Block A Unit 3 outline One port network Two port network Thevenin transformation Norton transformation Source transformation Two port network Hybrid (H) parameters Reciprocity theorem One port networks: G. Rizzoni, “Fundamental of EE”, Chapter 3.6 – 3.7 EE2301: Block A Unit 3

One-Port Network Represented as: +v– One-port Network A one port network is simply a two terminal device (that we can think of as a block or black box), which can be described by its I-V (current-voltage) characteristic. +v– One-port Network EE2301: Block A Unit 3

Thevenin’s Theorem One-port network Any one-port network composed of ideal voltage and current sources, and linear resistors, can be represented by an equivalent circuit consisting of an ideal voltage source VT in series with an equivalent resistance RT One-port network RT +- VT This is an extremely power method is you want to find the load condition for maximum power transfer. i.e. what is the RL you need to achieve maximum power transfer for a given one port network? EE2301: Block A Unit 3

Norton’s Theorem One-port network Any one-port network composed of ideal voltage and current sources, and linear resistors, can be represented by an equivalent circuit consisting of an ideal current source IN in parallel with an equivalent resistance RN One-port network IN RN This is an extremely power method is you want to find the load condition for maximum power transfer. i.e. what is the RL you need to achieve maximum power transfer for a given one port network? EE2301: Block A Unit 3

Applies to both Thevenin & Norton Finding RN or RT Applies to both Thevenin & Norton Method Summary Remove the load (assumption: you need to know what the load is). Never forget this step! Zero all independent voltage & current sources Short-circuit voltage sources (0V across source) Open-circuit current sources (0A through source) Calculate the total resistance seen across the load terminals to obtain equivalent resistance RN or RT Combine resistors in series and parallel (REF previous unit) EE2301: Block A Unit 3

That is as if the source never existed Zeroing the Sources That is as if the source never existed + - No voltage => Short Voltage Source Vs No current => Open Current Source IN EE2301: Block A Unit 3

Illustration of finding RN or RT Step 1: Remove load R1 R3 R1 R3 + - + - vs R2 RL vs R2 Step 2: Zero all independent sources Step 3 Compute total resistance between the terminals Thevenin Resistance: RT = R3 + R1||R2 R1 R3 R2 EE2301: Block A Unit 3

Equivalent R: Example 1 R1 R2 VS R2 VS R1 +- +- Rth = R1 || R2 EE2301: Block A Unit 3

Equivalent R: Example 2 R1 R2 IS R2 IS R1 R1 Rth Rth R2 Rth = R2 EE2301: Block A Unit 3

Finding Thevenin voltage VT Definition of VT - The Thevenin equivalent voltage is equal to the open-circuit voltage present at the load terminals (with the load removed) Method Summary Remove load (never forget this step) Define open circuit voltage Voc across the open load terminals Solve for Voc using any preferred method to obtain the Thevenin voltage VT = Voc EE2301: Block A Unit 3

Finding Thevenin VT R1 R3 Step 1: Remove load! R1 R3 vs R2 RL vs R2 + - + - + Voc - vs R2 RL vs R2 Step 2: Define open circuit voltage Voc across the open load terminals Step 3: Solve for Voc using any preferred method Note that current through R3 = 0 due to open circuit (ie no voltage drop across R3) Therefore voc = voltage across R2 Voltage divider rule: voc = [R2 / (R1 + R2)] vs *Thevenin voltage vT = voc EE2301: Block A Unit 3

Thevenin example 1 Problem 3.55 Find the Thevenin equivalent circuit as seen by the resistor RL EE2301: Block A Unit 3

Thevenin example 1 solution Mesh 1: 10 = 2000i1 - (1000)(0.01)  i1 = 10mA 1Ω 3Ω 1kΩ 1kΩ Rth = 504Ω Mesh 2: -V = (1001)(0.01) - (1000)(0.01)  V = -10mV Note that with RL removed, no current runs through 3Ω resistor Vth = V = -10V 504Ω + - -10mV This slide is meant to blank EE2301: Block A Unit 3

Thevenin example 2 Problem 3.52 Find the voltage across the 3Ω resistor by replacing the remainder with the its Thevenin equivalent - V1 + EE2301: Block A Unit 3

Thevenin example 2 solution Equivalent resistance seen by 3Ω resistor is simply 4Ω resistor Mesh 1: V1 = 2(4+2) - i (2)  2i + V1 = 12 Vth = (1.5)(2) - 8.5 = -5V 4Ω Mesh 2: 3 = i(2+2) - 2 (2)  i = 1.75A + - -5V  V1 = 8.5V V3Ω = 3/(3+4) * -5 = -5.14V This slide is meant to blank EE2301: Block A Unit 3

Finding Norton current IN Definition of IN - The Norton equivalent current is equal to the short-circuit current that would flow if the load were replaced by a short circuit Method Summary Replace load with short circuit (SC) Define SC current isc to be Norton equivalent current Solve for isc using any preferred method to obtain IN = isc EE2301: Block A Unit 3

Step 1: Short circuit load Finding Norton IN (1) R1 R3 Step 1: Short circuit load A R1 R3 + - + - vs R2 RL vs R2 isc Step 2: Define short circuit current isc Step 3: Solve for isc using any preferred method Apply KCL at node A: EE2301: Block A Unit 3

Norton example 1 Problem 3.51 (modified) Find the Norton equivalent circuit to the left of the 3Ω resistor EE2301: Block A Unit 3

Norton example 1 solution 1Ω 5Ω 4Ω RN = 3.22Ω Norton current is given by the current through 1Ω resistor This slide is meant to blank EE2301: Block A Unit 3

Norton example 2 Problem 3.53 Find the Norton equivalent circuit to the left of the 2Ω resistor EE2301: Block A Unit 3

Norton example 2 solution 1Ω 3Ω 1Ω 3Ω Rth = 4.75Ω KCL at node 1: KCL at node 2: Solve for V2: V2 = -1.263V IN = V2 / 3 = -0.42A This slide is meant to blank EE2301: Block A Unit 3

Source Transformation RT We can transform between the two equivalent circuits, observing each time that: VT = IN RT + - VT IN RT For example using one of the previous circuits as shown below: R1 R3 R3 + - R2 R1 R2 VS I1 VS and R1 form a Thevenin circuit which we can transform to a Norton circuit defined by current source I1 = VS/R1 and parallel resistance R1. Note that R1 and R2 and be combined. EE2301: Block A Unit 3

Transformation illustration R1||R2 R3 + - R1||R2 V1 I1 The parallel combination of R1 & R2 (R1||R2) and current source I1 form a new Norton circuit (with a different value of parallel resistance than in the first instance), which can in turn be transformed back into a Thevenin circuit as follows. This Thevenin circuit comprises of a series resistor R1||R2, and a voltage source of V1 = I1(R1||R2). Finally, it will now become obvious that this Thevenin resistor is in series with R3, and therefore can be easily combined. Therefore, in summary we see that VT = V1 = [R2/(R1+R2)]VS and RT = R1||R2 + R3 (same result as before) Note that V1 is NOT the same as VS. Hence the transformed circuit values are not to be confused with the original values – never make this mistake! EE2301: Block A Unit 3

Source Transformation: Proof Comes directly from the theorem: Thevenin to Norton – Any network can be composed of a current source in parallel with a resistor; this would therefore also include a Thevenin circuit Norton to Thevenin – Any network can be composed of a voltage source in series with a resistor; this would therefore also include a Norton circuit EE2301: Block A Unit 3

Two-port network Looking back A port refers to a pair of terminals through which a current may enter or leave a network We have focused only on one-port networks so far, where we consider the voltage across or current through a single pair of terminal The rest of this unit deals with two-port networks A two-port network is an electrical network with two separate ports for input and output We will see examples of two-port networks (op amps and transistor circuits) later on in this course Like for a one-port network, knowing the parameters of a two- port network enables use to treat it as a “black-box” placed within a larger network. In a two-port network, we need to relate V1, V2, I1, I2 The terms relating these currents and voltages are known as parameters EE2301: Block A Unit 3

Impedance parameters V1 = z11I1 + z12I2 V2 = z21I1 + z22I2 As shown in the figure, a two-port network may be voltage-driven or current-driven. The terminal voltages and currents represent 4 variables, of which two are independent. The terminal voltages can be related to the terminal currents V1 = z11I1 + z12I2 V2 = z21I1 + z22I2 The voltages V1 and V2 are the dependent variables in this case It can also be expressed in matrix form: These z terms are known as impedance parameters (since there are defined by V over I), or simply z parameters. EE2301: Block A Unit 3

Handling parameter subscripts 11: Input from port 1 and output back to port 1 21: Input from port 1 and output to port 2 12: Input from port 2 and output to port 1 22: Input from port 2 and output back to port 2 EE2301: Block A Unit 3

z parameters The values of the parameters can be obtained by setting: I1 = 0 (input port open-circuited) or I2 = 0 (output port open-circuited) Therefore: z11: Open-circuit input impedance z12: Open-circuit transfer impedance from port 1 to port 2 z21: Open-circuit transfer impedance from port 2 to port 1 z22: Open-circuit output impedance EE2301: Block A Unit 3

Deriving z parameters We can see from the definition of the z parameters, we obtain z11 and z21 by connecting V1 to port 1 and leaving port 2 open-circuited (I2 = 0) [Referring to Figure (a)]: z11 = V1/I1, z21 = V2/I1 Likewise, we obtain z12 and z22 by connecting V2 to port 2 and leaving port 1 open-circuited (I1 = 0) [Referring to Figure (b)]: z12 = V1/I2, z22 = V2/I2 z11 and z22 are known as driving-point impedances z21 and z12 are known as transfer impedances EE2301: Block A Unit 3

Z parameter example 1 Determine the Z-parameters for the following circuit We first apply V1 at port 1 and open-circuit port 2 (I2 = 0) [Figure (a)] z11 = V1/I1 = (20 + 40)I1/I1 = 60Ω z21 = V2/I1 = 40I1/I1 = 40Ω Next we apply V2 at port 2 and open-circuit port 1 (I1 = 0) [Figure (a)] z12 = V1/I2 = 40I2/I2 = 40Ω z22 = V2/I1 = (40 + 30)I2/I2 = 70Ω EE2301: Block A Unit 3

Z parameter example 2 Determine the Z-parameters for the following circuit z11 = 14, z12 = z21 = z22 = 6Ω EE2301: Block A Unit 3

Hybrid parameters V1 = h11I1 + h12V2 I2 = h21I1 + h22V2 The z parameters do not exist for all two-port networks, so there is a need to develop an alternative set of parameters to describe such types of two-port networks. This particular set of parameters is based on making V1 and I2 the dependent variables. V1 = h11I1 + h12V2 I2 = h21I1 + h22V2 These h terms are known as hybrid parameters, since there are a hybrid (mix) of ratios, or simply h parameters. This set of parameters is very useful for describing electronic devices such as transistors (which we will cover in Block C); namely because they are easier to measure compared to z or y parameters. The h parameters are defined as: EE2301: Block A Unit 3

h parameters We can see that: h11 represents an impedance; h12 represents a voltage ratio h21 represents a current ratio; h22 represents an admittance This is why they are known as hybrid (which means a mixture) parameters h11: Short-circuit input impedance h12: Open-circuit reverse voltage gain h21: Open-circuit forward current gain h22: Short-circuit output admittance To find h11 and h21: Short-circuit port 2 and apply I1 to port 1 To find h22 and h12: Open-circuit port 1 and apply I2 to port 2 EE2301: Block A Unit 3

h parameter example 1 Find the h parameters for the following circuit. To find h11 and h21: Short-circuit port 2 and apply I1 to port 1 [Fig (a)] h11 = V1/I1 = (2 + 3||6)I1/I1 = 4Ω h21 = I2/I1 = -2/3 (current divider) To find h12 and h22: Open-circuit port 1 and apply V2 to port 2 [Fig (b)] h12 = V1/V2 = 6/(6+3) = 2/3 (voltage divider) h22 = I2/V2 = I2/(3+6)I2 = 1/9S EE2301: Block A Unit 3

h parameter example 2 Find the h parameters for the following circuit. h11 = 1.2Ω, h12 = 0.4, h21, = -0.4, h22 = 0.4S EE2301: Block A Unit 3