Recall that voltage refers to the potential difference between two points in a circuit. A voltmeter has two connection terminals, one typically denoted with a “V” and the second denoted “COM” for the common terminal. These two terminals allow one to measure a difference in potential. A voltage measurement requires us to connect the meter between the two points of interest without disturbing the behavior of the circuit. That is, when we make a voltage measurement, the meter should not change how the circuit operates.

Let’s consider the circuit of Figure 1 consisting of an ideal independent voltage source and three resistors in series.

Figure 1: A series circuit consisting of ideal elements. V_S = 9V, R₁ = 10Ω, R₂ = 20Ω, and R₃ = 30Ω.

Let’s say that we wish to know the voltage difference between nodes B and C, that is the voltage across resistor R₂. Before describing the voltage measurement, we should be clear that each of the four elements shares the same current. The value of I in the circuit is 150 mA in this case (V_S/R_total =9V/60Ω = 0.15A). By Ohm’s law we know that the voltage across a given resistor is the product of the current associated with the resistor and its resistance. It is critically important therefore, that the current in the circuit is not changed when we place the voltmeter in the circuit to measure a potential difference. Voltage measurements are made in parallel with the element or elements across which we want to know the potential difference. Clicking on Figure 1 will show a voltmeter properly connected to measure the voltage drop across resistor R₂.

With Figure 1 showing the voltmeter, notice the current labeled I_meter. For the meter to not alter the current in the circuit, I_meter must equal zero. This observation gives us insight into the model for an ideal voltmeter -- the ideal voltmeter must behave like there is an open circuit between its two terminals. While an open circuit can sustain a voltage difference, it will carry no current. Clicking on Figure 1 again will show the meter replaced with its ideal model. Since we know that the current in the circuit is 150 mA, Ohm’s law reveals that the voltage drop across R₂, that measured by the voltmeter, will be 3 Volts.

Figure 2 again shows the voltmeter connected to measure the voltage drop across resistor R₂, this time the measured voltage is displayed. The voltmeter has two terminals -- does it matter which terminal we connect to node B and which to C? Remember that potential difference has a magnitude and a polarity and voltmeters allow us to determine this polarity. Click on Figure 2 to switch the meter connections in measuring the voltage between nodes B and C. In doing so, we see that the meter now reads - 3V. A positive reading on a voltmeter indicates that its common terminal (typically labeled COM) is connected to the lower potential node whereas a reading of a negative voltage indicates that the meter’s common terminal is connected to the higher potential node. In the next tab we consider how a DMM set to measure current behaves.

Figure 2: Measurement of the voltage drop across R₂ in a simple series circuit consisting of ideal elements. Notice that the voltmeter displays a positive potential difference as its COM terminal is at the lower potential end of R₂. The meter is probing V_BC. V_S = 9V, R₁ = 10Ω, R₂ = 20Ω, and R₃ = 30Ω.

An ammeter has two connection jacks, one typically denoted with an “A” and the second denoted “COM” for the common terminal. Recall that current is a measure of charge flow -- we want an ammeter to observe this charge flow without interfering with it. A properly functioning ammeter could not act like an open circuit for an open circuit allows no charge flow and thus no current. What element ideally allows charge flow without impeding it? A simple wire -- a short circuit! Therefore, an ideal ammeter may be modeled with a short circuit between its two terminals. Let’s say that we wish to measure the current in the series circuit we have been considering. Clearly, we don’t want to introduce a new path in which charge could flow and so we must place the meter in the path of the charge flow we wish to probe. Therefore, current measurements are made in series requiring us to break the circuit and to use the ammeter to complete the circuit once again. This is shown in Figure 3 which displays the circuit of Figure 1 with an ammeter inserted to measure current

Figure 3: A series circuit consisting of ideal elements. We wish to measure the current I, and so break the loop and insert an ammeter to complete the loop once again.

There is only one current in the circuit of Figure 3 and so we could break the circuit anywhere and insert the meter. In the case of Figure 3, the meter has been placed between resistors R₂ and R₃. Clicking on Figure 3 will show the ammeter replaced with its ideal model -- a short circuit between its terminal terminals. Clearly, an ideal ammeter will not perturb the circuit.

Figure 4 shows the circuit with the meter displaying the current, a positive 150 mA. This current is simply the source voltage divided by the equivalent resistance seen by the ideal voltage source. The ammeter has two terminals – does it matter which terminal we connect to the leg of R₂ and which to the leg of R₃? Recall that current has a magnitude and a direction and ammeters allow us to determine the direction. Click on Figure 4 to switch the meter connections in measuring the current in the circuit. In doing so, we see that the meter now reads - 150 mA. A positive reading on an ammeter indicates that current leaves the meter’s common terminal (typically labeled COM), whereas a reading of a negative current indicates that the current is directed into the meter’s common terminal.

Figure 4: Measurement of the current in a simple series circuit consisting of ideal elements. In effect, the meter is probing current assuming it is clockwise. V_S = 9V, R₁ = 10Ω, R₂ = 20Ω, and R₃ = 30Ω.

We have looked at the models of ideal voltmeters and ammeters. All meters have some impact on circuit performance but unless we are measuring very small voltages and currents, the impact of the meter should be minimal. In the final tab you will have a chance to solve a couple of circuits to determine the values shown on the meters. Remember that the first step in solving such circuits is to redraw them, replacing the meters with their ideal models.

Consider the circuit of Figure 5 in which a voltmeter is wired in series with a circuit consisting of a voltage source and three resistors. We should immediately see that this is the incorrect way to wire a voltmeter as the meter changes the functioning of the circuit. Nevertheless, let’s see if we can figure out what voltage will be displayed by the meter. The first step is to redraw the circuit, replacing the voltmeter with its ideal model. The values for the elements are given in the figure’s caption. Determine the voltage that will be displayed and submit your response using the provided box.

Figure 5: A chance for you to test yourself. Determine the voltage displayed on the meter assuming ideal elements and V_S = {V_S} V, R₁ = {R₁} Ω, R₂ = {R₂} Ω, and R₃ = {R₃} Ω.

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