Basic Electrical Measurements E80 /7/09 - Professor Sarah Harris Oeriew Electrical Buildg Blocks:, L, C Impedance: Z oltage Diision Experimental Plots Time Doma s. Bode Plots Instrumentation Signal generators Measurement struments Instruments affect the measurement! Electrical Buildg Blocks Electrical buildg blocks are characterized by their current-oltage (I-) relationship. esistor: ( i( i( + ( - Example: Gien Ω and i( s(4, then ( i( s(4 Figure. esistor: Current and oltage are phase.
Capacitor: ( i( C i( C + - ( Example: GienC μf and ( s(4, then ( i( C μ F[4 cos(4]. 0 cos(4. 0 s(4t + ) Inductor: Figure. Capacitor: Current is 90 ahead of oltage. i( ( L i( L + - ( Example: Gien L μh and i( s(4, then i( ( L μ H[4 cos(4]. 0 cos(4. 0 s(4 t + ) Figure 3. Inductor: oltage is 90 ahead of current.
Impedance (Z) Impedance is the ratio of the oltage to current: Z I Suppose current and oltage are represented as complex exponentials: jθ ( e, i( Ie, where and I are phasors, e.g. of the form Ae. esistor: ( i( e j ω t Ie Z I Capacitor: ( i( C e Ie C Ie C[ jω e ] Z C I j ω C Inductor: i( ( L Ie e L e L[ jω Ie ] Z L I jωl Phasor plots Im Im Im ω I I e I jωc ω e jωli ω I e esistor: Current and oltage are phase. Capacitor: Current is 90 ahead of oltage. Inductor: oltage is 90 ahead of current. Figure 4. Phasor representation of current and oltage. The impedance and phasor analysis gies the same results as the earlier time doma plots. 3
oltage Diision We show two methods for solg for the put oltage, ( the put oltage (. 40Ω, of Circuit terms of ( 0Ω ( Figure. Circuit Method : Sce the circuit is purely resistie, we can use simple oltage diision: 0 ( ( ( + 0 + 40 Method : For any circuit, we can fd the frequency response function, FF, usg impedances. The FF, where and are the put and put phasors of the circuit. The FF gies the ga and phase of the put relatie to the put. For the gien circuit: ( FF Z Z + Z 0 0 + 40 Sce the FF is completely real this case, there is no phase shift of the put relatie to the put and the put is / th the magnitude of the put. 4
We now use Method to sole for the put oltage of a more complex system, Circuit shown below. 00Ω ( C mf ( Figure 6. Circuit Z C jωc FF + Z C Z jωc + jω + jωc The ga and phase depend on the frequency of the put. (0.) Case : Suppose ( s( (0), then j0 j j e e 0.03e ω(0.) (0)(0.) 3 j j + j + j + 3 3e Thus, the magnitude of ( is 0.03 smaller than ( and its phase is shifted by ab. ( 0.03s( (0) t ) + Case : Suppose ( s( (0.0), then j0 j0 e j0 jω(0.) + j (0.0)(0.) + e j0.03+ e Thus, the magnitude and phase of ( are the same as (. ( ( s( (0.0)
Experimental Plots Plotted experimental results gie sights to the characteristics of a system. Here we discuss seeral useful experimental plots: time doma plots, s. and Bode plots. Time Doma For Circuit Figure 6, we plot ( and (. If ( s( (0), ( 0.03s( (0) t ). The experimental results can be obtaed usg a signal generator as the put and an oscilloscope to measure the put. Figure 7. Time doma plots s For Circuit Figure 6, we can plot s at arious frequencies. Below we show s for ω (0) t. The magnitude of the put is measured for arious put magnitudes, as shown for 0 to 0. The slope of the le (0.3/0 0.03) gies the ga of the system at the set frequency, this case 0 Hz. 6
Figure 8. s Bode Plots A system s response at different frequencies is clearly displayed by its Bode plot. Below is the Bode plot of Circuit. A Bode plot can be experimentally determed by measurg the put oltage and phase on an oscilloscope while aryg the put frequency. ecall that the magnitude plot gies the Log Magnitude, Lm 0log0. Figure 9. Bode plots. Note that this plot the x-axis is rad/s. emember that to make an experimental Bode plot you must measure both the put and the put. 7
t Instrumentation Instrumentation is used to generate and measure signals and deices. Some useful struments are: signal generators, power supplies, multimeters (ohmmeters, ammeters, oltmeters), and oscilloscopes. Specifications and operation of a specific strument are found its manual. Signal generators (e.g., HP/Agilent 330A) Signals: Signal generators (also called function generators) generate se waes, square waes, triangle waes, etc. Because the signals change oer time, the signal generator is as an AC (alternatg curren supply. Parameters: The terface allows the user to select the signal amplitude (Ampl), frequency (Freq), etc. Output Impedance: The signal generator has an ternal resistance, sometimes referred to as the put resistance or impedance or source resistance or impedance. A signal generator can be modeled as shown below, with the put/source resistance labeled s. pp: The Agilent 330A allows you to set the peak-to-peak oltage pp. This measures the signal from its maximum to its mimum. So, the peak oltage is twice the signal s magnitude: pp m. For example, pp for s ( 3s( (0) is 6. Out Term: The signal generator allows you to select among two loads: High Z (large impedance) or 0 Ω. A load is the circuit to which you connect the signal generator. Notice that the figure below, the signal source s ( s( is actually the same, only the signal generator settgs hae changed. The signal generator can t tell what kd of load you actually use, so if you set the Out Term to 0 Ω and then use a ery large (relatie to 0 Ω) load, the measured amplitude won t match the displayed amplitude. If you are drig a load that is neither ery large nor 0 Ω, you ll need to choose one of the settgs and then model the entire circuit to fd what oltage will be deliered to the load. Signal Generator Signal Generator Signal Generator s s s ) t ( s ) s ( ) s( t L OUT TEM High Z s ( ) s( t L 0 Ω OUT TEM 0 Ω Figure 0. Signal generator with arious loads PP 0 PP Power supplies (e.g., HP/Agilent 636) Power supplies generate a DC (direct current or unchangg) signal. Note: The COM port is the common reference (usually referred to as your circui for all of the generated power supplies. The GND port is the ground reference comg from the wall. COM and GND are not connected ternally. 8
Multimeters (e.g., Simpson 60 (analog), Elenco (digital)) Usg different settgs, multimeters can measure oltage, resistance, or current. Ammeter: measures current. Place multimeter leads series with circuit. Internal impedance is low to aoid changg the behaior of the measured circuit. For example, the Simpson 60 can measure current as low as 0µA. Digital multimeters can usually measure lower currents. oltmeter: measures oltage. Place multimeter leads parallel with circuit. Internal impedance is large to aoid changg the behaior of the measured circuit. For example, the Simpson 60, DC impedance is 0 kω per olt. So with a 0 scale, the impedance is 00 kω. Digital oltmeters hae a fixed impedance, typically ~0MΩ. Ohmmeter: measures resistance. Place multimeter leads parallel with resistor. An ohmmeters use an ternal battery to supply a oltage and then measure the current through the meter, so disconnect the resistor from any other external power supply. Oscilloscopes (e.g., Tektronix ) AC signals: It is most straightforward to use an oscilloscope to monitor an AC signal as a function of time. When an oscilloscope is not aailable, multimeters can also be used to measure the AC oltage (or curren. Internal impedance: Just like multimeters, an oscilloscope (or scope ) has fite ternal impedance. These numbers can often be read directly off the port of the oscilloscope. x and 0x probes: A x probe reads the measured signal directly (with some strument loadg effect due to the probe and ternal impedance). A 0x probe creases the impedance of the probe and scope by 0 times (0x), thus decreasg the strument loadg effect. This also decreases the current through the probe by a factor of 0, so the signal read on the scope is decreased by a factor of 0. Some oscilloscopes hae a settg to dicate the kd of probe beg used. Instruments affect the measurement! Both signal generation and signal measurement (scopes, multimeters) are nonideal. To accurate model our systems, we must take non-idealities to account. Below are some examples of how the struments themseles change our measurements. Example : What is the put oltage across the load resistor gien the followg settgs: Out Term 0 Ω, PP? From the manual, we read that the put (or source) impedance, s, is 0 Ω. (For now, assume an ideal scope probe for measurg the put oltage.) 9
Signal Generator s s ( t ) s( t ) L 0 Ω OUT TEM 0 Ω ( PP Figure. Example Sce s L, half of the put oltage is dropped across the 0 Ω load resistor. L s s + L S So we expect that (.s(. The peak-to-peak oltage of the put, PP, is, as displayed on the signal generator. Example : What is the put oltage across the load resistor gien the followg settgs: Out Term 0 Ω, PP? From the manual, we read that the put (or source) impedance, s, is 0 Ω. (For now, assume an ideal scope probe for measurg the put oltage.) Signal Generator s ) s ( t ) s( t L 0 Ω OUT TEM 0 Ω ( PP Figure. Example L s s L + S 6 So we expect that ( s(. The peak-to-peak oltage of the put, PP, is. 6 3 Note, this is not the alue displayed on the signal generator. 0
Example 3: What is the put oltage across the load resistor gien the followg settgs: Out Term High Z, PP? From the manual, we read that the put (or source) impedance, s, is 0 Ω. (For now, assume an ideal scope probe for measurg the put oltage.) Signal Generator s ( t ).s( t ) s L 0 Ω OUT TEM High Z ( PP Figure 3. Example 3 Sce we set the Out Term to High Z, the pp alue would be put only if we actually had a large (approximately fite) load impedance. Howeer, as aboe, only /6 th of the signal gets transferred to the load, so ( s(. The peak-to-peak oltage of the put, PP, is. Aga, this is not the alue displayed on the signal generator. 6 Example 4: Now let s exame another place where we troduce strument loadg effects: signal detection. First consider the ideal case: Signal Generator s 0 Ω ) s ( t ) s( t 0 Ω 40 Ω ( Figure 4. Example 4 s + + So we expect that ( s(. Now, suppose our scope and probe hae a s s combed impedance of M Ω, as shown the Figure. What is the measured put oltage?
Signal Generator s ( t ) s( t ) s 0 Ω 0 Ω Oscilloscope 40 Ω ( str MΩ Figure. Example 4 with Oscilloscope Sce ( str s s str ) + + s str is large compared to, ( s( t. result as before: ) str is ab equal to. So we hae the same Now suppose and are replaced by larger resistors, as shown below. Signal Generator s ( t ) s( t ) s 0 Ω 0 MΩ Oscilloscope 40 MΩ ( str MΩ Figure. Example with Oscilloscope Now we hae: str ( ) + str + s s str str + s With the strument loadg effect ( + s + s + 4 s str ), we would hae expected:
Measurg AC signals Oscilloscopes are best struments to use to measure AC signals. Howeer, analog multimeters (like the Simpson 60) can also be used. The Simpson 60 measures the root-mean-squared or rms oltage. rms T 0 T 0 ( dt So, for example for a se wae, rms ( dt s T T 0 T 0 0 T0 0 ( dt T 0.707 Side note: the rms oltage is a useful alue for calculatg aerage power dissipation. For example, the power through a resistor is: P i With an AC oltage, you could measure the stantaneous power (usg the equation aboe) or the aerage power, which is usually most useful, usg the equation below. P ae ae ( ) ae T 0 T 0 ( dt rms The Simpson 60 actually measures the oltage by first fdg the aerage alue of the rectified signal (absolute alue of the signal) and then scalg the measured aerage alue to produce the rms alue. The aerage alue is physically produced by rectifyg the put signal, and storg the put onto a capacitor, which then aerages the signal. ae ( dt s( T T T 0 T 0 0 T0 0 dt 0.637 0.707 The Simpson 60 scales its ternally measured alue by. (. ) to produce 0.637 the rms alue. So, for example if a susoidal oltage of amplitude is put to the Simpson 60, the displayed rms oltage will read 0.707. The ternally measured alue (not pu is the aerage alue of the rectified signal (0.637 ), and that alue is ternally multiplied by. to produce the readg of 0.707. 3
Note: If the AC put signal is not a se wae, it will still take the aerage alue of the rectified signal and scale it by.. The analog oltmeter will not report the correct rms oltage for anythg other than a se wae. Howeer, if you know the shape of the put, you can derie the correct alue from the reported number. 4