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Niels Bohr Institutet, November 4, 2008 (J.H.M.) Fysik 6, Laboratory Exercise 6 Nonlinear circuit elements In this exercise we will investigate some common nonlinear circuit elements diodes and bipolar transistors. We will not attempt to understand in detail how those elements are made up and work – you would need advanced knowledge of solid state physics – but rather see how they behave and can be used for typical applications like rectiﬁers and ampliﬁers. 1 Linear and Nonlinear circuit elements So far in all the electronics experiments we used passive linear elements like resistors, capacitors, and inductors. Linear means here, that the voltage drop across such an element is directly proportional to the current ﬂowing through the element. The proportionality constant - the impedance - in general depends on frequency. We have deﬁned for convenience impedance as a complex quantity, so that we can treat time-dependent voltages and currents easily by looking at amplitudes and phase-shifts of harmonically varying signals of a given frequency and then adding the results together to construct arbitrary time- dependent signals. We are only allowed to do so, because the single elements in a more complicated network behave linearly, i.e. voltage drop depends linearly on the current through an element. This is, of course, only an idealization and all real devices are nonlinear to some extent. A resistor, for example, converts electrical energy into heat when current ﬂows through it - if this heat is not removed from the body of the resistor the temperature will increase and with a change in temperature the resistance value will change. This means that the resistance value depends on the current that ﬂows (and on the current at past times). Most of the time this is not desired, so one chooses materials for resistors with a low temperature coeﬃcient of resistivity and shapes the resistors such that they are eﬃciently cooled by ambient air. In other cases, this nonlinearity is exactly what you want, for example in a fuse, where a wire junction breaks as soon as you send too high a current through it. If we have a circuit with nonlinear elements in it, we are not allowed to use the superposition principle anymore or simply scale up voltages and currents by a constant factor. It also means that a nonlinear circuit driven by a sinusoidally varying voltage of frequency at the input will show in general at the output a response also at multiples 0 f,f, f, f,... of the input frequency. On the other hand, the Kirchhoﬀ rules the sum of currents into a node is zero the sum of voltage drops around a loop equals the sum of emf in the loop can – and must – still be used to analyze the behavior of a circuit, because they are directly derived from the conservation laws for charge and energy and independent of the relation between current and voltage in a given device. All we need to do to analyze a circuit, is to satisfy Kirchhoﬀ’s rules and the possible nonlinear relations between current and voltage simultaneously. Figur 1: Circuit symbol of a diode with anode terminal to the left and cathode terminal to the right. 2 Diodes and transistors In most electronics circuits you will encounter diodes and transistors. A diode is a two-terminal device with one terminal named cathode and the second terminal named anode . The circuit symbol of a diode is shown in Figure 1. The terminal connected to the base of the triangle is the anode and the terminal connected to the crossbar is the cathode. On a real device the cathode terminal is marked with a ring. The relation between current diode through the diode and the voltage diode between anode and cathode terminal is well described by

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Nonlinear circuit elements diode rs (exp ( diode αU th 1) (1) where rs , the so-called reverse saturation current, is a constant which depends mainly on material and geometry of the device, th kT/e is the thermal energy of an electron, and is a dimensionless correction factor with a value between 1 < 2. The value of th at room temperature is 25 mV By inspection of equation 1 you see that current ﬂows in the diode much easier from anode to cathode (forward current) than the other way around (reverse current). Already at a forward voltage of 0 the current is orders of magnitude bigger than for the same voltage applied in the opposite direction. This property makes diodes useful to steer currents around in a circuit. The function is similar to a ball-valve in water pipes. In the experiment you will see this property and you will verify the nearly exponential dependence of current on voltage. To do this, you build a circuit with a series connection of a diode and a resistor, drive the system with an input voltage and measure the current through the diode by monitoring the voltage drop across the resistor (see Fig.2). Figur 2: Circuit to measure the current voltage characteristics of a diode. Very useful devices in a water pipe system are regulating valves, where a piston is used to vary the eﬀective cross-section and thus the conductance of the pipe - you use that every day when you open and close a faucet. In an hydraulic system the position of the piston can be regulated by the pressure and ﬂow in another smaller system of pipes. A transistor is essentially a translation of this concept into an electrical circuit. Figur 3: Circuit symbol of a bipolar npn-transistor showing the collector, base, and emitter terminals. A bipolar transistor is a 3-terminal device. The terminals are named collector base , and emitter The circuit symbol of a bipolar transistor is shown in Fig.3. Transistors come in diﬀerent varieties. In the experiment we use npn-type transistors, which stands for a certain material composition of the transistor. In analogy to a regulating valve, with a transistor we can control the conductance, and thus the current into the collector terminal by setting the voltage between the base and the emitter terminals. Under normal operating conditions the potential at the collector is at least 0 more positive than the potential at the emitter terminal and the base terminal is about 0 more positive than the emitter terminal. The control input - the base emitter junction - behaves similar to a diode. You can measure

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Nonlinear circuit elements that by shorting the collector to the base and measuring the current from base to emitter as a function of the base emitter voltage (see Fig.4). The control law connecting the current into the collector with Figur 4: Circuit to measure the current voltage characteristics of the base-emitter junction. the voltage between base and emitter be is given by rs (exp ( be th 1) (2) so it looks very similar to the diode equation, but here it is the current into the collector and not the current into the base, which is described. The current into the base is typically much smaller than the current into the collector, which makes the transistor such a useful device, because it allows us to use a small power to control a much bigger power. Kirchhoﬀ’s rules tell us that the current out of the emitter terminal needs to be the sum of the currents into base and collector. In typical transistors the ratio between the base current and the collector current is several 100. Figur 5: Circuit to measure the basic relation between base-emitter voltage and collector current in a bipolar transistor circuit. A circuit where we can check the validity of equation 2 and which can work as a power ampliﬁer is shown in Fig.5. When we apply a voltage between points and in the circuit, we can measure as outputs ﬁrst the voltage BG . Knowing the values of and we can determine the current into the base terminal. Measuring for the same input the voltage across allows us to determine the current out of the emitter and combining the two measurements we also know the voltage between the base and the emitter. You will observe that the output voltage between emitter and ground is lower than the voltage between base and ground, so at ﬁrst sight the advertised performance as an ampliﬁer is disappointing.

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Nonlinear circuit elements The important point to realize is, however, that the current through the emitter resistor is much higher than the current into the base. To see the principle of the ampliﬁer better we extend the circuit somewhat by adding a dedicated control input to the circuit as shown in Fig.6. We connect an independent input source through a capacitor to the base terminal of the transistor. We add a resistor in series that allows us also to measure the input current. When the supply voltage to the collector terminal is on, we see that on top of a DC oﬀset a copy of the input voltage appears across the emitter resistor with a time varying current ﬂowing through the emitter resistor much higher than the current ﬂowing through at the input. Figur 6: Input circuitry added to the circuit from Fig.5. The ampliﬁer delivers more power to the resistor than it takes from the source connected to the input.

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Nonlinear circuit elements 3 Experimental program IMPORTANT: For points 1-3 below you should use the Picoscope input coupling as DC . Remember you are measuring on nonlinear circuit elements, where knowledge of the DC voltage is essential to measure the current/voltage characteristics! 1. Build a circuit with a diode and a resistor in series combination as shown in Fig.2. Use the pulse generator set to ”Squarewave”with a convenient frequency (around = 100 Hz ) as a voltage source to the input. Measure the voltage drop across the resistor for various input voltages between in ... 10 (Take at least 5 points in that interval!). Start with a resistor of = 1 Ω, and repeat the measurements with = 10 Ω and = 100 Ω. The use of diﬀerent resistors allows you to vary the current over a large range, while still keeping good sensitivity for the current measurement at low current values. Use Kirchhoﬀ’s rules to ﬁnd the current through the diode from your measurement of the voltage across the resistor and to ﬁnd the voltage drop across the diode from your measurement of the input voltage and the voltage across the resistor. Verify with your data the exponential dependence of the current ﬂowing through the diode on the voltage drop across the diode! 2. Connect a sinusoidal voltage from the waveform generator with an amplitude around 1 and a frequency of your choice to the circuit and observe the output signal. Look at input and output signal with the spectrum analyzer (set a frequency range which allows you to see up to 10 times the input frequency) and explain what you see! Is it evident from your observation that a diode is a nonlinear circuit element? 3. Built the circuit shown in Fig.5 with the following values of components: ,R = 51 Ω, Ω. Use the pulse generator set to squarewave with 100 Hz and measure BG ,U EG for various input voltages between in = 1 ... 10 . Repeat for = 10 Ω and verify eq.2 with your data. To do this you need to determine the voltage diﬀerence between base and emitter from your measurements of BG and EG , the sum of collector and emitter current from your measurement of EG , and the base current from your measurement of in and BG . Make a plot of vs be ! How big is the ratio between collector current and the base current What happens if you replace by a short circuit? 4. If you have time left, extend the circuit as shown in Fig.6, connect the waveform generator (sinu- soidal) with amplitude around 1 and frequency set between = 2 kHz.... 100 kHz to the input and observe the gain behavior of the ampliﬁer. Use a value for the load resistor of = 5 Ω. (For these measurements it is convenient to use AC-coupling of the Picoscope.) Try to measure the input current drawn from the waveform generator by measuring the voltage drop across the sense resistor (convenient values for are between 1 Ω and 5 Ω). How big is the input impedance of the ampliﬁer? Can you explain why it has this value?

HM Fysik 6 Laboratory Exercise 6 Nonlinear circuit elements In this exercise we will investigate some common nonlinear circuit elements diodes and bipolar transistors We will not attempt to understand in detail how those elements are made up and work ID: 23847

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Page 1

Niels Bohr Institutet, November 4, 2008 (J.H.M.) Fysik 6, Laboratory Exercise 6 Nonlinear circuit elements In this exercise we will investigate some common nonlinear circuit elements diodes and bipolar transistors. We will not attempt to understand in detail how those elements are made up and work – you would need advanced knowledge of solid state physics – but rather see how they behave and can be used for typical applications like rectiﬁers and ampliﬁers. 1 Linear and Nonlinear circuit elements So far in all the electronics experiments we used passive linear elements like resistors, capacitors, and inductors. Linear means here, that the voltage drop across such an element is directly proportional to the current ﬂowing through the element. The proportionality constant - the impedance - in general depends on frequency. We have deﬁned for convenience impedance as a complex quantity, so that we can treat time-dependent voltages and currents easily by looking at amplitudes and phase-shifts of harmonically varying signals of a given frequency and then adding the results together to construct arbitrary time- dependent signals. We are only allowed to do so, because the single elements in a more complicated network behave linearly, i.e. voltage drop depends linearly on the current through an element. This is, of course, only an idealization and all real devices are nonlinear to some extent. A resistor, for example, converts electrical energy into heat when current ﬂows through it - if this heat is not removed from the body of the resistor the temperature will increase and with a change in temperature the resistance value will change. This means that the resistance value depends on the current that ﬂows (and on the current at past times). Most of the time this is not desired, so one chooses materials for resistors with a low temperature coeﬃcient of resistivity and shapes the resistors such that they are eﬃciently cooled by ambient air. In other cases, this nonlinearity is exactly what you want, for example in a fuse, where a wire junction breaks as soon as you send too high a current through it. If we have a circuit with nonlinear elements in it, we are not allowed to use the superposition principle anymore or simply scale up voltages and currents by a constant factor. It also means that a nonlinear circuit driven by a sinusoidally varying voltage of frequency at the input will show in general at the output a response also at multiples 0 f,f, f, f,... of the input frequency. On the other hand, the Kirchhoﬀ rules the sum of currents into a node is zero the sum of voltage drops around a loop equals the sum of emf in the loop can – and must – still be used to analyze the behavior of a circuit, because they are directly derived from the conservation laws for charge and energy and independent of the relation between current and voltage in a given device. All we need to do to analyze a circuit, is to satisfy Kirchhoﬀ’s rules and the possible nonlinear relations between current and voltage simultaneously. Figur 1: Circuit symbol of a diode with anode terminal to the left and cathode terminal to the right. 2 Diodes and transistors In most electronics circuits you will encounter diodes and transistors. A diode is a two-terminal device with one terminal named cathode and the second terminal named anode . The circuit symbol of a diode is shown in Figure 1. The terminal connected to the base of the triangle is the anode and the terminal connected to the crossbar is the cathode. On a real device the cathode terminal is marked with a ring. The relation between current diode through the diode and the voltage diode between anode and cathode terminal is well described by

Page 2

Nonlinear circuit elements diode rs (exp ( diode αU th 1) (1) where rs , the so-called reverse saturation current, is a constant which depends mainly on material and geometry of the device, th kT/e is the thermal energy of an electron, and is a dimensionless correction factor with a value between 1 < 2. The value of th at room temperature is 25 mV By inspection of equation 1 you see that current ﬂows in the diode much easier from anode to cathode (forward current) than the other way around (reverse current). Already at a forward voltage of 0 the current is orders of magnitude bigger than for the same voltage applied in the opposite direction. This property makes diodes useful to steer currents around in a circuit. The function is similar to a ball-valve in water pipes. In the experiment you will see this property and you will verify the nearly exponential dependence of current on voltage. To do this, you build a circuit with a series connection of a diode and a resistor, drive the system with an input voltage and measure the current through the diode by monitoring the voltage drop across the resistor (see Fig.2). Figur 2: Circuit to measure the current voltage characteristics of a diode. Very useful devices in a water pipe system are regulating valves, where a piston is used to vary the eﬀective cross-section and thus the conductance of the pipe - you use that every day when you open and close a faucet. In an hydraulic system the position of the piston can be regulated by the pressure and ﬂow in another smaller system of pipes. A transistor is essentially a translation of this concept into an electrical circuit. Figur 3: Circuit symbol of a bipolar npn-transistor showing the collector, base, and emitter terminals. A bipolar transistor is a 3-terminal device. The terminals are named collector base , and emitter The circuit symbol of a bipolar transistor is shown in Fig.3. Transistors come in diﬀerent varieties. In the experiment we use npn-type transistors, which stands for a certain material composition of the transistor. In analogy to a regulating valve, with a transistor we can control the conductance, and thus the current into the collector terminal by setting the voltage between the base and the emitter terminals. Under normal operating conditions the potential at the collector is at least 0 more positive than the potential at the emitter terminal and the base terminal is about 0 more positive than the emitter terminal. The control input - the base emitter junction - behaves similar to a diode. You can measure

Page 3

Nonlinear circuit elements that by shorting the collector to the base and measuring the current from base to emitter as a function of the base emitter voltage (see Fig.4). The control law connecting the current into the collector with Figur 4: Circuit to measure the current voltage characteristics of the base-emitter junction. the voltage between base and emitter be is given by rs (exp ( be th 1) (2) so it looks very similar to the diode equation, but here it is the current into the collector and not the current into the base, which is described. The current into the base is typically much smaller than the current into the collector, which makes the transistor such a useful device, because it allows us to use a small power to control a much bigger power. Kirchhoﬀ’s rules tell us that the current out of the emitter terminal needs to be the sum of the currents into base and collector. In typical transistors the ratio between the base current and the collector current is several 100. Figur 5: Circuit to measure the basic relation between base-emitter voltage and collector current in a bipolar transistor circuit. A circuit where we can check the validity of equation 2 and which can work as a power ampliﬁer is shown in Fig.5. When we apply a voltage between points and in the circuit, we can measure as outputs ﬁrst the voltage BG . Knowing the values of and we can determine the current into the base terminal. Measuring for the same input the voltage across allows us to determine the current out of the emitter and combining the two measurements we also know the voltage between the base and the emitter. You will observe that the output voltage between emitter and ground is lower than the voltage between base and ground, so at ﬁrst sight the advertised performance as an ampliﬁer is disappointing.

Page 4

Nonlinear circuit elements The important point to realize is, however, that the current through the emitter resistor is much higher than the current into the base. To see the principle of the ampliﬁer better we extend the circuit somewhat by adding a dedicated control input to the circuit as shown in Fig.6. We connect an independent input source through a capacitor to the base terminal of the transistor. We add a resistor in series that allows us also to measure the input current. When the supply voltage to the collector terminal is on, we see that on top of a DC oﬀset a copy of the input voltage appears across the emitter resistor with a time varying current ﬂowing through the emitter resistor much higher than the current ﬂowing through at the input. Figur 6: Input circuitry added to the circuit from Fig.5. The ampliﬁer delivers more power to the resistor than it takes from the source connected to the input.

Page 5

Nonlinear circuit elements 3 Experimental program IMPORTANT: For points 1-3 below you should use the Picoscope input coupling as DC . Remember you are measuring on nonlinear circuit elements, where knowledge of the DC voltage is essential to measure the current/voltage characteristics! 1. Build a circuit with a diode and a resistor in series combination as shown in Fig.2. Use the pulse generator set to ”Squarewave”with a convenient frequency (around = 100 Hz ) as a voltage source to the input. Measure the voltage drop across the resistor for various input voltages between in ... 10 (Take at least 5 points in that interval!). Start with a resistor of = 1 Ω, and repeat the measurements with = 10 Ω and = 100 Ω. The use of diﬀerent resistors allows you to vary the current over a large range, while still keeping good sensitivity for the current measurement at low current values. Use Kirchhoﬀ’s rules to ﬁnd the current through the diode from your measurement of the voltage across the resistor and to ﬁnd the voltage drop across the diode from your measurement of the input voltage and the voltage across the resistor. Verify with your data the exponential dependence of the current ﬂowing through the diode on the voltage drop across the diode! 2. Connect a sinusoidal voltage from the waveform generator with an amplitude around 1 and a frequency of your choice to the circuit and observe the output signal. Look at input and output signal with the spectrum analyzer (set a frequency range which allows you to see up to 10 times the input frequency) and explain what you see! Is it evident from your observation that a diode is a nonlinear circuit element? 3. Built the circuit shown in Fig.5 with the following values of components: ,R = 51 Ω, Ω. Use the pulse generator set to squarewave with 100 Hz and measure BG ,U EG for various input voltages between in = 1 ... 10 . Repeat for = 10 Ω and verify eq.2 with your data. To do this you need to determine the voltage diﬀerence between base and emitter from your measurements of BG and EG , the sum of collector and emitter current from your measurement of EG , and the base current from your measurement of in and BG . Make a plot of vs be ! How big is the ratio between collector current and the base current What happens if you replace by a short circuit? 4. If you have time left, extend the circuit as shown in Fig.6, connect the waveform generator (sinu- soidal) with amplitude around 1 and frequency set between = 2 kHz.... 100 kHz to the input and observe the gain behavior of the ampliﬁer. Use a value for the load resistor of = 5 Ω. (For these measurements it is convenient to use AC-coupling of the Picoscope.) Try to measure the input current drawn from the waveform generator by measuring the voltage drop across the sense resistor (convenient values for are between 1 Ω and 5 Ω). How big is the input impedance of the ampliﬁer? Can you explain why it has this value?

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