Transformer. V1 is 1.0 Vp-p at 10 Khz. William R. Robinson Jr. p1of All rights Reserved

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Morgan Poole
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1 V1 is 1.0 Vp-p at 10 Khz Step Down Direction Step Up Direction William R. Robinson Jr. p1of 24

2 Purpose To main purpose is to understand the limitations of the B2Spice simulator transformer model that I am using. Below I highlight the following limitations of the model: The Model does not account for the inductance of the secondary winding o If we add this inductance in series with the output The resonance response is correct The bandwidth is not good o If we add this inductance in parallel with the output The resonance response is poor The bandwidth is not good When significant currents and/or large internal resistances are involved the B2Spice model can show significantly lower output than a real transformer will. o In a real transformer power loss in the primary and secondary are distributed across the inductances but in the model RP and Rs in series leading to higher voltage loss than in the real transformer o If the Resistances are removed then the model would draw more current than the real transformer Frequency response of the model is not precise. o Determination of LA is important to frequency response and can be involved. Does not model iron losses well o Should have XP and Xs to model iron loses, but model doe not contain Xs Flipping a step down model to produce a step up model (or vice versa) gives an incorrect frequency response. William R. Robinson Jr. p2of 24

3 Theory and Design Because a transformer can be used in either direction I studied it in both directions Step Down Direction (3 terminal as primary) Step Up Direction (2 terminal as primary) Gain (ideal transformer) Step Down Direction (3 terminal as primary) Ns 1 o Vout Vin Np Ns = number of turns on primary Np = number of turns on the secondary Ns/Np is also known as the turns ratio Step Up Direction (2 terminal as primary) o Ns and Np swap values o Gain Step Up Direction = 1/gain forward) Input Impedance (ideal transformer) Assuming the ideal transformer is lossless than the power out of the transformer must equal the power out of the transformer Step Down Direction (3 terminal as primary) o Vout Iout Vin Iin o Substituting the equation above for Vout Ns Vin Iout Vin Iin Np o Dividing both sides by Vin Ns Iout Iin Np o Substituting Rload/Vout for Iout Ns Vout Iin Np Rload o Substituting for Vout from the top equation Ns Vin Ns Iin Np Rload Np o Rearranging 2 Vin Np Rload Iin Ns o Rin = Vin/Iin therefore 2 Np Rin Rload ref2 Ns Step Up Direction (2 terminal as primary) o Np and Ns swap values and the above equation holds William R. Robinson Jr. p3of 24

4 B2 spice Model (non-ideal transformer) The B2Spice model X1_X1_TransPP_param_0 is shown below Center Tap Audio Transformer, for push-pull tube operation, without screen grid taps TransPP Primary Terminals P1, P2 plate connections B primary center tap (+Vpp) Secondary terminals Sp1, Sp2 speaker connections Parameters : RP one half primary winding resistance LA one half primary winding inductance (series) LB one half primary inductance, (parallel) RA impedance ratio, primary plate-plate to secondary RS secondary winding resistance the default parameter values given are for an 8K to 8ohm transformer, and 30 to 30KHz 1 db frequency response (15 to 60KHz 3dB response).subckt TransPP P1 B P2 Sp1 Sp2 PARAMS: RP=25 LA= LB= RA=1000 RS=0.8 primary RP1 P1 1 {RP} La1 1 2 {LA} Lb1 2 B {LB} Lb2 B 5 {LB} La2 5 6 {LA} RP2 6 P2 {RP} LPA 3 4 {RA} R u R u secondary Rs Sp1 9 {RS} LSA 9 Sp2 1 coupling Kcore LPA LSA ENDS TransPP Notes: 1. The model s RA is Np/Ns which is the inverse of Ns/Np discussed in the theory section. 2. The center tap is on the primary side (many IF transformers have a tap on secondary side. 3. RP primary winding resistance cannot be equal to 0 otherwise William R. Robinson Jr. p4of 24

5 Vsense = Vsource (primary winding is just a short so no voltage drop across the primary winding) Vout =0 as 0 Volts across primary winding time the turns ratio = 0 4. Although the circuit will simulate, if a symmetric {model (RA=1) RS = RP} is Step Up Directiond in the simulation it does not give the proper result. Therefore in simulation the transformer should not be flipped to provide a center-tapped secondary. Below is my attempt at making a schematic from the spice model above P1 RP1 RP La1 LA LB1 LB R1 1p LPS RA Rs RS LSA RA Sp1 LB2 LB RP2 La2 R2 1p P2 Sp2 Rp LA The model at Wikipedia 2 is very similar and the discussion is good and applies to the B2spice model. Also see b2 spice article. 3 The B2Spice model has no equivalent of Rc therefore Iron losses are not accounted for. The B2Spice model has no equivalent of Xs therefore leakage inductance of the secondary is not accounted for. The physical limitations of the practical transformer may be brought together as an equivalent circuit model (shown below) built around an ideal lossless transformer. [41] Power loss in the windings is current-dependent and is represented as inseries resistances R P and R S. Flux leakage results in a fraction of the applied voltage dropped without contributing to the mutual coupling, and thus can be modeled as reactances of each leakage inductance X P and X S in series with the perfectly coupled region. Iron losses are caused mostly by hysteresis and eddy current effects in the core, and are proportional to the square of the core flux for operation at a given frequency. [42] Since the core flux is proportional to the applied voltage, the iron loss can be represented by a resistance R C in parallel with the ideal transformer. A core with finite permeability requires a magnetizing current I M to maintain the mutual flux in the core. The magnetizing current is in phase with the flux; saturation effects cause the relationship between the two to be non-linear, but for simplicity this effect tends to be ignored in most circuit equivalents. [42] With a sinusoidal supply, the core flux lags the induced EMF by 90 and this effect can be modeled as a magnetizing reactance (reactance of an effective inductance) X M in parallel with the core loss component. R C and X M are sometimes together termed the magnetizing branch of the model. If the secondary winding is made open-circuit, the current I 0 taken by the magnetizing branch represents the transformer's no-load current. [41] The secondary impedance R S and X S is frequently moved (or "referred") to the primary side after multiplying the components by the impedance scaling factor (N P /N William R. Robinson Jr. p5of 24

Transformer equivalent circuit, with secondary impedances referred to the primary side The resulting model is sometimes termed the "exact equivalent circuit", though it retains a number of

[41] Analysis may be simplified by moving the magnetizing branch to the left of the primary impedance, an implicit assumption that the magnetizing current is low, and then summing primary and

6 Transformer equivalent circuit, with secondary impedances referred to the primary side The resulting model is sometimes termed the "exact equivalent circuit", though it retains a number of approximations, such as an assumption of linearity. [41] Analysis may be simplified by moving the magnetizing branch to the left of the primary impedance, an implicit assumption that the magnetizing current is low, and then summing primary and referred secondary impedances, resulting in so-called equivalent impedance. The parameters of equivalent circuit of a transformer can be calculated from the results of two transformer tests: opencircuit test and short-circuit test. I note the following differences between the Wikipedia discussion and the B2Spice model o B2Spice does not have Xs o B2Spice moves Rs to the secondary side o B2 spice does not have Rc o B2 spice distributes Rp, Xl and Xm (probably due to the center tap Frequency The ideal transformer has a flat frequency response however for the model o RP, LB form a high pass circuit R 4 F cutofflower = 2L William R. Robinson Jr. p6of 24

7 2 RP F cutofflower = 2 (2 LB) o LA, (Rs + Rin) form a low pass filter Rin 4 F cutoffupper = 2L Np Rload Ns F cutoffupper = 2 (2 LA) Resonance The secondary winding has inductance. If a capacitor is put in parallel with this inductance than a tank circuit is for med. This fact is often used in IF transformers. 5 See IF_Transformers.doc 1 6 The Resonant frequency is Fr 2 LC 2 William R. Robinson Jr. p7of 24

8 Calculated Gain (ideal transformer) Step Down Direction (3 terminal as primary) o I was unable to find a data sheet for the transformer I used in the real circuit, based on using it forward and backwards I determined that its turns ration was 0.07 o Gain Step Down Direction = 0.07 Step Up Direction (2 terminal as primary) o Gain Step Up Direction = 1/gain forward) o Gain Step Up Direction = 1/0.07 o Gain Step Up Direction = 14.4 Input Impedance (ideal transformer) Step Down Direction (3 terminal as primary) 2 Np Rin Rload o Ns o When Rload = 1K Rin = 1K (14.4/1) 2 Rin = 207 K Step Up Direction (2 terminal as primary) o When Rload = 1K Rin = 1K (0.07/1) 2 Rin = 4.9 ohms Frequency See the real section for the measured values used in the calculations below Step Down Direction (3 terminal as primary) 2 RP o F cutofflower = 2 (2 LB) 2 55 F cutofflower = 2 (2 500mH ) F cutofflower = 17.5 hz Np Rload Ns o F cutoffupper = 2 (2 LA) LA = 0 so conceptually F cutoffupper = infinity 2 Step Up Direction (2 terminal as primary) 2 RP o F cutofflower = 2 (2 LB) William R. Robinson Jr. p8of 24

9 2 0.8 F cutofflower = 2 (2 1.8mH ) F cutofflower = 70.7 hz Np Rload Ns o F cutoffupper = 2 (2 LA) 1 1K 14.4 F cutoffupper = 2 (2 11uH ) F cutoffupper =34.9 Khz 2 Resonance To test resonance we replace Rload with C1=0.1uF across the output terminals Step Down Direction (3 terminal as primary) 1 6 o Fr 2 LC 1 6 o Fr mH 0.01uF o Fr = 26.5 Khz 2 Step Up Direction (2 terminal as primary) 1 6 o Fr 2 LC 1 6 o Fr mH 0.01uF o Fr = 1.59 KHz William R. Robinson Jr. p9of 24

10 Simulation (B2 Spice) See the real section for the measured values used in the calculations below Gain Step Down Direction (3 terminal as primary) vm(vout1) vm(vsense) Transformer Step Down Direction(3 terminal as primary)-small Signal AC-4-Graph m m m m m m m m m m m m m m 5.000m m k k k 1.000M M M 1.000G Frequency o Gain Step Down Direction = This is very close to the expected value Calculations did not account for internal resistance(s) The drop across the secondary resistance RS o V drop_secondary = VoutRS/(RS+Rload) o V drop_secondary = / o V drop_secondary = 1uV so this is of no significance The drop across the two RPs in the model o Iout = 0.07V/1K = 70 ua o In = 1/turns ratio Iout = 70uA/14.4 = 4.9 ua o V drop_primary = 2 Rp In = u = o The model drops about 5 uv so this is of no significance either Step Up Direction (2 terminal as primary) vm(vout1) Transformer Step Up Direction (2 terminal as primary)-small Signal AC-1-Graph k k k 1.000M M M 1.000G Frequency o Gain Step Up Direction = 9.9 William R. Robinson Jr. p10of 24

11 The Gain 9.9 is significantly lower than ideal transformer but calculations did not account for internal resistance Calculations did not account for internal resistance(s) The drop across the secondary resistance RS o V drop_secondary = VoutRS/(RS+Rload) o V drop_secondary = /( ) o V drop_secondary = 1.43 V quite significant The drop across the two RPs in the model o Iout = 14.4V/1K = 14 ma o In = 1/turns ratio Iout = 14.4mA/(1/14.4) = 207 ma o V drop_primary = 2 RP In = mA = 0.331V So V drop_secondary and V drop_primary drops about 1.8V In a real transformer power loss in the primary and secondary are distributed across the inductances but in the model RP and Rs in series leading to higher voltage loss than in the real transformer If the Resistances are removed then the model would draw more current than the real transformer When significant currents and/or large internal resistances are involved the B2Spice model can show significantly lower output than a real transformer will. Input Impedance (ideal transformer) Step Down Direction (3 terminal as primary) o The input impedance is calculated below vm(vsense) Transformer Step Down Direction(3 terminal as primary)-small Signal AC-7-Graph n n n n n n n n n k k 1.000M M M 1.000G Frequency Iin = Vsense/ Rsense Iin = 48nV/0.01 ohm Iin = 4.8uA ua Rintotal = (Vin Vsense)/Iin Rin = (1V- 48nV)/4.8uA Rin = 208 K Rin (refected) = Rintotal RP2 William R. Robinson Jr. p11of 24

12 Rin = 208K 255 Rin = 208K Step Up Direction (2 terminal as primary) o The input impedance is calculated below Transformer Step Up Direction (2 terminal as primary)-small Signal AC-4-Graph vm(vsense) 6.500m 6.000m 5.500m 5.000m 4.500m 4.000m 3.500m 3.000m 2.500m 2.000m 1.500m 1.000m u u k k k 1.000M M M 1.000G Frequency Iin = Vsense/ Rsense Iin = 1.44mV/0.01 ohm Iin = 144 ma Rin = (Vin Vsense)/Iin Rin = (1V- 1.44mV)144mA Rin = 5.9 ohms Rin (refected) = Rintotal RP2 Rin = Rin = 4.3 ohms Frequency Step Down Direction (3 terminal as primary) o F cutofflower = 18.2 hz o F cutoffupper = 77,900 Khz Step Up Direction (2 terminal as primary) o F cutofflower = 90 hz o F cutoffupper = 53 Khz Resonance To test resonance we replace Rload with C1=0.1uF across the output terminals Step Down Direction (3 terminal as primary) William R. Robinson Jr. p12of 24

1 6 o Fr 2 LC 1 6 o Fr 2 21.8mH 0.01uF o Fr = 1.

13 1 6 o Fr 2 LC 1 6 o Fr mH 0.01uF o Fr = 1.1 Mhz This is way off from calculated and real circuit When the inductance is added to the secondary circuit the resonant frequency is correct at 26.4 Khz William R. Robinson Jr. p13of 24

14 The Model does not account for the inductance of the secondary winding If we add this inductance is series with the output o The resonance response is correct o The bandwidth is not good If we add this inductance is parallel with the output o The resonance response is poor o The bandwidth is not good Step Up Direction (2 terminal as primary) o Fr = 20.8 Khz See notes above for step down direction William R. Robinson Jr. p14of 24

15 Real Circuit I measured the following parameters (step down direction) for the transformer and used these values for the model o ½ Primary resistance = 55 Ohms o Secondary resistance = 1.6 ohms o ½ Primary inductance = 250 or 500 mh o Because the two halves mutually induct, the total inductance is 4 times as large as the inductance of ½ of the turns o Secondary inductance = 3.6 mh LA was determined by the method discussed in reference 7, Figure 5 suggests that the Xin = RP + RS + X LA with the secondary shorted. o Step Down Direction (3 terminal as primary) Vsense was to small to measure so I used 0 mh for LA o Step Up Direction (2 terminal as primary) By measuring the Vsense verses frequency with the secondary shorted I got the following plot for Xin Zin verses Frequency, Step Up Direction (2 terminal as primary) Impedance ohm short Frequency Hz Choosing the data point at 100 Khz 7 Xin = X LA and Xl 2FL RP and RS are much smaller that X LA, so we can ignore them Substitute and solve for L L= Xin/(2piF) William R. Robinson Jr. p15of 24

16 L = 13.3/(2pi100 Khz) L= 21.7 uh LA = 1/2L = 10.6uH With source Vin conveniently set up to be at 1.0V Gain (ideal transformer) Step Down Direction (3 terminal as primary) Measured Gain vs Frequency, Step Down Direction (3 terminal as primary) Gain open 1K Frequency Khz o For Frequencies below 80KHz (audio range is Khz) o Gain = 0.07 Step Up Direction (2 terminal as primary) William R. Robinson Jr. p16of 24

17 Measured Gain vs Frequency Step Up Direction (2 terminal as primary) Gain open 1K Frequency Khz o For Frequencies below about 20 Khz o Gain = 10.6 (in audio range) Input Impedance Step Down Direction (3 terminal as primary) o The voltage across a 1 ohm Rsense resistor was to small to measure o This is consistant with the calculated and simulated resistance of 200K with a 1Vp-p input this would yield a 1V 1/207K = 4.8uVp-p signal Rin = infinity Step Up Direction (2 terminal as primary) o The input impedance is calculated below Vsense over 1 ohm resistor was measured at 150 mv Iin = Vsense/ Rsense Iin = 150mV/1 ohm Iin = 150 ma Rin = (Vin Vsense)/Iin Rin = (1V- 150mV)150mA Rin = 5.7 ohms Rin (refected) = Rintotal RP2 Rin = Rin = 4.1 ohms Frequency William R. Robinson Jr. p17of 24

18 Step Down Direction (3 terminal as primary) o F cutofflower = <100 o F cutoffupper = 16,000 Khz Step Up Direction (2 terminal as primary) o F cutofflower = 9.3 hz o F cutoffupper = Khz Resonance To test resonance we replace Rload with C1=0.1uF across the output terminals Step Down Direction (3 terminal as primary) o Fr = 330 Khz No peak found a t calculated position Peak at 6Mhz has disappeared Peak at 13 Mhz is still there Step Up Direction (2 terminal as primary) o Fr = 1.3 KHz William R. Robinson Jr. p18of 24

19 Comparison Transformer Step Down Direction (3 terminal as primary) o Frequency response of the model is not precise. Note real circuit peaks at about 6Mhz and 14 Mhz these imply one or more tuned LC circuits going to resonance but there is no capacitance in the model. This is supported also by reference 7 you should always be aware that these seemingly simple structures are in fact very complex elctromagnetic devices. Linear circuit models are a very crude approximation to the real component, and most of the elements of the circuit models have strong nonlinearities in them. For this reason, you can only expect very limited results in trying to run circuit simulators on power supplies. You should always make extended frequency response measurements on transformers when you are developing components. This will show increase in resistance with frequency, and change in leakage inductance, allowing you to properly specify the test conditions for a tightly-controlled part. The changes in the winding resistance and leakage inductance will be strongly dependent upon the physical winding layouts of the transformer, and great care should be taken to control this as tightly as possible during design and manufacturing. Comparision Measured vs Simulated, Step Down Direction(3 terminal as input) Gain K Simulated 1K Frequency Khz Real-Measured Simulation Calculated Gain Rin K N/A but large F cutofflower hz < F cutoffupper Khz 16,000 77,900 infinity Fr with 0.01uf Khz 330 1, with 3.6mH inductor 26.4 William R. Robinson Jr. p19of 24

20 in series William R. Robinson Jr. p20of 24

21 Step Up Direction (2 terminal as primary) o Gain is off see notes above. Comparision Measured vs Simulated, Step Up Direction(2 terminals as input) Gain K Simulated 1K Frequency Khz Real-Measured Simulation Calculated Gain Rin ohm F cutofflower hz < F cutoffupper Khz Fr with 0.01uf Khz with 3.6mH inductor in series 1.59 William R. Robinson Jr. p21of 24

22 Additional Test Flipping the model When entering the circuit into B2Spice for the Step up transfer it is tempting to simply flip the terminals in spice rather than finding and changing the transformers parameters, after all this is donw all the time with real transformers. Gain vs frequency plot for new model vm(vout1) Transformer Step Up Direction (2 terminal as primary)-small Signal AC-17-Graph k k k 1.000M M M 1.000G Frequency Vsense vs frequency for new model Transformer Step Up Direction (2 terminal as primary)-small Signal AC-19-Graph vm(vsense) 6.500m 6.000m 5.500m 5.000m 4.500m 4.000m 3.500m 3.000m 2.500m 2.000m 1.500m 1.000m u u k k k 1.000M M M 1.000G Frequency Gain vs frequency for flipped model vm(vout1) Transformer Step Up by flip of Step Down-Small Signal AC-2-Graph k k k 1.000M M M 1.000G Frequency William R. Robinson Jr. p22of 24

23 Vsense vs frequency for the flipped model vm(vsense) Transformer Step Up by flip of Step Down-Small Signal AC-0-Graph 6.500m 6.000m 5.500m 5.000m 4.500m 4.000m 3.500m 3.000m 2.500m 2.000m 1.500m 1.000m u u k k k 1.000M M M 1.000G Frequency Step Down Direction (3 terminal as primary) o The flipped model gives the same gain o The flipped model gives the same nominal input impedance o Flipping a step down model to produce a step up model (or vice versa) gives an incorrect frequency response. William R. Robinson Jr. p23of 24

24 References 1. UNKNOWN,, The ARRL Handbook For Radio Communications, (ARRL 2010) P2.62, (Eq. 134) 2. UNKNOWN The ARRL Handbook For Radio Communications, (ARRL 2010) P2.63, (Eq. 137) 3. Morehouse, Harvy, Magnetics Transformer Modeling online, accessed UNKNOWN, The ARRL Handbook For Radio Communications, (ARRL 2010) P Robinson, William, IF_Transformer, Another study from this series. 6. UNKNOWN, The ARRL Handbook For Radio Communications, (ARRL 2010) P Dr. Ray Ridley, (Power Systems Design Europe January/February 2007), High Frequency Power Transformer Measurement and Modeling, online, accessed William R. Robinson Jr. p24of 24

Center_Tapped_Transformer

Center_Tapped_Transformer V1 is 1.0 Vp-p at 10 Khz William R. Robinson Jr. p1of 17 Purpose The Bspice transformer model only provides for a center tapped transformer. In addition, in the study Transformers.doc 1 I discovered that