Measuring Gain and Phase Difference

           A series circuit of two resistors are connected to a sinusoidal voltage, and its phase difference

between input and output is measured by a oscilloscope.

V(t) = Vm sin(ωt + Φ)  , Vm = 2v , Φ = 0°
 

Figure 1: Schematic of Series Circuit of two Resistors

Figure 2: Series Resistor Circuit with 1 KHz Frequency

f = 1/T , f = 1/(1 ms)  , f = 1 KHz

Figure 3: Series Resistor Circuit with 5 KHz Frequency

f = 1/ (200 μs) , f = 5 KHz

Figure 4: Series Resistor Circuit with 10 KHz Frequency

f = 1/T , f = 1/(100μs)  ,  f = 10 KHz

There are no phase difference between input and output in a pure resistive circuit.

Rtotal = R1 + R2 , Rtotal = 45.9 + 97.7 = 143.6 Ω

According to fig 2 and scale, 50 mA = 40 mm → I(total) = (10 mm)*(50 mA/40 mm) = 12.5 mA


Vout = R2*I(total)  , Vout(Measure) = (97.7Ω )*(12.5 mA) = 1.22 v 

G = Vout/Vin , G = 1.22/2 , G = 0.61

G(Theory) = Vout/Vin = R2/(R1+R2) , G(Theory) = 97.7/(97.7+45.9) = 0.68 

Gain Percent Error = [(0.68-0.61)/0.68]*100% = 10.3%

Figure 5: Series RL Circuit with an AC Voltage

Figure 6: Graph of Voltage and Current in an Series RL Circuit with 1 KHz Input Voltage

According to the Figure 6 and scale, Vout = (24 mm)*(200 mv/ 20 mm) = 240 mv (Blue graph)

VR1 = (35 mm)*(1v/ 20 mm) = 1.75 v , I(total) = (16 mm)*(50 mA/ 20 mm) = 40 mA

Inductor voltage (blue graph) leads current (small red graph) by 90 degree , and two red graph are in

phase because those show current and voltage of R1.

|XL| = (2π*1000)*(1mH) = 2π Ω , G(Measure) = Vout/Vin , G = 0.24/2 = 0.12 

G(Theory) = XL/Z , Z = 45.9 + j(2π) , G(Theory) = 0.136 ˂ 82.2°


Gain Percent Error = [(0.136 - 0.12)/0.136]*100% = 11.8%

Figure 7: Graph of Voltage and Current in an Series RL Circuit with 5 KHz Input Voltage

According to the Figure 7 and scale, Vout = (40 mm)*(500 mv/ 20 mm) = 1 v (Blue graph)

VR1 = (32 mm)*(1v/ 20 mm) = 1.6 v , I(total) = (14 mm)*(50 mA/ 20 mm) = 35 mA

Inductor voltage (blue graph) leads current (small red graph) by 90 degree , and two red graph are in

phase because those show current and voltage of R1.

 |XL| = (2π*5000)*(1mH) = 10π Ω  , G(Measure) = Vout/Vin , G = 1/2 = 0.5 

G(Theory) = XL / Z , Z = 45.9 +j(10π) , G(Theory) = 0.565 ˂ 55.6°

When frequency increases, XL, Vout, and G increases and vice versa.

Figure 8: A Practical Series RC Circuit

Figure 9: A Series RC Circuit

Figure 10: A Series RC Circuit with 1 KHz Frequency

According to the Figure 10 and scale, Vout = (29mm)*(1v/ 30 mm) = 0.97v (Blue graph)

 |XC| = 1/(2π*1000)*(0.41μF) = 388.4 Ω    ,   G(Measure) = Vout/Vin  , G = 0.97/2 = 0.485

 Z = 46.1 - j(388.4) , G(Theory) = XC / Z , G(Theory) = 0.993 ˂ -6.8°

Gain Percent Error = [(0.993 - 0.485)/0.993]*100% = 51.2%

Figure 11: A Series RC Circuit with 5 KHz Frequency

According to the Figure 11 and scale, Vout = (35 mm)*(1v/ 30 mm) = 1.17v (Blue graph)

 |XC| = 1/(2π*5000)*(0.41μF) = 77.68 Ω    ,   G(Measure) = Vout/Vin  , G = 1.17/2 = 0.585

G(Theory) = XC / Z , Z = 46.1 - j(77.68) , G(Theory) = 0.859 ˂ -30.7°

Gain Percent Error = [(0.859 - 0.585)/0.859]*100% = 31.9%

Figure 12: A Series RC Circuit with 10 KHz Frequency

According to the Figure 12 and scale, Vout = (45 mm)*(1v/ 40 mm) = 1.13 v (Blue graph)

 |XC| = 1/(2π*10000)*(0.41μF) = 38.9 Ω    ,   G(Measure) = Vout/Vin  , G = 1.13/2 = 0.565

G(Theory) = XC / Z , Z = 46.1 - j(38.9) , G(Theory) = 0.645 ˂ -49.9°

Gain Percent Error = [(0.645 - 0.565)/0.645]*100% = 12.4%

When frequency increases, XC, Vout, and G decreases and vice versa.