage divided by the resistance.

02:31 So plug in our numbers, 100 volts, divided by 3 .88 times 10 to the fourth.

02:40 And so the current of the circuit is 0 .0025 amps.

02:51 So now we can use this current to find the voltage that the voltmeter will read at the 30k resistor.

02:59 So we know from oms law, voltage is equal to the current times of resistance.

03:06 So the current we just found, which is .0025 times the combined resistance of the voltmeter and the 30k, which we said was 1 .88 times 10 to the fourth.

03:28 And so the voltage that the volt meter will read with the 50k internal resistance is v is equal to 48 .52 volts...

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