Impedance and Admittance Parameters

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Abigail Arnold
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1 1/31/011 mpedance and Admittance Parameters lecture 1/ mpedance and Admittance Parameters Say we wish to connect the put of one circuit to the input of another. #1 put port input port # The terms input and put tells us that we wish for signal energy to flow from the put circuit to the input circuit.

2 1/31/011 mpedance and Admittance Parameters lecture / Energy flows from source to load n this case, the first circuit is the source, and the second circuit is the load. #1 (source) POWER # (load) Each of these two circuits may be quite complex, but we can always simply this problem by using equivalent circuits.

3 1/31/011 mpedance and Admittance Parameters lecture 3/ Load is the input impedance For example, if we assume time-harmonic signals (i.e., eigen functions!), the load can be modeled as a simple lumped impedance, with a complex value equal to the input impedance of the circuit. in in V in in V in # (load) in V in in in V in L in

4 1/31/011 mpedance and Admittance Parameters lecture 4/ Equivalent s The source circuit can likewise be modeled using either a Thevenin s or Norton s equivalent. 0 This equivalent circuit can be determined by first evaluating (or measuring) the oc open-circuit put voltage V : #1 (source) oc V sc And likewise evaluating (or measuring) the short-circuit put current sc : #1 (source) V 0

5 1/31/011 mpedance and Admittance Parameters lecture 5/ Thevenin s oc sc From these two values ( V and ) we can determine the Thevenin s equivalent source: V oc oc g g sc V V V g g V V V g g Vg V g

6 1/31/011 mpedance and Admittance Parameters lecture 6/ Norton s Or, we could use a Norton s equivalent circuit: V oc sc g g sc V g g g g V ( ) V g g

7 1/31/011 mpedance and Admittance Parameters lecture 7/ Model Thus, the entire circuit: #1 (source) V # (load) Can be modeled with equivalent circuits as: V g g V L Please note again that we have assumed a time harmonic source, such that all the values in the circuit above (V g, g,, V, L ) are complex (i.e., they have a magnitude and phase).

8 1/31/011 mpedance and Admittance Parameters lecture 8/ Two-Port circuits Q: But, circuits like filters and amplifiers are two-port devices, they have both an input and an put. How do we characterize a two-port device? A: ndeed, many important components are two-port circuits. For these devices, the signal power enters one port (i.e., the input) and exits the other (the put). 1 V 1 nput Port Output V Port

9 1/31/011 mpedance and Admittance Parameters lecture 9/ Between source and load These two-port circuits typically do something to alter the signal as it passes from input to put (e.g., filters it, amplifies it, attenuates it). We can thus assume that a source is connected to the input port, and that a load is connected to the put port. 1 V g g V 1 Two-Port V L

However, at least one of these components must be a source of energy.

10 1/31/011 mpedance and Admittance Parameters lecture 10/ How to characterize? Again, the source circuit may be quite complex, consisting of many components. However, at least one of these components must be a source of energy. Likewise, the load circuit might be quite complex, consisting of many components. However, at least one of these components must be a sink of energy. Q: But what ab the two-port circuit in the middle? How do we characterize it? A: A linear two-port circuit is fully characterized by just four impedance parameters! 1 -port V 1 V

11 1/31/011 mpedance and Admittance Parameters lecture 11/ Do this little experiment Note that inside the blue box there could be anything from a very simple linear circuit to a very large and complex linear system. Now, say there exists a non-zero current at input port 1 (i.e., 1 0 ), while the current at port is known to be zero (i.e., 0). 1 V 1 -port 0 V Say we measure/determine the current at port 1 (i.e., determine 1 ), and we then measure/determine the voltage at the port plane (i.e., determine V ).

12 1/31/011 mpedance and Admittance Parameters lecture 1/ mpedance parameters The complex ratio between V and 1 is know as the trans-impedance parameter 1 : 1 V 1 Note this trans-impedance parameter is the Eigen value of the linear operator v t : relating current ( ) 1 it to voltage ( ) { } ( ) L i( t) V G ( ω ) v t Thus: G ( ω ) 1 1 Likewise, the complex ratio between V 1 and 1 is the trans-impedance parameter 11 : 11 V1 1

13 1/31/011 mpedance and Admittance Parameters lecture 13/ A second experiment Now consider the opposite situation, where there exists a non-zero current at port (i.e., 0 ), while the current at port 1 is known to be zero (i.e., 0). 0 1 V 1 -port V The result is two more impedance parameters: 1 V1 V Thus, more generally, the ratio of the current into port n and the voltage at port m is: Vm (given that 0 for k n ) mn k n

14 1/31/011 mpedance and Admittance Parameters lecture 14/ Open circuits enforce 0 Q: But how do we ensure that one port current is zero? A: Place an open circuit at that port! Placing an open at a port (and it must be at the port!) enforces the condition that 0. Now, we can thus equivalently state the definition of trans-impedance as: Vm (given that port k n is open - circuited ) mn n

15 1/31/011 mpedance and Admittance Parameters lecture 15/ What s the point? Q: As impossible as it sounds, this hand is even more pointless than all your previous efforts. Why are we studying this? After all, what is the likelihood that a device will have an open circuit on one of its ports?! A: OK, say that neither port is open-circuited, such that we have currents simultaneously on both of the two ports of our device. Since the device is linear, the voltage at one port is due to both port currents. This voltage is simply the coherent sum of the voltage at that port due to each of the two currents! Specifically, the voltage at each port can is: V V 1 1

16 1/31/011 mpedance and Admittance Parameters lecture 16/ They re a function of frequency! Thus, these four impedance parameters completely characterizes a linear, - port device. Effectively, these impedance parameters describes a -port device the way that describes a single-port device (e.g., a load)! L But beware! The values of the impedance matrix for a particular device or circuit, just like, are frequency dependent! L

17 1/31/011 mpedance and Admittance Parameters lecture 17/ A complete equivalent circuit Now, we can use our equivalent circuits to model this system: 1 V g g V 1 V V 1 1 V L Note in this circuit there are 4 unknown values two voltages (V 1 and V ), and two currents ( 1 and ). Our job is to determine these 4 unknown values!

18 1/31/011 mpedance and Admittance Parameters lecture 18/ Let s do some algebra! Let s begin by looking at the source, we can determine from KVL that: V V g g 1 1 And so with a bit of algebra: Vg V1 1 ( look, Ohm s Law!) g Now let s look at our two-port circuit. f we know the impedance matrix (i.e., all four trans-impedance parameters), then: V V 1 1

A: Be very careful with the notation. Current is defined as positive when it is flowing into the two port circuit.

19 1/31/011 mpedance and Admittance Parameters lecture 19/ Watch the minus sign! Finally, for the load: V L Q: Are you sure this is correct? don t recall there being a minus sign in Ohm s Law. A: Be very careful with the notation. Current is defined as positive when it is flowing into the two port circuit. This is the notation required for the impedance matrix. Thus, positive current is flowing of the load impedance the opposite convention to Ohm s Law. This is why the minus sign is required.

20 1/31/011 mpedance and Admittance Parameters lecture 0/ A very good thing Now let s take stock of our results. Notice that we have compiled four independent equations, involving our four unknown values: Vg V1 1 g V L Q: Four equations and four unknowns! That sounds like a very good thing! V V 1 1 A: t is! We can apply a bit of algebra and solve for the unknown currents and voltages: L 1 V V 1 g g ( g )( L) ( g )( L) ( ) 11 L 1 1 L 1 V V V V 1 g g ( g )( L) ( g )( L)

21 1/31/011 mpedance and Admittance Parameters lecture 1/ Admittance Parameters Q: Are impedance parameters the only way to characterize a -port linear circuit? A: Hardly! Another method uses admittance parameters. The elements of the Admittance Matrix are the trans-admittance parameters Y, defined as: mn m Y (given that V 0 for k n) mn k V n Note here that the voltage at one port must be equal to zero. We can ensure that by simply placing a short circuit at the zero-voltage port! 1 V 1 -port V 0 Note that Y mn 1 mn!

22 1/31/011 mpedance and Admittance Parameters lecture / Short circuits enforce V0 Now, we can equivalently state the definition of trans-admittance as: Vm Ymn (given that all ports k n are short - circuited ) n Just as with the trans-impedance values, we can use the trans-admittance values to evaluate general circuit problems, where none of the ports have zero voltage. Since the device is linear, the current at any one port due to all the port currents is simply the coherent sum of the currents at that port due to each of the port voltages! Y V Y V Y V Y V 1 1

The Impedance Matrix

The Impedance Matrix 0/0/09 The mpedance Matrix.doc /7 The mpedance Matrix Consider the -port microwave device shown below: z ( z ) z z port z z port 0 -port 0 microwave 0 device P z z P z port z P z ( z ) z port 0 ( z ) z