Name1 Name2 12/2/10 ESE 319 Lab 6: Colpitts Oscillator Introduction: This lab introduced the concept of feedback in combination with bipolar junction transistors. The goal of this lab was to first create a circuit that had known poles, and then introduce feedback and view the effect on the frequency of the poles. When poles are selected at interesting frequencies, the transistor will magnify that frequency by creating high gain. This socalled Colpitts Oscillator is the name of the circuit that takes no input (just a power source) and produces a sinusoidal signal at the output. We will investigate the components needed to create this system and what other parasitics effect the system. Theory: The idea of this oscillator comes from the idea of positive feedback, which is described by Figure 6.1. Figure 6.1: Positive Feedback The system is described of having a closed loop gain of: A cl V ( j! ) = V o i A( j! ) = 1 " A( j! ) And so when we set the open loop gain A(jω) = e j2πn, n Ζ, we seemingly have some instability, which causes the oscillation. In order to create this kind of reaction, we will first describe the system in open-loop form, as shown in Figure 6.2. Figure 6.2: Open Loop Colpitts Oscillator
From detailed node analysis, we find the open loop gain of this system to be: And, like before, we want to set this gain A(jω) = 1, and in doing so, we get a set of criteria to satisfy our constraints. First, our oscillating frequency ω x is set by the equation: And in order to have these poles sit on the jω axis (edge of instability), we also must have the condition: We simplified this set up by first equating the capacitors (C 1 = C 2 ) (which is designated on Figure 6.2 by just C), so that g m R = 1. In practice, however, since we do not control R specifically (it consists of internal transistor resistances), we make g m R > 1 to ensure oscillation startup, and note the g m = I c /V T. Thus, by altering the bias current, we can make sure the system oscillates. Furthermore, by changing L, we can guarantee a certain frequency within a certain error. The other components, such as RFC, R 1, and R 2 have to deal with the DC biasing only. We made the RFC inductor high so it acts like an open circuit at DC, and short at high frequencies. We
picked a 100 mh inductor. For R 1 and R 2, we kept in mind that the DC current flowing down the resistor network needed to be small, and so we picked R 1 + R 2 = 50 kω. Because R is in the range of > 25 kω, we were safe to pick a bias current of greater than 1 ma. g m = I c /V T =.001/.025 = 40 ma/v, and so then g m R > 1. To set a bias current of about 1 ma, we made the assumption that V BE of the transistor was about.7 V, and setting R E = 1 kω (small enough for high gain), V B =.7 + 1mA*1kΩ = 1.7 V. With V cc being 12 V, the correct division of R 1 and R 2 to set this bias was R 1 = 43 kω and R 2 = 7 kω. Finally, we wished to set an oscillating frequency of.5 MHz. We picked capacitors of C 1 = C 2 = 1 nf and the inductor(s) that gave us that frequency (via the formula above). We desired a 200 µh inductor, but there are no inductors that custom value. So we used two 100 µh (L 1 and L 2 ) in series to create a 200 µh inductor. The summary of the components can be found in Table 6.1. Table 6.1: Theoretical Component Values Component Value C 1 1 nf C 2 1 nf L 1 100 µh L 2 100 µh R E 1 kω R 1 43 kω R 2 7 kω RFC choke 100 mh We then set out to verify this design via Multisim simulation. We first tested for the DC biasing of the circuit, as shown in Figure 6.3. Figure 6.3: Multisim DC Bias with Theoretical Components
The bias current is higher than expected (1.069 ma). However, it still successfully allows the transistor to work in forward-active mode, and so we continued forward. After verifying this design, we closed the loop to find the frequency of oscillation. We measured at and V o (which is now connected to the base, and equals V B ). This is shown in Figure 6.4. Figure 6.4: Closed Loop Simulation with Theoretical Components
With the knowledge of the oscillating frequency at about 474 khz (within 10% of.5 MHz), we implemented a 400 mv peak-to-peak sinusoidal wave at the input V i. We then measured the open loop gain and phase at V c and V o, as shown in Figure 6.5. Figure 6.5: Open Loop Simulation with Theoretical Components
Voltage at Collector vs Input: Voltage at Output vs. Input The gain of both of these outputs was about 6 V/V (2.5/.4), which is what g m R must equal (gain for any common emitter amplifier with no R c ), while the both phases were slightly out of phase with the input but only by small margins.
Finally, to make sure that the parasitics of the protoboard and scope probes would not alter the oscillator further than a small error, we implemented some parasitic components on Multisim. These include capacitors between adjacent rows and a capacitor-resistive load as a model of the scope probe. We took the closed loop system and re-measured the oscillating frequency. This is shown in Figure 6.6. Figure 6.6: Closed Loop Parasitic Simulation
With the knowledge that the parasitics do not affect this system past a certain error (gain constant and oscillating frequency still within 10% of.5 MHz), we moved on with implementing and simulating with real components, as shown in Experimental Setup. Experimental Setup: We began by setting up the circuit according to the schematic seen in Figure 6.2 (in Theory), and then measured the real values of these components. The values of the circuit components are seen in Table 6.1. Table 6.1: Circuit Component Values Component Nominal Value Actual Value R 1 43 kω 42.7656 kω R 2 7 kω 7.2457 kω R E 1 kω 0.992 kω RFC 100 mh 105.1 mh L 2 * 100 µh = 200 µh 201.9 µh C 1 1 µf 0.9971 µf C 2 1 nf 1.006 nf C 3 1 nf 0.9983 nf C 4 1 µf 1.05 µf The transistor used was the 2N3904. The Pin connection of the 2N3904 is given in Figure 6.7. Figure 6.7: 2N3904 Pin Connections
We tested both the open loop gain at V B and V o, and then closed the loop with feedback and measured these two values again, as seen in Experimental Data. Experimental Data: After assembling the circuit on our protoboard we first verified the biasing. We found: I C = 1.011 ma and I R1 = 0.242 ma These values are close enough to our requirements and match the simulations, and are thus acceptable. We then put the circuit in the open loop configuration and took some measurements and also ran a simulation. The actual measurements were taken with a function generator connected to the input node of the circuit through a 1 µf capacitor. The schematic for the circuit is the same as that of the Multisim simulation and is seen in Figure 6.8. Figure 6.8: Multisim Simulation of Open Loop Configuration
The frequency of the input sine wave was varied using the function generator and the output was measured. The readings thus obtained are tabulated in Table 6.2. Table 6.2: Open Loop Circuit Analysis
Frequency (Hz) Vi (V) Vo (V) Phase (degrees) Vo/Vi (V/V) Gain (db) 100000 0.125 2.563 96.5 20.504 26.23677186 250000 0.126 4.937-40.6 39.18253968 31.86185164 400000 0.126 1.625 87.7 12.8968254 22.2096564 500000 0.125 10.44 2 83.52 38.43580971 600000 0.123 1.062-80 8.634146341 18.72438811 750000 0.125 0.2937-86.3 2.3496 7.41987867 1000000 0.123 0.14-114 1.138211382 1.124458485 From these readings we may plot the Bode diagrams for the open loop configuration. These are seen in Figure 6.9. Figure 6.9: Open Loop Bode Plots
We also took a screenshot of the oscilloscope to record the form and shape of the output wave. This is seen in Figure 6.10. Figure 6.10: Oscilloscope Screen Capture As it can be seen, the above figure compares quite well with the one seen in the simulation (Figure 6.8). We also analyzed our simulation to obtain the simulated theoretical frequency response of the system. This is seen in Figure 6.11. Figure 6.11: Simulation Frequency Response
These too are seen to match the actual experimental results (Figure 6.9) fairly well. We also confirm that for the open loop configuration self-oscillation will occur i.e. the gain from input to output is greater than 1 when the phase shift is 0 degrees (The gain is 83.52 for a phase shift of 2 degrees). We then move on to testing the closed loop configuration. The circuit schematic for the closed loop configuration is the same as the one for its multisim simulation and is seen in Figure 6.12. Figure 6.12: Multisim Simulation of Closed Loop Configuration
We then displayed the output V o on the oscilloscope, as shown in Figure 6.13. Figure 6.13: V o from the Oscilloscope As can be seen the circuit is oscillating at a frequency of 512.8 khz. This compares reasonably well with the value of 473.23 khz from the simulation (Figure 6.12). Figure 6.14 shows us V C and V O. Figure 4.7: V O (1) and V C (2)
These two figures show that there is sizeable gain for the two points (V C being higher than V O ) and the two almost completely out of phase. The experimental oscillation frequency is ~= 511 khz, whereas the calculated oscillation frequency is 500.384 khz (calculated using actual component values). We then reduce R 2 till I C becomes 0.1 ma. R 2 is now 2.764 kω. The circuit continues oscillating as before as seen in Figure 6.15 (albeit with a slightly lower gain and smaller phase difference). Figure 6.15: Oscilloscope Screen Capture for I C = 0.1 ma
We continue to reduce R 2 and find that we can reduce it to 0.942 kω before the oscillations are extinguished. At this point we have essentially reached I C = 0 A i.e. the cut-off point for the transistor. When in the closed loop configuration with I C =.1 ma is used we measure: V C = 13.91 V and V O = 9.688 V And in the open loop we measure V O = 525 mv. We simulated this circuit again for the closed loop and open loop with R 2 = 2.764 kω, as seen in Figures 6.16 and 6.17. Figure 6.16: Closed Loop Simulation with R 2 = 2.764 kω Figure 6.17: Open Loop Simulation with R 2 = 2.764 kω Discussion: In this lab we wished to see if we could amplify a frequency using this Colpitts Oscillator and perpetuate it with positive feedback. First, however, we proved that the open loop had sizeable gain at the desirable frequency. We chose this frequency,.5 MHz using specific LC components. Our measurements (Figure 6.9) prove that there is a jump in gain around 500 khz. From that point, we closed the loop removed the signal source and found that amazingly we had a high voltage sinusoid at the collector (14 V) and the output (9.9 V). The output voltage is lower
than the collector voltage because the LC filter between the two is passive, and so no voltage can be gained there. Interestingly, the voltage of this sinusoid surpassed the power source of the system (V CC = 12 V). This is because the positive feedback increases the voltage until it hits its peak. We found that the sinusoid had a frequency well within plausible error (511-500/500 = 2.2%) but had lower voltage than simulated. We simulated a voltage of up to 20 V, but only measured a voltage of 14 V. This most likely had to deal with the actual limitations of our real transistor as well as the parasitic values embedded in the system. Also, we noticed that this closed loop output jumped from high to lower voltages and vice versa in small periods of time. This is because of the instability of the poles, and this could also play a part in not being able to have a constant high voltage. We also took note of the shape of the oscillations. They were not perfect sine waves, but distorted with a base frequency of about 500 khz. This is because other frequencies are being amplified as well, causing a non-symmetrical output. Finally, we proved that the oscillation would continue until we had almost no current flowing through the transistor. While the open loop gain decreased, the closed loop gain did only slightly. We needed g m R > 1 to startup the oscillation, so if we started I c at 1 ma and decreased it slowly, there would still be oscillation until cutoff, and in fact there was oscillation with still high gain (~14 V). The simulations verified this fact. To improve the system, we would need to one of a few things. First, we could obtain components that are exact values needed to hit exact values for g m R and ω x (expensive). We could also try to eliminate the parasitic capacitances and inductances in the protoboard and its wires, with items such as Faraday shields and corrective reactive components. Lastly, we used many approximations in our calculations of our oscillator. We unfortunately cannot account for all the microscopic values in the system. However, overall we created a steady high gain sinusoid at a pre-chosen frequency, so our Colpitts Oscillator was a success.