AMPLIFIERSA circuit containing only capacitors, amplifiers (transistors) and resistors may resonate.A circuit containing only capacitors and resistors may not.Why does amplification permit resonance in a circuit with only one kind of storage element?Amplification arises from static behavior, not dynamics and energy storage.Amplification is fundamentally dissipative.Amplifiers are basically multiport resistorsHow does the addition of a (multiport) dissipator enable resonance? We will see that• Amplification is fundamentally a non-equilibrium phenomenon.• Resistors far from equilibrium may contain a "hidden" junction structure that includes a gyrator.• This gyrator can cause resonance with only one kind of storage element.NODICITYElectrical amplifiers are nodic.(1) Continuity of flow: The sum of flows into the system is zero. This means that the element behaves as a node characterized by a Kirchhoff current law. (2) Relativity of effort: Each flow depends only on the difference of applied efforts. The same effort may be added to all inputs without changing the output. The constitutive equations of a nodic three-port resistor may be written as follows.Nodicity implies that the behavior of the element is independent of any absolute reference frame.A nodic three-port (n-port) contains an “embedded” two-port (n-1-port)characterized by two constitutive equations (not three).EXAMPLE: SEMI-CONDUCTOR DIODE.EMBEDDED JUNCTION STRUCTURESIn general a multi-port resistor may contain a "hidden" junction structure coupling the power flows on its ports.A two-port resistor may contain a "hidden" or "embedded" gyrator. The anti-symmetric component describes an ideal power-continuous gyrator which is "embedded" in the resistive two-port. Power must be dissipated and not generated. EXAMPLE: NPN TRANSISTORNote:The controlled-source representation is widely usedthis conductance representation is not.The conductance representation reveals that the h-parameters are constrainedthe conductance matrix must be positive definite.The controlled-source model hides this constraint. Evaluate parameters:Note the extreme asymmetry of the conductance matrix.Its symmetric and anti-symmetric components are:The symmetric component is purely dissipative. The anti-symmetric component describes a gyrational coupling between input (base) and output (collector). Because of this gyrational coupling, a resonant circuit may be constructed using this transistor, two capacitors and no inductors. BiasingWhy does "biasing" matter?The device only amplifies if operated far from equilibriumLARGE-SIGNAL TRANSISTOR BEHAVIOR: THE EBERS-MOLL MODEL.(describes static characteristics only — ignores important dynamics such as charge storage on the junction regions)The Ebers-Moll model describes a transistor as a pair of coupled back-to-back diodes. The coupling is usually represented as a pair of current-controlled current sources.Note:AMPLIFICATION IS A NON-EQUILIBRIUM PHENOMENONLinearizing the Ebers-Moll model reveals an interesting property of amplifiers in general: operated near equilibrium, amplifiers cease to amplify. Conclusion:If the transistor is operated about equilibrium, the locally linearized conductance matrix becomes symmetric. As a result, the transistor behavior is purely dissipative without any gyrational coupling between input and output. Relating this to the controlled-source model of the transistor, — the forward current gain is identical to the reverse voltage gain. Operated about equilibrium, the embedded gyrator disappearsthe transistor fails to amplifyAMPLIFIERS A circuit containing only capacitors, amplifiers (transistors) and resistors may resonate. A circuit containing only capacitors and resistors may not. Why does amplification permit resonance in a circuit with only one kind of storage element? Amplification arises from static behavior, not dynamics and energy storage. Amplification is fundamentally dissipative. Poutput ≤ Psupply Poutput = Psupply + Pinput - Pdissipated Amplifiers are basically multiport resistors How does the addition of a (multiport) dissipator enable resonance? Mod. Sim. Dyn. Sys. Amplifiers page 1WE WILL SEE THAT • Amplification is fundamentally a non-equilibrium phenomenon. • Resistors far from equilibrium may contain a "hidden" junction structure that includes a gyrator. • This gyrator can cause resonance with only one kind of storage element. Mod. Sim. Dyn. Sys. Amplifiers page 2NODICITY Electrical amplifiers are nodic. Assuming conductance causality the equations of the three-port resistor may be written as follows. f1 = Γ1(e1,e2,e3) f2 = Γ2(e1,e2,e3) f3 = Γ3(e1,e2,e3) Mod. Sim. Dyn. Sys. Amplifiers page 3Nodicity means that the efforts and flows at the ports are constrained so that they satisfy two conditions: (1) Continuity of flow: The sum of flows into the system is zero. This means that the element behaves as a node characterized by a Kirchhoff current law. (Note the implicit assumption of the "power positive in" sign convention) (2) Relativity of effort: Each flow depends only on the difference of applied efforts. The same effort may be added to all inputs without changing the output. Mod. Sim. Dyn. Sys. Amplifiers page 4The constitutive equations of a nodic three-port resistor may be written as follows. f1 = Γ1[(e1 – e3),(e2 – e3)] f2 = Γ2[(e1 – e3),(e2 – e3)] f3 = –(f1 + f2) Nodicity implies that the behavior of the element is independent of any absolute reference frame. A nodic three-port (n-port) contains an “embedded” two-port (n-1-port) characterized by two constitutive equations (not three). Mod. Sim. Dyn. Sys. Amplifiers page 5EXAMPLE: SEMI-CONDUCTOR DIODE. A semi-conductor diode has two ports (two wires) but, in common with all electronic devices it is well described as a nodic element and is characterized quite accurately by a single constitutive equation such as i = Is[e(e1–e2)/Vt – 1] where Vt: thermal voltage = kT/q = 25.3 mV at 20°C k: Boltzmann's constant T: absolute temperature q: charge on the electron Is: reverse saturation current Note that the element satisfies the conditions for nodicity even though both the thermal voltage and the reverse saturation current depend on absolute temperature. Mod. Sim. Dyn. Sys. Amplifiers page 6EMBEDDED JUNCTION STRUCTURES In general a multi-port resistor may contain a "hidden" junction structure coupling the power flows on its ports. A