BME3013 Lecture 1 Topics: . The DMM as an ohmmeter . The DMM as a voltmeter Voltage divider equation Kirchhoff’s Voltage Law (KVL) Effect of component values . The DMM as a current meter (Ammeter) Current divider equation Kirchhoff’s Current Law (KCL) . RC circuits Time constant . The Bode plot . AC measurements using the oscilloscope

1 Kirchhoff’s Voltage Law (KVL) = 0

Σ𝑉𝑉

The voltage drop created by an element has the polarity of + to – in the direction of current flow R I

For example: + V −

R1 R2 + 10 Ω − + 20 Ω − = + =R1I + R2I

V V 𝑠𝑠 1 2 1 2 30𝑉𝑉 =10I𝑉𝑉 + 20I𝑉𝑉 = Closed loop + − e.g. V2 (i.e. the voltage drop𝐈𝐈 across𝟏𝟏 𝑨𝑨 R2) is equal to R x I = 20 Ω x 1 A = 20 V I 2 VS = 30 V 2 Kirchhoff’s Current Law (KCL)

3 A 6 A I2 I1

I 3 = 0 II == ? 9 A

(a) Σ𝐼𝐼 (b) (a) (b)

(a). Kirchhoff’s current law states that the sum of the currents entering a node is 0. (b). Two currents entering and one “negative entering”, or leaving.

3 Current Division Z can represent either a resistance or any other passive component such as a capacitance that has am impedance which depends on frequency

KCL: Z1Z2 Ztotal = Z1 + Z2

4 Potentiometer: Adjustable voltage divider (attenuator) B B

A A A B C

C C

B A C

The Voltage Divider Equation

To use a potentiometer as a variable resistor, connect terminals A and B, Voutput R2 = OR A and C to your circuit; Note that the resistance between terminals B Vinput R1 + R2 and C is fixed, independent of the position of the wiper! 5 The Ground Point:  You must always be aware that your voltage measurements are relative to ground potential

 This ground is identical to the safety ground (third pin) of the 110 V power socket that the measuring equipment is plugged into

 For safety, the shells of the BNC connectors are also connected internally to this power system ground, as are the ground clips of an oscilloscope probe

 Note that if the circuit under test has a ground point (which it may not always have if it operates from a battery), then the voltmeter’s ground must be connected to the same ground point

For this reason, you need to pick a single ground point on your protoboard and make sure that all instrument grounds connect to it and not to any other point in the circuit. 6 The DMM as an Ohmmeter

 Ohmmeters cannot function if the circuit is connected to a power supply

 In order to measure the resistance of a given resistor in a circuit, the resistor must be removed or disconnected from the circuit and probed independently!

 The ohmmeter then passes a small current through the circuit component of interest and subsequently measures the voltage produced, and using principles based on Ohm’s law, displays the resistance of the component

 Note that resistance is an unsigned quantity. Therefore, it does not matter which colored probe from the DMM is connected to either side of the resistor

 Note that probing a powered circuit with an ohmmeter will likely damage the meter!

7 The DMM as a Voltmeter

 In order to measure voltage across a given component in your circuit, the voltmeter is connected in parallel to that component

 Power must be supplied to the circuit from an external source

 Because the voltmeter provides a parallel pathway, it should pass as little current as possible, so as not to short circuit the component across which it is measuring

 Hence, a voltmeter has a very high internal resistance

 Can be used to measure DC and AC voltages

8 The DMM as an Ammeter

 The ammeter should be connected in series with the rest of the components in your circuit

 This allows the current flowing through the circuit to pass through the ammeter as well

 Ammeters should not alter the behavior of the circuit whose current they are measuring, and thus, to avoid causing a voltage drop across them, an ammeter should have a very low internal resistance.

 Power must be supplied to the circuit from an external source

 Can be used to measure DC and AC currents

9 Measuring Voltages with the Oscilloscope

 The oscilloscope channels can only display voltages relative to the power-system ground reference. Also, if the black-shrouded alligator clips of the probes are connected to two different points in a circuit, these two points have now been shorted together through this internal connection of the oscilloscope!

 For this reason, you need to pick a single ground point on your prototype board and make sure that all instrument grounds connect to it and not to any other point in the circuit.

10 Effect of Component Values (Loading)  Measuring an electrical signal inevitably affects that signal. This applies to all measurements, including the display of an oscilloscope waveform

 Affecting the signal cannot be totally eliminated, but it can be minimized sufficiently that the effect is unimportant. Then the measured result is a sufficiently accurate representation of the real signal. Voltage loading effect Consider the following circuit:

Suppose an ideal voltmeter, which presents an open circuit to the measurement circuit, is used to measure the voltage. There is no current flowing through an ideal voltmeter, so there is no voltage drop across the resistor, and the voltage at the terminals of the voltmeter in this example is 1V. The voltmeter shows a reading of 1 V which is the correct result. 11 Effect of Component Values (Loading) Let’s assume now that the same measurement is attempted with a voltmeter that presents a load of say, 2MΩ to the circuit. Then, the 1MΩ internal resistance and the 2MΩ voltmeter resistance Rmeter form a voltage divider and the meter reading will be 0.666 volts. This is a misleading result caused by the loading effect of the voltmeter.

12 Capacitors in DC Circuits

 When a capacitor is first connected to a DC power supply, current flows initially in the circuit while the capacitor is still being charged

 Charge quits moving (current flow stops) once the voltage across the capacitor is the same as the supply voltage.

+ + − −

The capacitor is charging The capacitor is fully charged (DC current flows) and acts as an open circuit element (no DC current)

13 Capacitors in AC Circuits

Capacitors in AC circuits allow current to flow continuously, but they do not act like short circuits. Their ability to pass current depends on the frequency (f) of the signal, and is called reactance (Xc).

Ω 1 XC is the capacitive reactance, in f X C = is frequency, in Hz 2πfC C is capacitance, in F

Energy is not lost due to reactance (like it is to resistance). Instead, the energy is stored in the electric field of the capacitor when charged, and released to the circuit as current when discharged.

To find the current I in a capacitor for an applied voltage V, we use Ohm’s Law for AC circuits: =

Ohm’s Law for AC Circuits 𝐶𝐶 𝑉𝑉 𝐼𝐼𝑋𝑋 14 First-Order Response VC(t) Step input V 1 + vR − I 63% Charging VC(t) + R + V v C C − −

τ =τ RC t (a) (b)

KVL: VR + VC = V 1 + = for 0 = 𝑐𝑐 𝑡𝑡 𝑑𝑑𝑣𝑣 𝑉𝑉 −𝑅𝑅𝑅𝑅 𝑣𝑣𝑐𝑐 𝑡𝑡 ≥ 𝑉𝑉𝑐𝑐 𝑉𝑉 − 𝑉𝑉𝑒𝑒 𝑑𝑑𝑑𝑑 = 𝑅𝑅𝑅𝑅 and𝑅𝑅𝑅𝑅 = (Both decrease exponentially) 𝑡𝑡 𝑡𝑡 − − 𝑅𝑅𝑅𝑅 𝑉𝑉 𝑅𝑅𝑅𝑅 𝑉𝑉𝑅𝑅 𝑉𝑉𝑒𝑒 𝑖𝑖 𝑒𝑒 (a). Series RC circuit with voltage𝑅𝑅 step input at time 0. (b) Normalized voltage across the capacitor. 15 First-Order RC Circuits

Example: Let C= 1µF and R = 100 kΩ. Find the gain and phase at f = 2 Hz. Gain and Phase vs frequency (Bode) Plot Low-frequency asymptote

High-frequency asymptote

16 Using the oscilloscope to measure phase shift: An accurate way to use an oscilloscope to measure the phase shift between two sine- waves is the "zero-crossing method" described below: Note: This method uses measurements taken, where dV/dt is the largest, and is therefore more accurate than using the peaks themselves (where, dV/dt is zero).

• For the sample reference wave (A), we measure A1 at its first upward crossing with the zero line or A2 at the first downward crossing with the line. • For the phase-shifted wave (B), we measure B1 as the first upward crossing with the zero line or B2 as the first downward crossing with the line. • Since both waves have the same frequency and time period, and one full period is equal to 2π radians (or 360°), we can calculate the phase shift (usually expressed in degrees) by measuring the difference in time along the horizontal time axis between A1 and B1 or A2 and B2 by using the vertical time cursers of an oscilloscope.

A B

17