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Electronic Engineering

Circuit calculations using active non-linear devices

  • Circuit theorems such as
  • Kirchhoff’s Laws
  • Thévenin’s Theorem
  • Superposition 
  • Work only if the circuit components are linear i.e. if you double the voltage, you double the current. Components such as resistors, capacitors and inductors are, on the whole linear in nature. When we come to analysing circuits with non-linear components such as diodes, bipolar transistors and field effect transistors we must adopt one of two techniques
  • Electronic Engineering
    • Graphical
    • Equivalent Circuits
  • The graphical method uses plots of the input and output characteristics to determine the characteristics of the created amplifier or circuit. This requires a large amount of graphical information to be available especially if a design is being formulated with a wide range of possible devices to be considered.

Equivalent Circuits

  • The other approach is to use a circuit comprising linear components, which responds in the same way as the nonlinear active device. The equivalent circuit may not be perfect but will often give us a starting point when designing. Note that electronics on the whole is far from exact as we are working with components with relatively high tolerance.
  • Resistors typically 5% and Capacitors typically 10% as well as active devices which can vary dramatically in terms of their characteristics from one device to another.
  • The diode can be modelled using a resistor R, a voltage source E and an ideal diode
  • The diode D only conducts when the anode is positive with respect to the cathode. The supply E ensures that the anode of D only goes positive when the applied voltage reaches a certain positive level. The resistor R controls the current once the diode is conducting.
  • The result is a characteristic that looks like
    • This approximates the characteristic for a simple diode where E for Silicon is about 0.6v. Of course this is not exact but is fairly good over the limited range where the diode is conducting.
    • This type of model is called a small signal model as it has good approximation for a small range of inputs.  

Bipolar Transistor

  • The models for both NPN and PNP are the same. The models vary subtly for different configurations. We will examine the most common configuration – that of the common emitter.
  • The input on the left is between the base and the emitter and the output on the right is between the collector and the emitter.
  • The emitter is therefore common to input and output which gives the configuration the name. There are a number of models that exist for this device.
  • We will look in detail at two of these.

The Hybrid Parameter Network

  • This replaces the input and the output sides by conventional circuit theory equivalent circuits.
  • The input is replaced by a Thévenin equivalent i.e. a resistor in series with a voltage source.
  • The output part of the transistor is replaced by a Norton equivalent circuit comprising a current source in parallel with a resistor, as shown.
  • The reason for the choice of equivalent circuits is that the input is voltage driven whilst the output is associated with current flow.
  • If we now combine these we have the Hybrid Parameter Network
  • There are 4 parameters associated with this model, these being
    • hie – hybrid input common emitter 
      • This is a measure of the input resistance of the transistor and is measured in ohms. It is given by
    • hre – hybrid reverse gain common emitter
      • This is a measure of the effect of the output voltage on the input and is effectively a reverse voltage gain. It has no units. It is given by
    • hfe – hybrid forward gain common emitter 
      • This is a measure of the effect of the input current on the output and is effectively a forward current gain. It has no units. It is given by
    • hoe – hybrid output common emitter
      • This is a measure of the output conductance of the transistor and is measured in Siemens. It is given by:
  • The reason for the resistor on the output being expressed as a conductor will become apparent when we start to generate equations.
  • By looking at the input side and the output side we can generate two equations, these are
    • Input side

      • VBE = hie x IB + hre x VCE 

    • Output side

      • IC = hfe x IB + hoe x VCE 

  • The symmetry between the equations can be seen – this 
  • If we wish to measure the four parameters then we can see how this can be done using the equations
    • VBE = hie x IB + hre x VCE 
    • IC = hfe x IB + hoe x VCE 
    • hie = VBE/IB as long as VCE is zero 
      • this is written as
        • hie = VBE/IB |VCE = 0
  • What are the equations for the other three?
  • In any circuit containing a common emitter transistor, the transistor can now be replaced by the four.

NOTES

  • This is a small signal model and only works effectively over a limited range of input conditions.
  • This is an a.c. model and cannot be used to set up the initial d.c. conditions around the transistor (i.e. biasing)
  • This model does not take into account variations in frequency and can only be used within the normal operating frequencies of the amplifier.
  • All capacitors in the transistor circuit are considered to be short circuits when constructing the equivalent circuit.
  • All d.c. power supplies act as large capacitors and can therefore also be thought of as short circuits

  • We are now ready to start analysing transistor circuits but before we do here are some typical values for the parameters
  • This is for a BC107 – other transistor values can be found in manufacturer’s literature.
  • NOTE
    • The values are typical values for that device and will vary considerably. 

Example

  • Input is a voltage source with an internal resistance of 50Ω Output is a 120Ω loudspeaker 

The Hybrid π Model

  • This is model based on the physical construction of the transistor. A typical transistor has the following construction
  • From the diagram
    • rbb
      • this is the resistance from the base connection to the centre of the base region.
    • rbe
      • this is the resistance from the centre of the base to the emitter connection.
    • rb’c
      • this is the resistance from the centre of the base to the collector connection.
    • rce
      • this is the resistance from the collector connection to the emitter connection.
    • Cb’e
      • this is the junction capacitance of the base emitter junction.
    • Cb’c
      • this is the junction capacitance of the base collector junction.
  • The current generator has a value given by gM x Vb’e.
  • Typical values for the device are:
  • Parameter Value 
    • rbb’ 300 Ω 
    • rb’e 2000 Ω 
    • rb’c 1.5 MΩ
    • rce 25 kΩ 
    • Cb’e 8 pF 
    • Cb’c 4 pF 
    • gM 0.125 S
  • The electronic equivalent circuit is drawn in the following way
  • It is possible to draw some comparisons between the two models and in doing so we can equate certain components:
  • When we are working at low to medium frequencies, the capacitors will have relatively high values

    • Cb’e = 8 pF will have a reactance at 20 kHz of 1/2πfC = 995 kΩ This is so large compared to the other resistors both of the capacitors and rb’c are removed giving us

  • This model is now very similar to the original H parameter model.

Model at High Frequencies

  • At higher frequencies the reactance of the capacitors begins to drop and their effect increases. What they effectively do is reduce the current flow through Rb’e by allowing it to flow via other paths as shown.
  • We need to determine the flow through Cb’c 
  • Note. The transistor amplifier inverts the signal as it amplifies which means that as the input goes positive the output goes negative. The upshot of this is that the voltage across Cb’c is given by
    • Vb’e + Vce but Vce = gain x Vb’e so this gives us 
    • Vb’e + Vb’e x gain = Vb’e (1 + gain) 
  • an approximation for the gain is gM x RL so 
  • Voltage across Cb’c = Vb’e (1 + gM x RL)
  • Which means the current is Vb’e (1 + gM x RL) j ωCb’c
  • This has the same effect as a capacitor from b’ to e whose value is
    • Cb’c x (1 + gM x RL)
  • The input part of the circuit can therefore be redrawn as:
  • The two capacitors in parallel can now be combined to given a single capacitor CIN given by

    • Cb’e + Cb’c x (1 + gM x RL)

  • What has effectively happened is that the value of the feedback capacitor has been amplified and applied across the input. This is called the Miller Effect.

The Miller Effect

  • If we have an inverting amplifier with a capacitor connected between it’s input and output then this is equivalent to the amplifier with one capacitor connected from its input to ground and another between its output and ground.
  • The value of the input capacitor is C (A + 1) and the output capacitor is given by C (A + 1)/A

  • Going back to our amplifier
  • it is important to know the frequency at which the amplifiers gain begins to reduce due to the effect of the capacitors. The point at which we define the amplifier to be beyond it’s working limit, i.e. outside its bandwidth, is when the resistance of rb’e equals the reactance of CIN. (This is the amplifiers -3dB point) 
  • rb’e = 1/(2πfCIN) from which we can say that the break frequency for the amplifier is: f = 1/(2πCIN rb’e)

Field Effect Transistor FET

  • The equivalent circuit of an FET is relatively simple compared to the bipolar transistor. The circuit below shows the complete model.
  • At low frequencies the model simplifies to become
  • If we have load RL and RL << rds then
    • VDS = gM VGS RL Giving Gain = VDS/VGS = gM RL 
  • At high frequencies the Miller effect can once again be used to give us an equivalent input capacitance of
    • Cin = Cgs + Cgd (1 + gM RL )

  • If we have a Voltage Source connected to the input then the input circuit becomes
  • Reactance of the capacitor is
  • This produces a gain that rolls 1 off above a certain frequency.

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